Martin Flashman's Courses - Math 371 Spring, '04

Geometry Notes

[Work in Progress DRAFT VERSION 2-16-04]

Blue sections indicate tentative plans for those dates.
  1/21 Introduction
1/23 Continue discussion of what is "geometry"? 
Start on The Pythagorean Theorem
1/26 The Pythagorean Theorem plus...
1/28 Dissections
1/30 Finish Dissections.
2/2 M&I's Euclidean Geometry
Begin Constructions and the real number line.
2/4 The real number line.
 2/9 Inversion and Orthogonal Circles  2/11 Odds and ends, More on Inversion..
2/16 Classification of Isometries & More :)
2/18 Finish Classification of Isometries.
2/20 Proof of classification result for plane  isometries
2/23 Symmetry.Begin Similarity & Proportion
2/25 More on Proportion and Measurement
2/27 Inversion and Beginning to See The Infinite.
3/1 More on seeing the infinite.
The Affine Line and Homogeneous Coordinates.
3/3  More on The Affine Line and Homogeneous Coordinates.
3/5 More on The Affine Lineand Homogeneous Coordinates and the Affine Plane 
3/8 The Affine Plane and
Homogeneous Coordinates
3/10 Axioms and Finite geometries!
Connecting Axioms to Models. 
3/12 Connecting Axioms to Models. 

3/15 No class Spring Break!
3/17 No class Spring Break!
3/19 No class Spring Break!
Introduction to projective geometry with homogeous coordinates.
Z2 and  Finite Projective Geometry.Video "A non-Euclidean Universe."
Algebraic and Visual Models for Affine and Projective Geometries. 
Introduction to Desargues' Theorem- a result of projective geometry.
3/26  Desargues' Theorem- Proof of Desargues' Theorem in the Plane
Axioms for Synthetic Projective Geometry (see M&I)

3/29 More examples of proofs in synthetic projective geometry. 
3/31 No class CC Day 
4/2More examples of proofs in synthetic projective geometry. 
4/5 A look at duality and some applications.
4/7  Conics. Introduction to Pascal's Theorem
Sections and Perspectively related Figures
4/9 Some key duality, perspective reconsidered.
Video: Orthogonal Projection
4/12 Perspectivies as transformations.
The complete Quadralateral.
4/14 Quiz 2
Projectivities. Projective relations
Projective Line transformations: Synthetic Projectivities;
4/16  Start Isometries with Homog. Coord.
 More on Matrix Projective Transformations.
4/19 An Inversion Excursion.
4/21 Harmonics: uniqueness and coordinates for Projective Geometry. 
Planar transformations and Matrices
Quiz #3
Inversion Video.
4/26  4/28
5/3 5/5

[Side Trip] Moving line segments:

Case 1
Case 2
Case 3. The vertex is vk with k different from 3 or n. Then consider the polygonal regions Q3 = v1v2...vk which has k vertices (k<n) and Q4 = which has n-(k-2)<n vertices. By induction Q3 and Q4 can be triangulated, so the original polygon is triangulated using the triangulations of Q3 and Q4.
Case 3
Angle Bisection Euclid Prop 9

Line Segment Bisection
Euclid Prop 10

Construct Perpendicular to line at point on the line
Euclid Prop 11
Construct Perpendicular to line at point not on the line Euclid Prop 12
Move an angle Euclid Prop 23
Construct Parallel to given line through a point
Euclid Prop 31


We spent the class introducing Orthogonal Circles and The inverse of a point with respect to a circle as well discussing the concept of convexity of a geometric figure.

Odds and ends...and isometries !

(1) 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 a hole in the proof of Proposition 1 in Book I of Euclid.

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.

Sorry, this page requires a Java-compatible web browser. 
Definition: An isometry on a line l /plane π /space S is 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').

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.
You can read more about isometries by checking out this web site: Introduction to Isometries.

Some General Features of Isometries:

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.

Fact: If  T and S are isometries then ST is also an isometry where ST(P) = S(T(P))= S(P')  [T(P)=P'].

Proof :  Consider d(ST(P),ST(Q)) = d( S(T(P)),S(T(Q)) ) = d(S(P'),S(Q')) = d(P',Q') = d(P,Q) .

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?
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
] [ x ] = [ -y
1 0
Matrix for R90


[ a
] [ x ] = [ ax+by
Matrix for rotation by t degrees? 

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
] [ x ] = [ ax+by
Matrix for R(A,B)?

[ a
] [ x ] = [ ax+by
Matrix for rotation by t degrees? 

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
] [ x ] = [ ax+by
Matrix for R(A,B)?
Hint: Rotate, reflect, and rotate back!
Composition of Isometries corresponds to Matrix multiplication!

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.
Definition 5 (Byne's)
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.

3-3 And 3-5

Informal projective geometry and algebra 
Projective Geometry with a field with two elements
Another model for the 7 point geometry.
= F2 = {0,1}.
[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 indcates the point of the column is an element of the line of the row.


















Reconsider how we model geometry visually and algebraically:
Points:Ordinary coordinates(x,y) 
Lines: Ax+By+C=0
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]
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.
 Ax1+ By1+Cz1=0 and
Ax2+ By2+Cz2=0     

Sorry, this page requires a Java-compatible web browser.