Geometry Notes
Geometric Structures for the Visual
[Work in Progress DRAFT VERSION Based on
notes from 09 and 11]
Notes on Java and JavaSketchpad Malfunction:
The use of Java has become a browser and machine dependent issue.
In particular, the Java used in David Joyce's version of Euclid does not
work uniformly in Firefox on HSU computers, but does seem to work on
Chrome. Other Java works in Firefox, but not in Chrome. I will try to
indicate these dependencies when possible.
Readers who might have difficulty running the Java applets
are advised to use a book marklet that converts JavaSketchpad sketches
on this page (or anywhere else on the internet) to work completely
independently of Java.
Go to this site http://dn.kcptech.com/builds/804.12-r/cdn/bookmarklet.html
to install the small tool in your web browser OR
click on the following link to "fix" the java. Convert JavaSketch
Please give feedback,
whether bug reports or other, at the following email address:
wsp@kcptech.com
Green sectionsindicate
tentative plans for those dates.
1-20/22 Introductory Classes Summary (+).
What are different aspects of geometry? How is the study of geometry
organized? Analytic(numbers) , Synthetic(axiomatic),
Transformations (functions) are three ways to organize information and
the study of geometry. Also Projective and Differential geometry were
mentioned as alternative focuses for studying geometry.
Topological: General shape- especially holes and connectedness-
is important.
What is synthetic geometry? A geometry that focuses on connecting
statements
(theorems, constructions) to a foundation of "axioms" by using proofs.
What is analytic geometry? A geometry that focuses on connecting
statements
(theorems, constructions) to a foundation of number based algebra.
What is "structural geometry"? A geometry that focuses on
connecting statements
(theorems, constructions) to a foundation of structures (relations and
operations) on sets by using proofs.
transformations: tools that allow for changing figures:[the basis for studying different geometries in the Klein Erlangen Program. See Felix Klein - YouTubeFelix Klein (25 April 1849 – 22 June 1925)
Euclidean: translations,
rotations, and reflections.
Geometry has traditionally been interested in both results-
like
the
Pythagorean
Theorem- and foundations - using
axioms
to justify the result in some rigorous organization. We will be
concerned
with both results and foundations.
We will attempt to build an understanding of both the synthetic
and analytic approaches to geometry by using the tools of a structural
approach.
In the distinction between
synthetic and analytic geometry the key connecting
concept is the use of measurements.
Initially we will try to avoid when
possible the
use of measurement based concepts in the structures we use .
In one alternative proof for this theorem illustrated in the GeoGebra
sketch
below, follow the movie to consider 4 congruent right triangles and the square on the side of the hypotenuse arranged
inside of a square with side "b+c" and then
moving through the movie the
same 4 triangles and 2 squares arranged
inside of a square with side "b+c" . Can you explain how this
sketch
justifies the theorem?
1-27 Review of previous "proof of the PT.
Consider the on-line version of Euclid's
Geometry- especially Book
1:
The list (on-line) of initial definitions and
postulates. ( We will continue a discussion of these in more detail
later in the course.) The end results of Book I are The Pythagorean
Theorem and
its converse.
Consider Proposition 1 and that statement appear justified by the definitions and postulates! A more detailed
discussion of this proposition is found below.
Discuss: What would
we need in geometry to make a proof of the PT based on this activity?
What kind of structures and assumptions were
needed in the proof with the 4 triangles and squares?
Here are some considerations related to those assumptions:
What types of objects are being considered?
points
lines
line segments
angles
triangles
squares
What types of relations are being considered?
points on lines
Lines intersecting at points
Congruent line segments
congruent angles
congruent figures (triangles, squares)
Decomposing and recomposing a figure.
How could we justify identifying "equal" objects
(congruent figures)?
Back to Euclid for some further examination of the postulates:
Looking at the definitions and postulates, a key feature of
any geometric structure appears early- What defines a point and
a line?
An alternative- minimal geometry: points: A,B.
lines: {A,B}. That's all!
Postulate 1. To draw a straight line
from any point to any point.
Current
view: Two distinct points determine a unique straight line.
Issue: What
is a "straight" line?
Alternative
geometry- spherical. Points: points on a unit sphere in coordinate
space: { (x,y,z): x2 + y2
+ z2=1}
Straight lines: Great Circles on the sphere. Determined by a
plane in space through (0,0,0).
Fact: Two points on the sphere that are not antipodal- (Opposite)
determine a unique straight line. [Use the plane determined by the two
points and (0,0,0).] For this to make this spherical geometry satisfy
Post 1- redefine what "distinct" means: If the points on the unit
sphere are antipodal, then they are considered "equal" and thus not
distinct.
1-29:
Postulate 2. To produce a finite straight line continuously in a
straight line.
Current view:
A straight line has no "endpoints".
Issue: What
is the nature of a straight line not being "finite"?
Alternative Geometry. Points: all numbers in an open interval {x:
0<x<1}. Straight line: The interval (0,1) and all subintervals.
A "finite" straight line: Any closed sub-interval of (0,1).
Fact: Any closed sub-interval of (0,1) is properly contained in another
closed subinterval of (0,1).
Proof: Suppose 0<a,b<1. Then [a,b] is contained in the interval
`[a/2, (b+1)/2]` which is contained in (0,1).
Postulate 5. That, if a straight
line falling on two straight lines makes the interior angles on the
same side less than two right angles, the two straight lines, if
produced indefinitely, meet on that side on which are the angles less
than the two right angles.
Current view:
Under the given circumstances, the two straight lines are not
parallel.
Issue: How can you find the actual point
where the lines meet?
Here is the most distinctive of the postulates that appear in Euclid.
The definition
of parallel: Parallel straight lines are straight
lines which, being in the same plane and being produced indefinitely
in both directions, do not meet one another in either direction.
Looking at "Common Notions" we see the geometry has
language for equality and addition- a relation and an operation for the
objects.
2-1
Look at how the definitions and postulates are used in
Euclid:
These propositions demonstrate that Euclid did not treat
moving a line
segment as an essential property worthy of being at the foundations as
an axiom. However, this is a fundamental tool for all of geometry.
Note that in the proof of proposition 1 the point of
intersection
of circles is presumed to exist without reference to any of the
postulates.
This presumptions was left implicit for hundreds of years, but were
cleared
up in the late 19th century (see the work of David Hilbert) when
careful attention was given again to the axioms
as a whole system.
Notes on Existence:
Existence is an often
overlooked quality in a mathematical statement:
For example it is common to state that `sqrt{2}`
is an irrational number.
The proof starts by assuming `sqrt{2}` is a rational
number, say `sqrt{2} = p/q` where `p` and `q` are natural numbers, so
that
... `2 q^2 = p^2`. From this the usual proof
of
the statement deduces a contradiction. However, the statement
assumes
that there is a number the square of which is equal to `2`.
One can justify the existence of the square root
of `2` in many ways. One way presumes that any line segment has a length
measured by a real number. Then by the Pythagorean theorem, the
diagonal
of a unit square will have length `sqrt{2}`.
Alternative "rational
coordinate" plane geometry. [A geometry in which circles do not always
intersect.]
Points: ordered pairs of rational
numbers.
An example of a point not in the rational
coordinate plane is the point `(sqrt{2},0)`. This point can be
constructed
in the ordinary plane with straight edge and compass using the circle
with
center `(0,0)` and radius determined by the points `(0,0)` and `(1,1)`. This
circle will meet the X-coordinate axis at the point `(sqrt{2},0)`
The circle with center `(0,0)` and radius `1` and
the circle with center `(1,0)` and radius `1` meet in the ordinary plane at
the points with coordinates `(1/2, sqrt{3}/2)` and `(1/2, -sqrt{3}/2)` .
Since `sqrt{3}/2`
is not a rational number, this ordered pair does not correspond to a
point
in the rational coordinate plane, so the two circles do not have a
point
of intersection in the rational coordinate plane.
Note: Proposition 1 has a statement that was
not justified by the definitions and postulates!
Proposition 2 demonstrates a concern of Euclid about
establishing a "motion" or transformation for this geometry based on
fundamental postulates.
An example of the "web" of information formed in this
"axiomatic" synthetic geometry.
Not
just demonstrations- but the first two propositions are dedicated to
constructions! A focus on existence as well as conditions that
guarantee other conditions.
2-3 Some Comments on Problem Set #1:Intersections for families of convex figures:
Using the notation of M & I:
`cap \{ [P_0P_r : r>0\} = \{P_0\} ; cap \{
[P_{-1-r}P_{1+r} : r>0\} = \{P_{-1}P_1\}`
To show that `\D = `{`P` in plane where
`d(P,P^*) le 1`} is convex where `P^*` is a point in the plane,
recognize that `D = cap` { half
planes determined by tangent lines to the boundary of `D` that contain
`P^*`}. These half planes are all convex.
Note that M&I build an informal presentation of a
geometry structure based on using the real numbers.
To explore some of these issues, let's looked at the
Pythagorean
Theorem and its proofs.
Another
proof
using "shearing" illustrated in the GeoGebra visual proof by
choosing "Shear Pythogoras"and the Java sketch below taken from
a Geometers' Sketchpad example that can be connected to Euclid's proof.
(Based on Euclid's Proof) D. Bennett 10.9.9
Shear the squares on the legs by dragging point P, then point
Q, to the
line. Shearing does not affect a polygon's area.
Shear the square on the hypotenuse by dragging point R to
fill the
right
angle.
The resulting shapes are congruent.
Therefore, the sum of the squares on the sides equals the
square on the
hypotenuse.
[Side Trip?] Moving line segments: Can we move a line segment without changing its length.
2-5
We can look further at the foundations of the
proofs of the Pythagorean Theorem in two ways:
1. Dissections:
How
are
figures
cut and pasted together? Can be the proof be achieved using
dissections?
Solution!
2. Transformations: How are figures transformed? What
transformations will leave the measurements of "area", "lengths", and "angle measures" of figures
invariant
(unchanged)?
Note that in putting the pieces together to form any other
shape,
the area of that shape would be the same as the area of the square
unless
there is some overlap of the pieces in the shape.
Question: Is this necessary condition of equal areas
sufficient
to say that two polygonal regions could be decomposed (cut and pasted)
into smaller regions that would be congruent?
Comment: In a sense a positive (yes) response to this question means
that one could create a set of smaller shapes with which one could make
either of the two regions using precisely these smaller shapes. The
answer
to this question is yes (in fact this is a late 19th - early 20th
century result), which
is the basis for the remainder of this section
[The analogous problem in three dimensional geometry: volume equality
of polyhedra is a necessary but not sufficient condition for a similar
result. This was the third of the famous
23 "Hilbert problems" for the 20th century. This was first
demonstrated
by Dehn almost as soon as the problem was announced. Dehn used another
invariant of polyhedra related to
the lengths of the edges and the dihedral angles between the faces of
the
polyhedra.[
"Ueber den Rauminhalt," Math. Annalen, 55 (1902), 465-478].
(1) a. Parallelograms between a pair of parallel lines and on the
same
line segment are equal (in the sense of being able to decompose one to
reconstruct the other). Proposition
35.
b. Parallelograms between a pair of parallel lines and on
congruent
segments are equal (in the sense of being able to decompose one to
reconstruct
the other). Proposition
36.
(2) a. The line segment connecting the midpoints of two
sides
of a triangle is parallel to the third side and is congruent to one
half
of the third side. [The justification of this result is left
as an exercise in traditional Euclidean Geometry.]
b. By rotating the small triangle created by connecting the midpoints
of two sides of a triangle 180 degrees about one of the midpoints, we
obtain
a parallelogram. (This shows that the triangle's area is the area
of this parallelogram which can be computed by using the length of the
base of the triangle and 1/2 of its altitude- which is the altitude of
the parallelogram.) Compare this with Euclid
Prop. 42 and Prop. 44.
2-8
(ii) The triangulation of any polygonal region
in the plane is a key element in a proof of the equidecomposable
polygon
theorem. The proof of this proposition examines a more careful
characterization
of the polygonal regions being considered. The key idea of the proof
goes
by induction on the number n = the number vertices = the number of
sides
in the polygon. Proof of the
triangulation proposition.
Another component of the proof of the equidecomposable polygon
theorem is the ability to "add two parallelograms to form a single
parallelogram which
is scissors congruent to the two separate parallelograms". Here's how:
Intersect two pairs of parallel lines, l and l' with m and m'- one
from each of the given parallelograms. Draw a diagonal HI in the
resulting
parallelogram.
Cut and translate one parallelogram so that it is scissors congruent
to a parallelogram HIJK within the same parallel lines l and l' with
one
side being the diagonal.
Cut and translate the other parallelogram so that it is scissors
congruent
to a parallelogram HINO within the same parallel lines m and m' with
one
side being the diagonal and on the other side of the diagonal HI from
the
transformed first parallelogram. Now draw the parallel NO to the
diagonal
in the second transformed parallelogram HINO so that it intersects the
parallels l and l' from the first parallelogram at the points P
and
Q. This makes one larger parallelogram JKPQ which is scissors congruent
to the original two parallelograms. Compare this with Euclid
Proposition
45.
The film Equidecomposable Polygons also proves the result:
If two polygonal regions in the plane have the same area, then
there
is a decomposition of each into polygons so that these smaller polygons
can be moved individually between the two polygons by translations or
half
turns (rotations by 180 degrees).
2-10
Footnote on Equidecomposable Polygon Theorem:
The relation of being "scissors congruent" (sc) is an equivalence relation:
A sc A
If A sc B then B sc A.
If A sc B and B sc C then A sc C.
The theorem can be interpreted by saying for any A and B, polygonal regions in the plane, A sc B if and only if Area(A) = Area(B). General discussion of invariants:
Geometry (and Math in general) studies objects and how they are transformed.
Some features of the objects are preserved by the transformations. These
features are often described as invariants of the transformation.
In studying equidecomposable polygons, the objects are polygonal
regions in the plane, and the transformations are scissors congruences.
The area of the initial polygon is the same as the area of the transformed polygon, so area is an invariant of scissor congruence.
The general question: Given some objects and transformations, what are
the invariants of these transformations? Is there a collection of
invariants for the transformations so that if two objects have the same
invariants then they are equivalent in some sense based on the
transformations being studied.
The main impact of the Equidecomposable Polygon Theorem: For planar
polygons and scissors congruence, the area of the polygon is an
invariant that is sufficient to determine when two polygons are or are
not scissors congruent
.
Another geometric relation connected to transformations is congruence for triangles.
We will study further the transformations that are key to the congruence relation: translation. rotation, and reflection.
There are many invariants for these transformation: they include the
length of segments and the measurement of angles as well as area of the
triangles.
It should be clear that the area of two triangles is not enough to determine whether they are congruent.
There are some key invariants of a triangle which can determine congruence:
The three corresponding sides of two triangles are congruent: SSS
Two sides and the included angle of two triangles are congruent: SAS
Two angles and the included side of two triangles are congruent: ASA.
Read the definitions in M&I
section 1.1
REVIEW of basic plane geometry
concepts and definitions based on M&I.
M&I build their foundations for Euclidean geometry on a one to
one correspondence between points on a line and real numbers and the
ability
to match angles with numbers between 0 and 180.
Review materials defining rays, segments, angles, triangles,
and planes in M&I.
These definitions present a model for geometry based on the real
numbers. Points, rays, line segments, etc., are identified with real
numbers and intervals of real numbers. Notice that angles in the plane
are not oriented and are always considered to have a measure
between 0 and 180.
2-12
Review eight of the basic Euclidean
constructions
described in M&I
section 1.2.
Note that several of these constructions
rely on some foundations that assert the existence of points of
intersection
of circles. Thus these constructions will not be guaranteed to be
effective in a geometric structure where such points do not exist- such
as the geometry of the rational number plane.
Some comments about Constructions: It is important
to notice that constructions also require a justification
(proof) that
the construction has in fact been achieved.
In proving the constructions
we use some basic euclidean results, such as the congruence of all
corresponding
sides in two triangles is sufficient to imply the triangles are
congruent
(SSS). [Other basic Euclidean results are SAS and ASA congruence
conditions,
as well as the result that corresponding parts of congruent triangles
are
congruent (CPCTC).]
For these particular constructions to be
justified by the same arguments given by Euclid in a geometric
structure, the structure will need both an equivalence relation called
congruence for line segments, angles, triangles and propositions that
connect congruence of triangles to sufficient conditions
like SSS, SAS, and ASA.
The constructions play two
important but different roles in (euclidean) geometry:
(i) Construction allow us to "move", deconstruct and reconstruct
figures, while maintaining the magnitudes of the pieces and angles .
Thus constructions provide the tools for transformations such as
rotations and translations.
(ii) Constructions allow us to develop comparative measurements based
on a "unit" segment and the "straight" angle.
It should also be noted that the three
transformations
(translation, rotation, and reflection) commonly used in geometry are
connected
to constructions as well. For example, to translate a figure by a
vector
it would be useful to know how to construct parallelograms. We reviewed the connection between the Euclidean constructions and the three transformations (translation,
rotation, and reflection) and the fact that if T is either a
translation, determined by a vector ( an oriented line segment), a
rotation determined by a center and an oriented angle, of a reflection,
determined by a line, the image if a triangle `T( Delta ABC) = Delta
A'B'C'` is a triangle that is congruent to `Delta ABC`.
Other properties of these transformations, that preserve measure of line
segments and congruence of angles, as well as similarities will be
discussed further.
Note on midpoints: With the construction
of
midpoints in Euclidean Geometry, we can show that a Euclidean line
segment
has an infinite (not finite) number of distinct points. Furthermore, if
we think of approximating real number distance with points on a segment
after establishing a unit length, then we can construct the position of
a euclidean point as close as we want to the position where a real
number
might correspond to a point in that position.
The construction of points that
correspond to
numbers
on a line.
First construct `P_k` where `k` is an integer using circles.
We can construct points with fractions using powers of `2` for the
denominator
by bisection.
(However, with only bisection we could not construct a point for
`1/3`
although we could get very close to that point using a binary
representation
of that common fraction.) Using midpoints of segments we can
construct points that correspond to `n/(2^k)` for any `n` and `k`.
Unfortunately real numbers like `sqrt{2}` and `pi` cannot be
expressed in this way. To obtain a correspondence of points with all
real numbers using bisection we need a geometric property that will
give a point to correspond to any convergent sequence of numbers of the
form `n/(2^k)`.
The continuity axiom for a euclidean line:[Cantor's Axiom]
Any non empty family
of
nested segments
will have at least one point in the intersection of the family.
This axiom allows us to make a 1 to 1 correspondence between any real
number
and a point on a euclidean line once points have been determined to
correspond
to 0 and 1.
I.e., given P0 and P1
for any
real number x where a<x<b there is a point Px
where Px is between Pa
and Pbif
and only if the point `P_{n/(2^k)}` corresponds
to the number `n/(2^k) ` for any integer `n` and natural number `k`
.
[Review the construction of a line though a given point parallel to
a given line. See Euclid
I.31.] Brief Introduction to Similar Triangles: [More details later in the course.] Basic Result: Given `Delta ABC` and `Delta A'B'C'` with `<Acong<A'` and `<B cong <B'` then `<Ccong<C'` and
`{m(AB)}/{m(A'B')} = {m(BC)}/{m(B'C')} = {m(AC)}/{m(A'C')}` and we say that
`Delta ABC` is similar to `Delta A'B'C'` and write `Delta ABC ~
Delta A'B'C'`.
This fact is used repeatedly in M&I to justify the
construction of `P_{k/n}`, a rational point on the number line for the
rational number `k/n` from an integer lattice on the euclidean
plane.
We can construct `P_{k/n}` when `k/n` is a rational number using the theory
of similar triangles.
For example: We can bisect or trisect a line segment, giving us
the
ability to find points representing rational numbers with denominators
involving powers or 2 and 3, such as `5/6, 7/18`, etc.
The figure below gives two ways to achieve these constructions. One
can see how to generalize these to allow one to construct points to
represent
any rational number on the line so that the arithmetic of numbers is
consistent
with the arithmetic of geometry. [Adding segments and adding numbers,
etc.]
Consider M&I's constructions of the same
correspondence of
integer and rational points. These also rely on the ability to
construct
parallel lines.
Using the rational number points and the
continuity axiom for a euclidean line we can construct a unique point
`P_x` on the euclidean line for every real number `x` by using the
(infinite) decimal representation of the number. With this
correspondence the order relation of the real numbers corresponds to the
between-ness relation of points on a line segment and there is a
geometric construction for the arithmetic operations with real numbers
that matches all the arithmetic properties of the real numbers.
For example we can "add" `P_x` to `P_y` to obtain a new point `P_s` and
`s = x+y`. This can be done with circles or with parallel lines.
We will look at multiplication and division further in the next few
sessions. Historically this "isomorphism" between the real number
arithmetic and geometry constructions was identified by Descartes in his
work The Geometry.(1637) 2-19 Examples of using coordinates (and Vectors) for proofs in
euclidean geometry:
Proposition: An angle inscribed in a semicircle is a right angle.
Restated: If `AC` is the diameter of a circle and `B` is a distinct point
on the circle, then `<ABC` is a right angle.
Coordinate Proof: Using coordinates, assume the center of the circle has
coordinates `(0,0)` and that the point `A` has coordinates `(1,0)` and `C` has
coordinates `(-1,0)` while `B` has coordinates `(x,y)` where `x^2
+ y^2 = 1`. To show `<ABC` is a right triangle by application of the converse to
the Pythagorean theorem it suffices to show that `[(x - 1)^2 + y^2
]+
[(x + 1)^2 + y^2] = 2^2 = 4`.
This can be verified by expanding the squared terms involving `x`
giving : ` [x ^2 + 1 + y^2 ]+ [x
^2 + 1 + y^2] = 2+2 = 4`. EOP.
Vector Proof: Let `v` be the vector from `A` to `C` and `w` be the vector from
`B` to `C`. It suffices to show the dot product of these vectors is
`0`. Now `v cdot w = ( (x - 1)(x+1) + y^2 ] = x^2
-
1
+ y^2 = 0`. EOP.
Introducing Orthogonal Circles and The inverse of a
point with respect to a circle. Convexity of a geometric figure.
Tangents to circles.
In considering constructions of tangents to circles we use the
characterization
of a tangent line as making a right angle with a radius drawn at the
point
it has in common with the circle. ( Book
III
Prop.
16.) In our construction, not Euclid's (Book
III
Prop.
17), we also use the result that any angle inscribed in a
semi-circle is a right angle. ( Book
III
Prop.
31.)
Note on Three
Historical Problems of Constructions with Straight Edge and Compass (Euclidean Constructions) (1) Trisection of an angle: Since it
possible
to bisect and trisect any line segment and bisect any angle, the issue
is, is it possible to trisect any angle? Note: If you use folding and Origami construction it is possible to trisect an angle. [How to Trisect an Angle with Origami - Numberphile ...]
(2) Duplication of a cube [Wikipedia]: Since it is
possible
to construct a square with twice the area of a given square, the issue
is, is it possible to construct a cube with twice the volume of a given
cube?
(3) Squaring a circle: Since it is possible
to construct a square the same area as any given polygonal region, the
issue is, is it possible to construct a square with the same area as a
given circle.
[Squaring the Circle]
Doing arithmetic
with constructions in geometry. Note
that the construction above allows one to construct a point Px' from a
point Px as long as x is not 0 so that x' x = 1.[ Use the circle of
radius
1 with center at P0 to construct the inverse point for Px.]
2-22
The relation of the
inversion
transformation with respect to a circle and orthogonal circles. Proposition: If
C2 is orthogonal to C1 (with center O) and A is a point on
C2 then the ray OA will intersect C2 at the point A' where A and A' are
inverses with respect to the circle C1. Click
here for the proof.
We can use this proposition in the following
Constructions: 1. Construct a circle C2 through a given
point
B on a circle C1 and a point A inside the circle so that C2 is
orthogonal
to C1.
Solution: First construct the inverse A' of A with
respect
to C1 and then the tangent to C1 at B and the perpendicular bisector of
AA' will meet at the center of the desired circle.
2. Construct a circle C2 through two points A and B inside
a circle C1 so that C2 is orthogonal to C1.
Solution: This solution is demonstrated in the sketch below.
The continuity axiom can also be used to prove:
If a line, l, (or circle, O'A') has at least one point inside a given
circle
OA and one point outside the same given circle then there is of a point
on the line (circle) that is also on the given circle.
Proof outline for the line-circle:
Use
bisection between the points on the line l outside and inside the
circle
OA to determine a sequence on nested segments with decreasing length
approaching
0. The point common to all these segments can be shown to lie on the
circle
OA.
Proof outline for the circle-circle:
Draw the chord between the inside and outside points on the circle
O'A'.
Use bisection on this chord to determine rays that by the previous
result
will meet the circle O'A'. The bisections can continue to determine a
sequence
of nested segments with decreasing length approaching 0 and with
endpoints
determining one outside point and one inside on O'A' . The point common
to all the endpoints on the chord will determine a point on O'A'
that can be shown to also lie on OA.
Note: The circle-circle result fills in the
hole
in the proof of Proposition 1 in Book I of Euclid.
Isometries:
Definition: Anisometry on a line l /plane
π
/space
Sis a function (transformation), T, with
the property that
for any points P and Q, d(T(P),T(Q))=d(P,Q) or m(PQ)=m(P'Q').
Comment: Other transformation properties can be the focus of
attention- replacing isometries as the key transformation in a
geometrical structure. The connection between figures and the
selected transformations is made by the fact that the
transformations have a "group" structure under the operation of
composition. When the transformations form a "group" there is a
resulting equivalence relation structure on the figures of the
geometry. This is illustrated by the relation between isometries and
"congruence" in euclidean geometric structures in which distance and/or
measure has a role.
Facts: (i) If
T
and
S are isometries then ST is also an isometry
where ST(P) = S(T(P))= S(P') [T(P)=P'].
(ii) If T is an isometry, then the transformation S defined by S(P') =
P when T(P) = P' is also an isometry.
Comment: If G ={ T: T is an isometry of a structure}, then since
Id(P)=P is an isometry, Facts (i) and (ii) together the fact that
composition (product) of transformations is an associative operation
make G together with the operation of composition a group structure.
2-26 Convexity: Another brief side trip into
the
world
of convex
figures.
Recall the Definition: A figure F is convex if whenever A
and
B are points in F, the line segment AB is a subset of F.
The half plane example: Consider the half plane
determined by
a line l and a point P not on the line. This can be defined as
the
set of points Q in the plane where the line segment PQ does not meet
the
line l. Discuss informally why the half plane is convex.
Review the problems on convex figures in Problem Set 1.
Other convex examples: Apply the intersection property
[The intersection
of convex sets is convex.] to show that the interior of a triangle is
convex.
Show that the region in the plane where `(x,y)` has `y>x^2`
is convex using the tangent lines to the parabola `y=x^2` and
the
focus of the parabola to determine a family of half planes whose
intersection
would be the described region.
Some General Features of Isometries:
What information determines an isometry?
Proposition: For a planar isometry, T, where T(P) =
P', when we know T(A), T(B), and T(C) for A,B, and C three noncolinear
point , then T(P) is completely determined by the positions of A', B',
and C'.
Proof: In fact we saw that T(B)=B' must be on the
circle with center A' and radius= m(AB), and T(C)= C' must be on
the intersection of the circles one with center at A' and radius = m(AC)
and the other with center at B' and radius = m(BC). Once these points
are determined, then for any point P, P' must be on the intersection of 3
circles, centered at A', B', and C' with radii = to m(AP), m(BP), and
m(CP) respectively. These three circles do in fact share a single
common point because the associated circles with centers at A,B, and C
all intersect at P.
Proposition: If T is an isometry, then T
is 1:1 and onto as a function.
Proof: 1:1. Suppose that T(P)=T(Q). Then d(
T(P),T(Q))=0=d(P,Q)
so P=Q.
onto. Suppose R is in the plane. Consider A,B, and C in the
plane where C is not on the line AB. Then the points T(A),T(B), and
T(C)
form a triangle and using the distances d(T(A),R), d(T(B),R), and
d(T(C),R),
we can determine a unique point X in the plane where d(A,X)=d(T(A),R),
d(B,X) = d(T(B),R), and d(C,X)=d(T(C),R), so T(X) = R.
Now consider Euclid's treatment of the side-angle-side
congruence [Proposition
4] and how it relates to transformations of the plane that preserve
lengths and angles.
Such a transformation T: plane -> plane, has T(P)=P',
T(Q)=Q' and T(R)=R' with d(P,Q) = d(P',Q') [distance between points are
preserved] or m(PQ)=m(P'Q') [measures of line segments are
invariant].
Review briefly the outline of Euclid's argument for Proposition
4.
Notes:
The Side-side-side (SSS)
congruence of triangles
(If Corresponding sides
of two triangles are congruent, then the triangles are congruent)
This allows
one to conclude that any isometry also transforms an angle to a
congruent
angle.
The key connection between the congruence of figures in
the plane and
isometries:
Proposition: Figures
F and G are congruent if and only if there is an isometry of the plane
T so that T(F) = {P' in the plane where P'= T(P) for some P in F} = G.
We begin a more detailed study of isometries with a look at
Line Isometries:Consider briefly isometries
of a
line.
1) translations and 2) reflections.
How can we visualize them?
Mapping Diagrams (before and after
lines) T: P -> P';
Correspondence
figures on a single line:
Example: A reflection.
Graph of transformation.
Example: A reflection.
2-29
Coordinate function. x -> x' = f (x)
Examples: P x -> Px+5 a translation;
P x -> P-x a reflection.
Can we classify them? Is every line isometry either a
translation or
a reflection? Why?
Prop.: The only isometries of the line are reflections and translations.
Proof:
Given A and A', there are only two choices
for B '. One forces the isometry to be a translation, the other forces
the isometry to be a reflection.
Discuss further in class.
Line
isometries and coordinates:
Use T to denote both the geometric
transformation and the corresponding function transforming the
coordinates of the points. So ... T(x) = x + 5 for the
translation example and T(x) = -x for the
reflection example.
More generally, a translation Ta
:Px -> Px+a
would have Ta(x)
=
x + a and
Reflection about the origin can be denoted R0, R0(x)
=
-x . What about a general reflection about the point with
coordinate c? Rc(x)
=?
Use
a translation by -c, then reflect about 0, and
translate back to c. So Rc(x)
=
Tc(R0(T-c(x)))
=Tc(R0((x
-c))=Tc(-x+c)=
-x + 2c.
Remarks on line isometries:
(i) Any isometry of a line can be expressed as the product of at most 2
reflections.
(ii) The product of two line reflections is a translation.
Now we look at Plane Isometries: Consider isometries of a plane.
1) translations 2) rotations and 3) reflections.
How can we visualize them?
Mapping Diagrams (before and after planes);
[GeoGebra] [Add figures here.]
Correspondence figures
on a single plane; [GeoGebra]
Graph? [Visualizing 4 dimensions! ]
Coordinate functions? Remark: we have previously
shown
that an isometry of the plane is completely determined by
the correspondence of three non-colinear points.
The classification
of isometries.
There are (at least) four
types of isometries of the plane: translation,
rotation, reflection and glide reflection. [In fact , we will show that
any planar isometry is one of these four types.]
3-2
Proposition: Any
plane
isometry is either a reflection or the product of two or
three
reflections.
Proposition:
(i) The product of two
reflections that have the lines of reflection intersect at a point O is
a rotation with center O through an angle twice the size of the angle
between
the two lines of reflection.
(ii) The product of two reflections that have parallel lines of
reflection is a translation in the direction perpendicular to the two
lines
and by a length twice the distance between the two lines of reflection.
(iii) The product of three reflections is either a reflection or
a glide reflection. Proof: We can do this
geometrically or using analytic geometry and the matrices!
Note: The
four
types
of isometries can be characterized completely by the
properties of orientation preservation/reversal and the existence of
fixed
points. This is represented in the following table:
Orientation
Preserving
Orientation
Reversing
Fixed points
Rotations
Reflections
No Fixed points
Translations
Glide reflections
Note: The plane isometries form a
group: A set together with
an operation (the product or composition) that is (0) Closed under the operation - the product of two isometries
is
an isometry; (1) The operation is associative - R(ST) = (RS)T or for
any
point P, (R(ST))(P) = R( ST(P))= R(S(T(P))) = (RS)(T(P)) =
((RS)T)(P). (2) The identity transformation (I) is an isometry -
I(P)=P. (3) For any isometry T there is an (inverse) isometry S so TS =
ST = I.
The operation of composition is not necessarily commutative in the
sense that ST is not always the same transformation as TS:
For example
If R1 and R2 are reflections in intersecting lines l1 and l2
then
the
isometry R1R2 is a rotation about the point of intersection in
the opposite direction
to R2R1.
What about coordinates
and plane isometries?
Coordinates: (a la M&I
I.3) Use any two non-parallel lines in the plane with coincident 0.
Then you can determine "coordinates" for any point by using
parallelograms. [As
indicated previously, all rational coordinates can be constructed from
establishing
a unit. This is outlined more thoroughly in the reading in I.3.]
Isometry examples with coordinates in the plane:
(See M&I
I.5 and I.6)
Translation: T: P (x,y) -> P(x+5,
y+2)
is a translation of the plane by the vector <5,2>. If we use the
coordinates
for the point and T(x,y) = (x',y') then x' = x+5 and y' = y+2. We
can express this with vectors <x',y'> = <x,y> +
<5,2>. So translation
corresponds algebraically to the addition of a constant vector. Reflections: Across X-axis RX(x,y) = (x,-y); Across Y axis
RY(x,y)
= (-x,y); Across Y=X , R(x,y)=(y,x). Notice that these can be
accomplished
using a matrix operation. Writing the vectors as row vectors
(x,y)
(
1
0
0
-1
)
= (x,-y)
Matrix for RX
(x,y)
(
-1
0
0
1
)
= (-x,y)
Matrix for RY
(x,y)
(
0
1
1
0
)
= (y,x)
Matrix for R
Or writing the vectors as column vectors
[
1
0
]
[
x
]
=
[
x
]
0
-1
y
-y
Matrix for RX
and
[
-1
0
]
[
x
]
=
[
-x
]
0
1
y
y
Matrix for RY
and
[
0
1
]
[
x
]
=
[
y
]
1
0
y
x
Matrix for R
Rotations:
If R90 is rotation about (0,0) by 90 degrees, then R90(x,y)
= (-y,x).
Question: What is rotation about (0,0) by t degrees: R(t)?
Hint: What does the rotation do to the points (1,0) and (0,1)? Is this
rotation a "linear transformation?"
More general Question: What about reflection R(A,B) about the line
AX+BY
= 0?
[
0
-1
]
[
x
]
=
[
-y
]
1
0
y
x
Matrix for R90
[
a
b
]
[
x
]
=
[
ax+by
]
c
d
y
cx+dy
R(t):Matrix for
rotation by t degrees?
Hint:
Consider that P(1,0) will be transformed to
P'(cos(t), sin(t))= (a,c)
and Q(0,1) will be transformed to
Q'(-sin(t), cos(t))=(b,d)
[
a
b
]
[
x
]
=
[
ax+by
]
c
d
y
cx+dy
Matrix for R(A,B)?
3-4
More general Question: What about reflection R(A,B) about
the line AX+BY
= 0?
[
cos(t)
-sin(t)
]
[
x
]
=
[
xcos(t)-ysin(t)
]
sin(t)
cos(t)
y
xsin(t)+ycos(t)
R(t):Matrix for
rotation by t degrees.
See discussion above.
[
a
b
]
[
x
]
=
[
ax+by
]
c
d
y
cx+dy
Matrix for R(A,B)?
Hint: Rotate, reflect, and
rotate back!
Composition of Isometries corresponds to Matrix multiplication!
If B = 0 then the line of reflection is the Y axis and
we know the matrix for RY already. For all other B, let t be the
angle which has tan(t) = - A/B. Thus the reflection R(A,B)
has a matrix that must be the product [from left to right]of the
matrices for R(t), RX, and R(-t).
[
cos(t)
-sin(t)
]
[
1
0
]
[
cos(t)
sin(t)
]=[
`cos^2(t)-sin^2(t)`
2cos(t)sin(t)
]=[
cos(2t)
sin(2t)
]
sin(t)
cos(t)
0
-1
-sin(t)
cos(t)
2cos(t)sin(t)
`sin^2(t)-cos^2(t)`
sin(2t)
-cos(2t)
Here is the visualization of R(A,B) as a map in Winplot using:
(x,y)==>(cos(2t)x+sin(2t)y,sin(2t)x-cos(2t)y)
Before Reflection
After Reflection
More General Planar Isometries:
As with line isometries, a key idea is do the work at the
origin and then transfer the work elsewhere using "conjugacy": Any
transformation T at a general point or about a general line can be
investigated by first translating
the problem to the origin, S, performing the related
transformation at the origin,T', and then translating the result back to the
original position,
S-1. That is using the "conjugacy"
operation: T = S-1 T' S where T' is the relevant
transformation
at the origin. This works as well for rotation and reflections
through the origin because
the composition of these transformations corresponds to matrix
multiplication.
3-7
Here is a link to an
example of the matrix for the
isometry of the coordinate plane that is rotation by 90 degrees
counterclockwise about the point (1,2).
An Applications of Reflection:
(i) Here is a problem encountered frequently
in
first semester of a calculus course. The
Carom Problem Can you see how the solution is related to a
"carom"
(angle of incidence=angle of reflection)?
(ii) Here is a similar problem about triangles
that is also related to reflections. Fano's
Problem Can you find the
relationship?
Symmetry of a figure: S is
a symmetry
of a plane figure F if
S is an isometry with S(F)=F. Given
a figure F, the symmetries of F
form a subgroup of all the plane isometries, denoted Sym(F). Example: Consider the figure F,
a
given
equilateral
triangle. We looked at the six symmetries of this
figure and the table indicating the multiplication for these six
isometries.
The Group Table of Symmetries of an Equilateral
Triangle.
Discussion: What about isometries in three
dimensions?
These are generated by spatial reflections in a plane.
A spatial isometry is determined by the transformation of 4 points
not all in the same plane (which determine the simplest 3 dimensional
figure-
a tetrahdron!)
Any isometry of space can be expressed as the product of at most 4
reflections.
These results are proven in the same fashion as the comparable results
were proven in the plane.
Proportions
and similarity:
An example of the use of
similar triangles and proportions to constructing "square roots":
Mean Proportions in right triangles and Inverses:
Consider a right triangle ABC with hypotenuse AB. Notice that if the
altitude
CD is constructed with the hypotenuse AB as the base, the figure that
results
has 3 similar right triangles. ABC, ACD, and DCB. Using similarity of
these
triangles we see that there is a proportion of the segments of the
hypotenuse
AD, DB and the altitude CD given by AD:CD::CD:BD. If we consider the
lengths
of these segments respectively as a,h, and b then the numerical
proportion may be expressed as a/h=h/b or using common algebra $ab=h^2$.
Notice
this
says
that $ h= \sqrt{ab}$.
Application of this construction: Choosing a = 1, this
proportion becomes $h = \sqrtb$
Euclid's original
treatment of the "division
algorithm" :
If n and d>0 are integers
then there are integers q and r with
r=0 or 0<r<d where n = q*d +r.
Using q*OD to represent a segment that is made of q segments all
congruent
to OD,
Euclid's division algorithm is stated in geometry that of ON is
a segment and OD is a segment that is contained as a subsegment of ON,
then ON is congruent to q*OD with possibly a remaining
segment
RN which is congruent to a subsegment of OD.
Repeated use of this algorithm suggests the
Euclidean algorithm for finding a common unit to measure both n and d,
or ON and OD.
If r1=0 or R1N is a point, then OD will be a
common unit.
If not apply
the division algorithm to d or OD and r1 or R1N.
If this works to give
r2= 0 or R2N is a point, then R1N will be the common unit.
If not
apply the division algorithm to r1 or R1N and r2 or R2N.
If this works
to give r3= 0 or R3N is a point, then R2N will be the common unit.
If not
continue.
In common arithmetic since each remainder that is not zero is
smaller than the previous remainder, eventually the remainder must be 0
and the process will end- finding a common divisor of the original d
and
n.
In the application of the euclidean algorithm to the
diagonal and side
of a square, the procedure appears to stop.
However, we can show that because of the fundamental theorem of
arithmetic,
it would be impossible to find a segment with which to measure both the
side and the diagonal of a square.
The impact of this on geometry was that
one could not presume that all of geometry could be handled by using
simple
ratios of whole numbers for measurements.
[A geometry based on ratios alone
would not permit one to accomplish proposition 1 of Book I of Euclid!
since
this would mean that the geometry would have to be able to have ratio
involving
the sqr(3) - which like the sqr(2) is also an irrational number.]
3-9
A major
part of geometry before Descartes was Euclid's
(Eudoxus') resolution of the issue in Book
V
def'ns
1-5.(Joyce)
Definition 1
A magnitude is a part of a magnitude, the
less of the greater, when it measures the greater.
Definition 2
The greater is a multiple of the less when it
is measured by the less.
Definition 3
A ratio is a sort of relation in respect of
size between two magnitudes of the same kind.
Definition 4
Magnitudes are said to have a ratio to one
another which can, when multiplied, exceed one another.
Magnitudes are said to be in the same ratio,
the first
to the second and the third to the fourth, when,
if any equimultiples
whatever are taken of the first and third,
and any equimultiples
whatever of the second and fourth,
the former equimultiples alike
exceed, are alike equal to, or alike fall short of, the latter
equimultiples respectively taken in corresponding order.
Look at these definitions and note some key items:
* Ratios exist only between magnitudes of the same type. (This is
usually
described as "Homogeneity".)
* For ratios to be equal the magnitudes must be capable of
co-measuring.
* Euclid's axioms do not deny the existence of infinitesimals-
but
will
not
discuss equality of ratios that use them.
Briefly: An infinitesimal
segment
is a segment AB that is so short that it
would appear to be coincident points and for any k, k*AB cannot contain
any ordinary segment determined by points that we can see as distinct!
In the history of mathematics, infinitesimal segments we
used in the early
developments of the calculus. For example: Consider the ratio of {the
change in the area
of a square, A(s), when the length of a side, s, is changed by an
infinitesimal,
ds} to {the change in the length of the side, ds}. That is: What is
[A(s + ds) - A(s)]/ds ? Answer: [2s*ds + ds*ds]/ds
= 2s +ds. Thus the ratio would be indistinguishable as a number from
2s.
]
* The Axiom
of Archimedes that says that
for any two segments one can be used to measure the other. [There
are
no
infinitesimals
for an Archimedean geometry.]
The connection
[between Euclid's definition of proportionality
(equal ratios) and real number equality of quotients].
We can show the following Proposition: For segments A,B,C,and D with m(A)=a,m(B)=b,
m(C)=c, and m(D)=d:
Proposition
1
(Book VI) of Euclid provides the tool for the central similarity
transformation in Euclidean geometry. To perform the
transformation of the plane with center O and using two given line
segments to determine the magnification, first draw a ray from O and
locate P and P' on the ray so the given segments are congruent to OP
and OP'. To transform a point Q
draw the ray OQ, then the segment PQ. Through P' construct a line l
that is parallel
to PQ. Then l will meet OQ at a point Q' and by proposition 1,
Q' is the appropriate point to which Q is transformed by the central
similarity as required.
It is not difficult to show from this construction that a central
similarity will transform a line l to a line l' that
is
parallel
to the original line l. Thus one can prove the
related results for euclidean geometry:
Corollary: If T is a central similarity then
(i) T preserves the measure of angles and
(ii) T transforms a pair of parallel lines to parallel lines.
Since isometries also preserve the measure of angles and transform a
pair of parallel lines to parallel lines (since a reflection does) we
see that any composition of central similarities and isometries will
likewise preserve the measure of angles and transform a pair of
parallel lines to a parallel lines.
Exercise: Show such compositions will transform a triangle to a similar
triangle and ABC and A'B'C' is any pair of similar triangles, the there
is an isometry T and a central similarity S so that TS transforms ABC
to A'B'C'.
Similarity Transformations in Coordinate Geometry. [Not covered in class.]
Consider a similarity on a line with a center of similarity and a given
positive magnification factor. This leads to a consideration of the
effect
of a similarity on the coordinates of a point on the line.
If we use the point P0 for the center, then we see that the similarity
T with magnification factor of 2 would transform Px to P2x, or
T(Px)=P2x.
Removing the P from the notation we have T(x)=2x. Using T(Px)=Px', we
find
that T is described by the correspondence where x'=2x.
[More generally: If we use the point P0 for the center, then we see
that the similarity T with magnification factor of m would
transform Px to Pmx, or T(Px)=Pmx. Removing the P from
the notation we have T(x)=mx. Using T(Px)=Px', we find that T is
described by the correspondence where x'=mx. ]
With the center at another point, say P3, the transformation T* is
controlled by the fact that T*(Px) - P3 = 2(Px-P3).
So that for T* we have x' = 3 + 2(x-3).
[Again generally: With the center at point Pa, the
transformation T* is controlled by the fact that T*(Px) - Pa = m(Px-Pa).
So
that
for T* we have x' = a + m(x-a).]
We can also observe that if we let S(Px)=P(x-3)
and S-1(Px)=P(x+3)
then
Central similarities in the
plane:
In the plane, a similarity Tm with factor m and center
at
(0,0) will transform (x,y) to (mx,my). Thus x' = mx,
and y' = my are the equations for this transformation. This
transformation
can be represented using a matrix as follows:
[
m
0
]
[
x
]
=
[
mx
]
0
m
y
my
Matrix for Tm
Other central similarities T* with factor m
and center at (a,b)
can be recognized as related to Tm by using the translation
S(x,y)
= (x - a,y - b) and seeing that S-1TmS=T*.
[Try this yourself!]
Proposition: If l is a line in the
plane with equation
AX+BY=C with A and B not both 0, then the set l' =
{(x',y'):
Tm(x,y)=(x',y') for some (x,y) on l} is a line in the plane
that
is parallel to l (unless l = l' ). Proof(outline): Show that the equation of l'
is
AX+BY = mC. Suppose (x,y) satisfies the equation AX+BY=C.
Then
Tm(x,y) = (mx,my). But Amx+Bmy=m(Ax+By)=mC,
so
the
line l ' has equation AX+BY=mC,
and
therefore
is parallel to the line l.
We can
use coordinate geometry to describe a central similarity S of factor 5
with center at (3,2) as follows: (x,y) -> (x
- 3,y - 2) -> (5(x
-3) ,5(y - 2)) -> (5(x
-3) +3 ,5(y - 2) +2).
Thus S(x , y) = (5x
-12 ,5y - 8). Here is the
visualization of S(x,y) as a map in Winplot:
S(x
, y) = (5x -12
,5y - 8).
Before
After
View video (in Library #4376 ) on
"Central
similarities" from the Geometry Film Series. (10 minutes) View video (in Library #209 cass.2) on
similarity (How big is too big? "scale and form") "On Size and
Shape"
from the For All Practical Purposes Series. (about 30 minutes)
The Affine
Plane: A first look at an alternative geometry structure for parallel
lines.
In this
structure we consider all points and lines in the usual plane with the
exception of one special line designated as the "horizon" line. Points
on this line is not considered as a part of the geometrical points but
are used in defining the class of parallel lines in the reamining
plane.
A line in this structure is any line in the original plane with the
exception of the horizon line.
Two lines are called A-parallel (A for affine) if (i) they are
both parallel in the usual sence to the horizon line or (ii) they have
a point in common that lies on the horizon line.
At this stage we do not have a correspondence for points in this plane
and real number coordinates.
We can consider a figure to illustrate this geometry as below:
3-30 An introductory discussion of the role of
axioms and postulates
in mathematics and the sciences.
In the history of geometry, there have
been two main
approaches to the nature of the postulates or axioms for geometry.
1. The postulates are a part of an attempt
to model the
reality of measurements on planes and in space. They are a convenience
that allows one to verify only a few limited statements in the
modelling
process and then deduce from those postulates other properties of the
reality
of the model. Under this view, geometric axioms might be
considered similar
to other scientific statements that are supposed to reflect in a
platonic
world what we find in the empirical world of reality.
2. Axioms provide linguistic and
mathematical structures
useful for developing further logical relations that can be used to
understand
a variety of contexts in some abstraction efficiently.
An example of such
a view of axioms is found in the axiomatic description of a vector
space
and the study of abstract vector spaces, which can be modelled in
the physics view of vectors as things with magnitude and direction, or
in as abstract a setting as the collection of all real valued functions
on a given set S.
3. We have seen examples of axioms and postulates in Euclid's Elements
and Hilbert's axioms listed in the video.
Note: Descartes'
geometry: Algebra constructions for products and quotients.
Here are some examples of "structures" in a
variety of settings.
Projective
Geometry with a field with two elements:
Another model for the 7 point geometry. Instead of using real numbers for the prjective plane - use
Z2 = F2
= {0,1}.
A projective plane using F2 has
exactly
7 lines: AX + BY +CZ = 0 or [A,B,C] is a line using homogeneous
coordinates.
[0,0,1], [0,1,0], [0,1,1], [1,0,0], [1,0,1],
[1,1,0],
[1,1,1]. Note
that
<x,y,z> is a point on[A,B,C] if and only if
using the vector dot
product: (x,y,z)(A,B,C) = 0
This projective plane satisfies the geometric
structure
properties described previously in the "7 point geometry".
This can be verified using the following table where an X indicates the
point of the column is an element of the line of the row.
Lines\Points
<0,0,1>
<0,1,0>
<0,1,1>
<1,0,0>
<1,0,1>
<1,1,0>
<1,1,1>
[0,0,1]
X
X
X
[0,1,0]
X
X
X
[0,1,1]
X
X
X
[1,0,0]
X
X
X
[1,0,1]
X
X
X
[1,1,0]
X
X
X
[1,1,1]
X
X
X
We can visualize the projective plane
using F2
with the 7 points of the unit cube in ordinary 3 dimensional coordinate
geometry.
In doing this, many of the lines do correspond to
planes
through the origin in the usual geometry, such as [1,0,0] ( X=0) , but
others look a little strange because the use the arithmetic of Z2
, such as [1,1,1] (X+Y+Z=0) which has on it the three point
<0,1,1>,
<1,0,1> and <1,1,0> which does not appear to be a plane
through the
origin (0,0,0).
Models for geometry using algebra and coordinates for
euclidean and affine
geometry especially equations for lines.
We define a perspective relation:
Two points P and P' are perspectively
related by the center O if O is on the line PP" . Two triangles
ABC and
A'B'C' are perspectively related by the center O if O is on the lines
AA',
BB', and CC'.
Another aspect of Projection: Desargues'
Theorem in 3-space
and the plane.
4-11&13
Review of Constructions and the distinctions in our informal
discussion of euclidean,
affine and projective geometry.
Euclidean Line: `P_0` and `P_1`
Euclidean Plane: `P_{(0,0)}, P_{(1,0)}`, and `P_{(0,1)}`.
Axes at right angles.
Parallel lines have no point in common.
Affine Line: `P_0`, `P_1`, and `P_{oo}`.
Affine Plane: `P_{(0,0)}=<0,0,1>,
P_{(1,1)}=<1,1,1>,P_{(oo,0)}=<1,0,0>`,
`P_{(0,oo)}=<0,1,0>`
The Horizon or Ideal line.
Parallel lines meet at an infinite point on the horizon.
Projective Line: `P_{oo` is treated as an ordinary point. A
circle.
Projective plane: All points in the affine plane are treated as
ordinary
points. Homogenous coordinates determine all points and lines.
There are no parallel lines. All lines meet.
In an affine plane constructions using parallel lines in the euclidean
plane can be performed using the horizon line to determine appropriate
parallel lines in the constructions. This was demonstrated by
constructing
P(1,0) and P(2,0).
The structure established so far for the affine and
projective planes.
In an affine plane there is a special line, the horizon or ideal,
line containing all the ideal infinite points for lines in the ordinary
euclidean plane.
Any ordinary line has exactly one ideal infinite point
on it.
Two lines in the ordinary plane are parallel if they do not meet.
Two lines in the affine plane are parallel if they meet at the same
ideal
infinite point on the ideal line.
In the projective plane there are no
parallel lines!
We consider these statements in algebraic models for these
planes.
Reconsider how we model geometry visually and
algebraically:
Points: Ordinary `(x,y)` or generally homogeneous
`<x,y,z>`, `x,y,z` not
all `0`.
Lines: `Ax+By+C=0` or `Ax+By+Cz = 0` or `[A,B,C]`
Projective
Points: `<x,y,z> x,y,z` not all 0.
Lines: `Ax+By+Cz = 0` or `[A,B,C] \ A,B,C` not all `0`.
Homogeneous functions and in particular how a
circle and a parabola would be described with homogeneous coordinates in affine and projective geometry.
The Synthetic Projective Plane can be
modelled
by the
corresponding visual and algebraic concepts.
Postulates (Axioms)from M&I:
There exists at least one line.
There are at least three distinct points
on each line.
If A and B are any two distinct points,
there is at least one line AB on both A and B
If A and B
are any two distinct points, there is at
most one line AB on both A and B.
Not all points are on the same line.
If a line intersects two sides of a
triangle and does not contain a vertex of the triangle, then the line
intersects the third side of the triangle.
Some simple consequences of these postulates: [See
M&I for proofs.]
Theorem 4.1: If A and B are any two distinct points, there is one
and only one line AB on both A and B. Proof: Post 3 and 4. Theorem 4.2: A line is determined by any two of its points. Proof: Suppose m is a line with points A and B members of m. It's enough to show that m = AB. Certainly m is on A and m is on B, but there is only one line on both A and B, so m = AB.
Definition: If A,B and C are three distinct points with C
not on the line AB, then the triangle ABC is the set of
points A,B,and C and lines AB, AC, and BC, i.e., `Delta ABC = \{
A,B,C, AB, AC, BC\}`. Note: In this geometry we have only lines- "line segments" are not defined.
Definition:If `A,B` and `C` are three distinct points with `C` not on the
line `AB`, the plane `AB-C` is the set of all points on the
lines `PC` for any point `P` on the line `AB`.
Examples of interpretations in Cartesian Geometry, Affine Geometry and RP(2): Two points determine a line.
(Visually) Affine Geometry: Two ordinary points determine an ordinary line. An ordinary
point and an ideal point determine an ordinary line. Two ideal points
determine the horizon line.
Algebraically in RP(2):
The problem is to determine `A, B`, and `C` so that
`Ax_1+
By_1+Cz_1=0` and
`Ax_2+ By_2+Cz_2=0`
where `<x_1,y_1,z_1>` and `<x_2,y_2,z_2>` are
distinct points, so that they are not scalar multiples of each
other.
But this means we have have two homogeneous linear equations with
unknowns `A,B,C`.
By elementary linear algebra this system will have a family
of solutions determined by a single parameter.
Thus any two solutions `A,B,
C` and `A',B',C`' will have a scalar `t ne 0`, with `A'=tA, B'=tB`, and
`C'=tC`.
So there is a unique line `[A,B,C]` determined by the two points.
The 6th axiom from M&I .
Visually: This axiom can be seen to justify the
statement
that
there are no parallel lines
in the visual model
for this geometry. From a
euclidean viewpoint: This can be understood,
since
any pair of parallel lines can be organized to be the base and the line
connecting midpoints of a triangle. But Axiom 6 says that the line
determined
by midpoints of a triangle must meet the third side, so it cannot
be parallel to the base! Thus Axiom 6 would contradict the existence of parallel lines in euclidean geometry.
From the algebraic model the 6th Axiom is also
true.
If
`Ax + By + Cz = 0`
and
`A'x + B'y + C'z = 0`
are two distinct lines in the projective plane,
then
there are non-trivial solutions `(x,y,z)` to this pair of homogeneous
linear
equations.
These solutions will be on a line through `(0,0,0)`, so any
non-trivial solution will determine the homogeneous coordinates for a
point
`<x,y,z>` lying on both lines.
Where we're going in the next
few weeks. We
will be discussing the following topics: the definition of the
algebraic
model for the affine and projective plane and spaces, the connection of
these geometries to visualizations and transformations, and the construction of points with
corresponding coordinates in the affine line and plane using P0, P1 and
Pinfinity. The text materials in Meserve
and Izzo provides support for these issues and topics. The
relevant sections are listed in the
course reading assignments.
4-15
Notation: We'll use the notation R`P(2) = { <x,y,z> : x,y,z` real numbers and not all
zero}. `<x,y,z> = <x',y',z'>` means there is a non-zero number
`t` so that
`x'=tx, y'=ty`, and `z'=tz`.
Similarly R`P(3)= \{ <x,y,z,w> : x,y,z,w` real numbers and
not
all zero}.
`<x,y,z,w> = <x',y',z',w'>` means there is a non-zero number
`t` so that `x'=tx, y'=ty, z'=tz` and `w'=tw`.
Likewise Z2`P(2) = \{ <x,y,z> : x,y,z` in {0,1}and
not all zero}
`=
\{
<0,0,1>,<0,1,0>,
<0,1,1>,<1,0,0>,
<1,0,1>,
<1,1,0>,
<1,1,1>\}`.
Of interest to research mathematicians currently is
C`P(2)={
<x,y,z> : x,y,z` complex numbers and not all zero}.
`<x,y,z> = <x',y',z'>`
means there is a non-zero complex number `t` so that `x'=tx, y'=ty`, and `z'=tz`.
Notice that { `<x,y,z>` in RP(2) where `x^2 + y^2
+ z^2 = 0`} is an empty set,
whereas `<i,0,1>` is
a member of the comparable set { `<x,y,z>` in CP(2) where `x^2 + y^2 + z^2 = 0`} because this set allows the
use
of complex numbers such as i.
Proof: This is a statement of set equality.
By the
symmetry of the
equation, it will be enough to show that if X is a point in AB-C, then
X is a point in AC-B.
So suppose X is a point in AB-C.
Then by the
definition
of AB-C, there is a point P on the line AB so that X is on the line PC.
[What is required is to find a point Q on AC so that X is on QB.]
Now
consider
the triangle `Delta` PAC. B is on PA, X is on PC, so by Axiom 6, there is a
point
of intersection, Q where BX meets AC. End of Proof. [EOP]
Note: Why can it be assumed that B and X are not vertices of triangle `Delta` PAC?
Proposition: If F and G are points in AB-C
then FG is a subset of AB-C
Proof: Suppose X is on the line FG with F and G
in AB-C.
Since F is in AB-C,
there is a point P on AB where F is on PC.
Also by work done previously,
we have shown that FG intersects AB at a point Q.
To show that X is in
AB-C, it is enough to show that XC meets AB (at the point R). So
consider
the triangle PQF. X is on FQ, C is on PF, so by Axiom 6, XC meets PQ.
But
PQ = AB. EOP.
If F,G, and Q (not on FG) are in AB-C then
AB-C=FG-Q
This is done in two parts. (i) Show that FG-Q is a subset of AB-C.
(ii) Show that C is in FG-Q. [Then by previous work, A and B are also
in FG-Q and by part (i) AB-C is a subset of FG-Q.
Proof:
(i) Suppose X is FG-Q. Then there is a point Z on FG and XQ.
So Z is in AB-C (being on FG) and X is on ZQ
(which is in AB-C), so X is in AB-C. Thus FG-Q is a subset of AB-C.
(ii) Left as an exercise. [Hint: Use Postulate 6 to find a point on CQ
intersect FG.]
Proposition: Any two distinct co-planar
lines intersect in a unique point. [The proof
of this proposition is an exercise.]
Discussion of Postulates, interpretaions, models, the
"incompleteness of arithmetic" (Godel), and foundational issues with
connecting geometry to numbers.
A Start on Considering Transformations with homogeneous coordinates
of a line.
We will consider `T` from translations `T: x
-> x+a`, `R_0: x -> -x` for reflection about `P_0`, `S: x ->
\alpha x` for central similarity, and `I : x -> 1/x , x ne 0` for
inversion.
Consider
the isometry `T` of an ordinary (or affine) line with coordinates that
translates
a point 3 units to the right. This isometry is indicated using
the
ordinary coordinates by `T(P_x) = P_{x+3}` or `T(x) = x + 3`. If we wish to
understand
this isometry using the homogeneous coordinates of an ordinary point we
recognize that `P_x = <x,1>`. So for an ordinary point,
`T(<x,1>)
= <x+3,1>`.
Notice that the ideal point will be left fixed by this
translation,
that is, `T(<1,0>) = < 1,0>`. In fact the ideal point is the
only fixed
point of translation. We can see this also by looking at the
visualization
of the translation construction on an affine line.
But what is the algebra for this isometry
when using arbitrary homogeneous
coordinates for an ordinary point?
In summary, `T ( <a,b>) =
<a+3b,b>`.
Notice this formula works as well for the ideal point on the affine
line.
This work can be done using some ideas from linear algebra. Recall
that in matrix multiplication:
[
1
3
]
[
x
]
=
[
x+3
]
0
1
1
1
and
[
1
3
]
[
a
]
=
[
a+3b
]
0
1
b
b
So the isometry of translation can be considered from the point of view of algebra with homogeneous coordinates:
Treat the coordinates as a column vector, multiply by a square matrix,
and then using the components of the resulting column vector as the
homogenous
coordinates for the transformed point.
More on line transformations with homogeneous coordinates .
These transformations can be identified with matrix multiplications
since `A(cv)=cAv` for `A` a matrix, `c` a scalar, and `v` a column vector so
that
pairs of homogeneous coordinates for one point are transformed to
homogeneous
coordinates for a single point.
Translation again: `T(x)=x+3` became
[
1
3
]
[
a
]
=
[
a+3b
]
0
1
b
b
Reflection `R(x)= -x` uses the matrix
[
-1
0
]
0
1
Thus
[
-1
0
]
[
x
]
=
[
-x
]
0
1
1
1
and
[
-1
0
]
[
a
]
=
[
-a
]
0
1
b
b
Dilation by a factor of 5, `M(x)=5x`
uses the
matrix
[
5
0
]
0
1
Thus
[
5
0
]
[
x
]
=
[
5x
]
0
1
1
1
and
[
5
0
]
[
a
]
=
[
5a
]
0
1
b
b
Inversion `I(x)=1/x, x \ne 0` uses the matrix
[
0
1
]
1
0
Thus
[
0
1
]
[
x
]
=
[
1
]
1
0
1
x
and
[
0
1
]
[
a
]
=
[
b
]
1
0
b
a
Notice that composition of these matrix
transformations corresponds
to matrix multiplication.
Planar
duality
and
the Converse of
Desargues' theorem in projective geometry.
The dual of a statement or description in the context
of a projective
plane replaces the word "point" with the the word "line" and the
word "line" with the word "point".
Here are some examples of statements and the corresponding dual
statement:
Two distinct points
A
and B are (incident) on a unique lineAB.
Two distinct lines
a
and b are (incident) on a unique point `a nn b`.
If the point C is
not (incident to) on
the line AB then there are three lines
AB, AC, and BC.
If the line c
is not (incident
to) on the point `a nn b`then
there
are three points `a nn b`, `a nn c`,
and `b nn c`.
The lines AA',
BB' and CC' are incident
to the point O.
The points `a nn a'`, `b nn b'`, and `c nn c'`are incident to the line o.
One of the most important logical features of planar
projective geometry
is connected to the duality relation. Each of the dual statements for
the
postulates for planar projective geometry is a theorem of this
geometry.
For example:
Postulate: Given two distinct points there is a line with those points
on it.
Dual Statement and Theorem: Given two distinct lines there is a point
with those lines on it.
As a consequence of this feature, plane
projective has a special result
which is about the theorems of geometry and their dual statments.
The
Principle of Plane Projective
Duality:
Suppose S is a statement of plane projective geometry and S'
is
the planar dual statement for S. If S is a theorem of projective
geometry,
then S' is also a theorem of plane projective geometry.
The proof of this principle is a proof about
proofs.
The idea is that a proof consists of a list of statements about lines
and points.
Each statement in a proof is either one of the postulates,
a previously proven theorem, or a logical consequence of previous
statements.
So if we have a proof of a statement S, we have a
sequences of statements
A1,A2,...,AN=S.
Each of these statements is either one of the postulates,
a previously proven theorem, or a logical consequence of proevious
statements.
Now one can construct the sequence of dual
statements A1', A2', ...,
AN' = S'.
With a little argument it can be seen that each of these dual
statements is also either a postulate, a theorem,or a logical
consequence
of previous statements.
Here is an application of the principle of
duality to Desargues' Theorem.
Desargues' Theorem: (in the (projective) plane). If two
coplanar
triangles (determined by points) `ABC` and `A'B'C'` are perspectively
related
by the center `O`, then the points of intersection `X=(AB)nn(A'B');
Y=(AC)nn(A'C')
; and Z=(BC)nn(B'C')` all lie on the same line.
Since Desargue's Theorem uses the hypothesis of a perspective relation
between two triangles, we first look briefly at the dual concept. Dual of Desargues' Theorem: . If two coplanar triangles
(determined
by lines) `abc` and `a'b'c'` are perspectively related by the line `o`, then
the lines joining the points `x=(a nn b)*(a' nn b'); y=(a nn c)*(a' nn c')` ; and
`z=(b nn c)*(b' nn c')`
all pass through the same point.
Note on Duality and the concept of perspective:
We called two
point triangles ABC and A'B'C' perspectively
related
with respect to a point O, if O is
on the lines AA', BB' and CC'.
We define the dual concept by saying that two line triangles
abc and a'b'c' are perspectively related with
respect to a line o, if o passes
through
the points a#a', b#b' and c#c'.
Notice that the Dual of Desargues' Theorem is
also the logical converse
of Desargues' Theorem. Thus we can say, "The
converse of Desargues' theorem is true by the duality principle. 4-22
Quadrangles and quadrilaterals in the projective plane.
An important concept for projective geometry.
A Triangle is a figure composed of three points and three lines in the plane. A triangle is a planar self- dual figure.
This is not so for a quadrangle and quadralateral.
We look at the simple quadrangle and quadralateral,
and the complete quadrangle and dual quadralateral, which has three diagonal points.
The Complete Quadrangle: 4
points {A,B,C,D} determine
6 lines {AB,AC,AD, BC, BD, CD}
and three
additional
points
{X,Y,Z}.
The Complete Quadilateral: 4 lines
{AB, BC, CD, AD} determine
6 points {A,B, C, D, X,Y}
with three
additional
lines{AC,
BD, XY} .
Postulate: The three diagonal points in a complete quadrangle do not lie on the same line.
[This eliminates the 7 point geometry as a projective geometry with these axioms.] 4-25
Sections and Perspectively related figures in
a projective plane.
Motivation: In the study of surfaces
and solids in 3 dimension (as
in the 3rd semester of calculus) one key method for understanding a
fugure
in space is to consider a planar cross-section. This is a planar figure
determined by the intersection of the spatial figure with a plane. Of
course
we have examined just this kind of section in our introduction to the
conic
sections!
We can consider this as well as what happens when a 3 dimensional
object
passes through Flatland. Examples: A cube has cross sections that might be a point,
a
line segment, a triangle, a square, a rectangle, a quadrilateral,
a pentagon, or a hexagon.
A tretrahedron has a cross section that might be a point, a line
segment, a triangle, a square, a rectangle, or a quadrilateral.
In a plane we consider a figure to be made up of
points and lines. A section of a planar figure, F, by a
line l, where l is not
in the figure F, is a new figure consisting of the points P where
`P = l nn m` for any m, a line in the figure F.
Examples: The section of the triangle abc by the
line l,
is the set of three points on l , `{a nn l,b nn l, c nn l}`.
The section by a line l (not passing through the point O) of
a pencil of lines on the point O is the set of points on l with
one point for each line in the pencil.
The dual concept for section: A section
of a planar figure, F, by
a point L, where L is not in the figure F, is a new
figure
consisting of the lines p where p = L*Mfor any
M,
a point in the figure F. Examples: The section of the triangle ABC by the point L,
is
the set of three lines on L , `{AL,BL, CL}`.
The section by a point L(not lying on a line o) of a
pencil of points on the line o is the set of lines through L
with
one line for each point in the pencil.
Point Perspective in the Plane: Figure F is perspectively
related to Figure F' by the point O if there is a correspondence of the
points of F with those of F' so that for any corresponding points, A in
F and A' in F', the point O lies on the line A*A' . Notes: 1. If the figure F and the figure F' are similar because
of a central similarity, then F and F' are perspectively related. 2. If the figure F and the figure F' are congruent because of a
translation, then F and F' are prespectively related by the point on
the
horizon line determined by the translation vector. 3. If the figure F and the figure F' are congruent because of a
rotation by 180 degrees about the center O, then F and F' are
perspectively
related by the point O.
Line Perspective in the Plane: Figure F
is perspectively related
to Figure F' by the line o if there is a correspondence of the
line
of F with those of F' so that for any corresponding lines, m in
F and m' in F', the line o passes through the point `m nn m'`
. Notes:1. If the corresponding lines in two figures F and F' are
parallel, then in the affine plane, these corresponding lines will meet
on the horizon line. Thus these figures are perspectively related in
the
projective plane. 2. If the figure F and the figure F' are congruent because of a
reflection, then F and F' are prespectively related by the line of
reflection.
A look at perspective of planar figures with
respect to a center O and
its dual: perspective of planar figures with respect to an axis o.
Perspective relation as a transformation:
Given O and point ABC on a line l, and
a second line l', we can determine unique points A'B'C' on
l'
by `A' = OA nn l'`,
`B' =OB nn l'`, and `C' = OC nn l'`. Thus `A'B'C'`
are perspectively
related to `ABC` with center `O`.
For Reading: A side
trip to planar graphs and dual graphs.
The term duality is used in many different
mathematical contexts.In
the study of planar graphs we can find an example of the use of
"duality"
that illustrates some of the aspects of mathematical duality.
A planar graph consists of a finite set of
points called vertices, line
(straight or curved) segments with these vertices as endpoints called
edges,
enclosing planar sets called regions. We can think of these regions as
geographic states, the edges as boundaries between land sections, and
the
vertices and places where these boundaries meet.
So a planar graph G is a set of vertices,
edges, and resulting regions
in the plane.The dual graph of G is another graph, which for now we'll
denote D(G). D(G) consists of a vertex for each region in G, a region
for
each vertex in G, and an edge for each edge. If R is a region in G, we
choose a point in R, call it r, as a vertex of D(G). For each edge, E,
of G, with regions R1 and R1 bordering on E, choose an edge, e, between
r1 and r2 that crosses E. Finally, suppose V is a vertex of G. consider
the edges that end at V and the regions that border these edges. Then
these
regions and edges correspond to vertices and edges of D(G) that
surround
a region which we'll denote v.
The graph D(G) consists of the vertices, r,
edges, e, and regions v
just described.
One aspect of the dual graph is that information about it is revealed
by knowing information about the graph G. For example, if G has 5
regions, the D(G) has 5 vertices.
If
G
has
7 edges then D(G) also has 7 edges, and if G has 4 vertices,
then D(G) has 4 regions. The feature that duality exposes
here in these statements is the replacement
of the word "vertex" in the
statement
about G with the word "region" in
the
statement about D(G) and the word "region"
in
the
statement
about the graph G with the word "vertex"
in
the statement about D(G).
Duality and Sections in Space.
Duality in space: objects are
points, lines, and planes.
Consider duality with spatial figures: the platonic polyhedra,
tetrahedron,
cube, octahedron,dodecahedron, icosahedron.
Point - plane duality... leaves
line-line.
S: A point C and a line AB determine a plane (The join or span AB-C)
.
S': A plane ABC and a line l determine a point (the
intersection
ABC#l).
This duality is more like the graph
duality discussed earlier. Notice that as figures, the
cube and octahedron are dual, the octahedron
and icosahedron are dual, and the tetrahedron is dual to itself.
Sections
in
Space: Section by a plane: Suppose F is a figure in space
consisting
of points, lines, and planes. Suppose p is
a
plane that is not containing any element of the figure F. Then the
section
of F by p is the figure with lines
determined
by the intersection of the planes in F and p
, the points determined by the intersection of the lines in F with p
. Example. The figure F is a tretrahedron. p
is a plane that is distinct from the tretrahedron's elements. The
section
consists of 4 lines and the six points where those lines intersect in
the
plane of p . [ Include Figure here.]
Section by a point: Suppose F is a figure in space consisting
of points, lines, and planes. Suppose P is
a
point that is not contained in any element of the figure F.
Then
the
section
of F by P is the figure with lines
determined
by the join of the points in F and P
, the planes detemined
by the join of the lines in F with P.
Example. The figure F
is a tetrahedron. P is a point that is
distinct from the tetrahedron's elements.
The section consists of 6 planes
and the 4 lines where those lines join the vertices to the point P .
Duality and graphs in the
ordinary plane.
A basic principle is discussing graphs of equations and functions in
the plane is the following:
Any point described by coordinates is intersection of two lines, any
line is the join of two points.
Curves in the study of calculus are figures determined by points. For
example the line with equation y= 2x+1 is a figure made up of points
with
coordinates of the form (a, 2a+1). Each of these points is determined
by
the intersection of two lines: X= a and Y = 2a+1.
We
can dualize this figure to a figure made up of lines determined
by the join of points (on two separate axes) x=a and y = 2a+1.
Notice
that in the original line figure all the points lie on the same line,
while
in the dual figure all the lines will pass through a common point. Thus
this second figure shows a perspective relation between the
corresponding
points on the two axes. [Also the line in the graph of this
function gives a perspective relation between the pencils of lines
parallel to the X and Y axes!]
We can consider other function relations in this dual visualization
as perspective relations (or perspective functions).
For example `f(x) = x + 1` [or `f(x) = x`] are perspective between the
parrallel axes determined by the appropriate point at infinity in the
affine
(projective) view.
Definition: The
product (composition) of two
or more perpectivities is a called a projectivity.
Is we compose two perspectivities we can transform points on a line
to a second line and then back to the original line. Thus the map from a euclidean, affine, or projective line to
itself
can be a projective transformation.
Example: Translation of a euclidean
coordinate line by adding
1 unit.
T(Px) = Px+1 can be realized as a product of
perspectivities in the affine plane.
First use the perspectivity that transforms the point Px
on the X axis to the point P' = Qx+1 on the Y axis.
Then use the perpectivity that transforms the point Qx+1
on the Y axis back to Px+1 on the X axis.
Then if T represents the product of these two perspectivities, T(Px)
=
Px+1.
Other examples of projective line
transformations include central reflection [Use P' =Q
-x
for reflection about 0]and central similarities [Use
P'
=Q ax
for central similarity with scale a and center 0].
Curves are determined by lines.
Projective relation and projectivity between points on lines (and dual)
and then points on a single line.
A little about conic curves and coicndences: A brief introduction to
Pascal's Theorem/ Coincidence.
More on projectivities.
The set of projectivities can be considered a
group (as did isometries)
, i.e. a set together with an operation which satisfies certain nice
algebraic
properties: closure, associativity, an identity and inverses.
Projectivities and geometric transformations of a
line.
Isometries
Similarities
Inversions!
Definition:A general
projective transformation of a projective line is a transformation that
can be expressed using homogeneous coordinates for the points on the
line
and an invertible 2x2 square matrix.
If we let the matrix of these transformation be
denoted by
T= [
a
b
]
c
d
Because T is an invertible matrix, the
determinant
of T= ad-bc is not zero, and conversely if the
determinant
of T= ad-bc is not zero, then the matrix T
defines
a projective transformation.
[
a
b
]
[
x
]
=
[
ax+b
]
c
d
1
cx+d
and
[
a
b
]
[
x
]
=
[
ax+by
]
c
d
y
cx+dy
Thus x' = ax+by and y' = cx+dy.
We have shown now that all the transformations of the line we had
previously
discussed in the course were examples of this general type of
projective
transformation.
Notice that the transformation defined by the
matrix T
does
not depend on which of the homogeneous coordinates are used to
represent
a point, since
[
a
b
]
[
kx
]
=
k[
ax+by
].
c
d
ky
cx+dy
Furthermore the matrix kT
defines
the same transformation as T for a similar reason
since
k[
a
b
]
[
x
]
=
k[
ax+by
].
c
d
y
cx+dy
Noticing again that ad-bc
is not zero, at least
one of the entries, say a, in the matrix is not zero.
Using k
= 1/a we can assume that the matrix has the
form
T= [
1
b
].
c
d
Focus Question: Which
transformations leave the
point at infinity fixed? Analysis: Using homogeneous coordinates for the point
at infinity,
it can be expressed as <1,0>. The question is: For what matrices will this point be
transformed
to a point with homogenous coordinates of the form <x, 0> where x
is
not zero?
For this to be true, we can see from the
matrix equation
[
1
b
]
[
1
]
=
[
x
]
c
d
0
0
that c = 0. Thus the transformations that leave the
point
at infinity fixed are of the form,
T= [
1
b
].
0
d
Then recognizing that the determinant of T is
not 0, it follows
that d is also not 0.
Thus these transformations are of the form on a finite point:
:
[
1
b
]
[
x
]
=
[
x/d+b/d
]
0
d
1
1
and
[
1
b
]
[
x
]
=
[
x+by
]
0
d
y
dy
Thus this transformation involves a
translation if b is not
0, and a similarity of d which is not 0. Reminder: Next Quiz #3 on Wednesday May 4. - Sample will be available over
weekend.
An overview of what we've
considered
so far-
Euclidean Geometry lines/planes
Affine geometry lines/planes
Finite geometry lines[/Planes?]
Projective Geometry lines/planes
Axioms
Euclid
Hilbert
No Axioms Yet
A figure indicating
an ideal point or line
Axioms
7 points/7lines
A figure.
M & I Axioms
A figure indicating
all points and lines
Homogeneous Coordinates
with coefficients in {0,1}= Z 2
Homogeneous Coordinates
with real number coefficients
Transformations
Isometries
Similarities
Similarities
?
perspective and projective transformations.
Consider the construction of`P_{1/2}` from
`P_0, P_1` and `P_{oo}` on
the
projective
line. This begins a discussion of a harmonic
relation
between four points on a line. We observed that in fact the
relation
was present between `P_0, P_1, P_{oo}` and `P_{1/2}`.
Harmonic
Relation
of
4 points on a line.[This
will
be
the chief tool used to introduce (homogeneous) coordinates into
projective
geometry.]
Four points on a line l are harmonically related if the line
is determined by a pair of points from the intersection of lines in a
complete
quadrangle and the intersection of that line with the other two sides
of
the complete quadrangle.
[In the figure: The line XZ would determined two other points, `XZ nn AD=R`and `XZ nn BC=S`,
so that the points `XRZS` are harmonically related.] This would be
denoted H(XZ,RS).
Four points on a line that
are harmonically related:
Using the text notation we can show that if H(AB,CD) then
H(BA,CD),
and also conversely if H(BA,CD) then H(AB,CD). This is the
meaning
of saying "H(AB,CD) is equivalent to H(BA,CD)". Similarly we can show
H(AB,CD)
equiv. to H(AB,DC) and H(BA,DC).
One way to see this is to envision
relabeling the original quadrangle
to reverse the order in which the points on the line are organized. Test Java
Figures for harmonics.
Notice the relation between the
"double points"
in the figure and the "single points". Proposition: H(AB,CD) is equivalent to H(CD,AB) I.e., if H(AB,CD) then H(CD,AB). Link to
the proof (adapted from Meserve & Izzo).
4-27
Recall the construction on the affine
line of the
point `P_{1/2}` from `P_0`, `P_1`, and `P_{oo}`.
Notice that the figure used to
construct `P_{1/2}` showed that `H(P_0 P_1, P_{oo}P_{1/2})`.
Notice that the construction on the
affine line of the point `P_2` from
`P_0,P_1`, and `P_{oo}` looks
the same.
Notice that the figure used to construct `P_2` shows that `H(P_1 P_{oo},
P_0
P_2)`.
In general, following this construction of `P_2`, it is possible
to construct a fourth point, D, on a line given A,B and C already on
the
line so that H(AB,CD). We can use the construction of `P_2` to show
how
to construct the point D in general. This will be the key to giving a
correspondence
between points on a projective line and the real numbers with`oo`
as a single additional "number" added to extend the set of numbers.
One key issue then is: Is the point constructed from the
points A,
B and C uniquely determined by the fact that it is in the harmonic
relation
with A, B, and C? That is, if A,B, and C are three points
on a line and D and D* are points where H(AB,CD) and H(AB,CD*), then
must
D=D*?
This is the question of the uniqueness of the point D. We can prove
that in fact the point D is uniquely determined.
Discussion of the dual concept
of a harmonic relation between
four lines passing through a single point. Theorem: If A,B,C, and D are on a line l
with
H(AB,CD)
and O is a point making a section with these four points, consisting of
the four lines a,b,c and d, then H(ab,cd).
Proof: see M&I Theorem 5.4. Corollary: (By Duality) If a,b,c,
and
d
are on a point O with H(ab,cd) and l is
a
line making a section with these four lines, consisting of the four
points
A,B,C and D, then H(AB,CD).
Application:
We can think of a perspectivity between points ABCD on line l
and A'B'C'D' on the line l' with respect to the point O
as
being a section of the points ABCD by the point O followed by a section
by the line l' of the lines a,b,c, and d on
the point
O.
Applying the previous theorem and its corollary we see that: If
H(AB,CD) then H(ab,cd) and thus H(A'B',C'D').
Note: This application shows
that if four harmonically related points
on a line are perspectively related to four points on a second line,
then
the second set of four points is also harmonically related. Furthermore,
this result can be extended easily to points that are projectively
related. Thus the transformations of projectivity in projective
geometry
preserves the harmonic relationship between four points.
[This last note is comparable
to the fact that in Euclidean geometry,
isometries preserve length, and in affine geometry that similarities
preserve
proportions.]
Harmonic Conjugate as a
Transformation: Given A,B, and C points
on a projective line, we have shown that there is a unique point D so
that
H(AB,CD). D is called the harmonic conjugate of C with respect to AB.
In many ways this gives a
transformation of the point on the line to
other points that is similar in its nature to reflections and
inversions.
Notice that on a projective (affine) line two point will cut the line
into
two disjoint pieces, as does a single point for reflection and the
points
PR and P-R for inversion, where the
transformation
maps points in one set into the other while leaving the "boundary
points"
fixed.
With the existence and
uniqueness of the point D established, we can
now consider some examples illustrating how to establish a
coordinate
system for a projective line by choosing three distinct points to be
`P_0,
P_1`, and `P_{oo}`.
We can construct `P_2, P_{-1}, P_{1/2}`, (in two different ways).
Exercise: Construction `P_3` and `P_{1/3}`.
Show that with the choice of three points on a projective line
we can construct points using harmonics to correspond to all real
numbers
(as in our informal treatment of the affine line).
5-4 and 6
Conics revisited: Pascal
and Brianchon Theorems:
Point and line conics.
Consider lines connecting corresponding points in a
pencil of points on a line related by a projectivity (not a
perspectivity) and noticed that the envelope of these lines seemed to
be a conic, a line conic. Notice briefly the dual figure which
would
form a more traditional point conic. [Also notice how line
figures
might be related to solving differential equations e.g. `dy/dx=2x-1` with
`y(0)=3` has a solution curve determined by the tangent lines determined
by the derivative: `y=x^2-x+3` which is a parabola.]
Pascal's Theorem.
Duality for Pascal's Theorem.
Proof of Brianchon's Theorem.
Use of Pascal's Theorem to construct a conic from 5 points.
