Class summaries

 1/21 26 28 2/2 2/4 2/9 2/11 2/16 2/18 2/23 2/25 3/2 3/4 3/9 3/11 4/6 4/8 4/13 4/15 4/20 4/22

• 1/21 Introductory Class. We went over the Course Description. The difference between synthetic and analytic geometry was discussed. Other aspects of geometry were considered briefly, such as transformations, geometry as an empirical science, geometry as a formal system of information, geometry focused on special objects like triangles, or special qualities like convexity.  We looked at some geometry on the web and using GSP. We introduced the Pythagorean Theorem using GSP. Sketches done in class may be referenced or included in these summaries when convenient.

Next class we'll decide on whether we'll have a class notes system
.

• 1/26 Getting started... some motivational issues. 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. In the distinction between synthetic and analytic geometry the key connecting concept is the use of measurements. For this course initially we will try to avoid the use of measurement based concepts when possible.

To explore some of these issues we looked at proofs of the Pythagorean Theorem. We looked at Euclid's statement and proof of Proposition 47.
In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.
Note that Euclid's treatment in its statement or its "proof" never refers the traditional equation, a2+b2=c2.
The proof we looked used 4 congruent right triangles and 2 squares and the the same 4 triangles and the square on the side of the hypotenuse arranged inside of a square with side "a+b" .

We also considered the shearing proof on Geometer's Sketchpad.
(Based on Euclid's Proof)  D. Bennett 10.9.90
1. Shear the squares on the legs by dragging point P, then point Q, to the line. Shearing does not affect a polygon's area.
2.  Shear the square on the hypotenuse by dragging point R to fill the right angle.
The resulting shapes are congruent. The sum of the squares of the sides equals the square on the hypotenuse.

In considering the Pythagorean theorem we questioned what kind of assumptions were needed in the proof with the triangles and squares. These included assumptions on how we could identify "equal" objects (congruent figures), how object fit together,  and how movements may effect the shapes of objects.

We looked at Euclid's Proposition 1 and  Proposition 2. 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, but this was a fundamental tool  for all of geometry. In discussing these propositions we also noted that certain points of intersection of circles are presumed to exist without reference to any of the postulates. These presumptions are left implicit for hundreds of years, but are cleared up in the 19th century when careful attention is given again to the axioms as a whole system.

We will look further at the foundations of the proofs of the Pythagorean Theorem in two ways:
1. Dissections: How are figures cut and pasted together? What can be achieved using dissections?
2. Transformations: How are figures transformed? What transformations will leave the "area" and "lengths" of figures invariant (unchanged)?

Next Class: We'll look at the possiblilities of dissections (like Tangrams) and start using Geometer's Sketchpad in the lab time (2nd hour).

• 1/28 The lecture part of the class (the first hour) covered equidecomposable polygons. We began by using tangram pieces to make a square. It was observed 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 was some overlap of the pieces in the shape.

This led to a discussion of the question of whether this necessary condition of equal areas would be sufficient to say that two polygonal regions could be decomposed (cut and pasted) into smaller regions that would be congruent. In a sense this says one could create a set of smaller shapes with which one could make either of the two regions usiing prcisely these smaller shapes. The answer to this question is yes (in fact this is a 20th century result), which was the basis for the remainder of the class lecture.

We considered some of the background results which were known to Euclid: (1) parallelograms results and (2) triangle results. The justifications for these results were reviewed only briefly.

(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.
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 parallogram. (This shows that the triangle's area is  the area of this paralleogram 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.)

The film Equidecomposable Polygons was shown which proved 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 polgons can be moved idivually between the two polygons by translations or half turns (rotations by 180 degrees).

Reference was made to the analogous problem in three dimensional geometry where volume equality of polyhedra is a necessary but not sufficient condition for a similar result. This was first demonstrated by Dehn in the 1930's by using another invariant of polyhedra related to the lengths of the edges and the dihedral angles between the faces of the polyhedra.

The next hour was spent in the lab becoming familiar with several of the basic features of Geometer's Sketchpad. Reference was made to definitions in M&I section 1.1. By the end of the class we had constructed a sketch of Euclid's Proposition I in Book I.

• 2/2 After a discussion of some web resources for projects, we reviewed some basic definitions from Meserve and Izzo's 1.1.  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.

After reviewing materials defining, rays, segments, angles, triangles, and planes, we reviewed eight of the basic Euclidean constructions described in M&I. several of these constructions rely on some foundations that assert the existence of points of intersection of circles.

We discussed how constructions also require a justification (proof) that the construction has in fact been achieved. In proving the constructions we used some basic euclidean results, such as the congruence of all corresponding sides in two triangles is sufficient to imply the triangles are congruent (SSS). Also mentioned were SAS and ASA congruence conditions, and well as the result that corrrsponding parts of congruent triangles are congruent (CPCTC).

In considering constructions of tangents to circles we used 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 used the result that any angle inscribed in a semi-circle is a right angle. ( Book III Prop. 31.)

• 2/4 Construction of points to correspond to numbers on a line was discussed.

First Pk where k is an integer was constructed using circles.  After recognizing that we could construct points with fractions using powers of 2 for the denominator by bisection, we discussed how to contruct Pk when k is a rational number using the theory of similar triangles. (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.) We reviewed the construction of a line though a given point parallel to a given line.
We then considered M&I's constructions of the same correspondence of integer and rational points that relied on the ability to construct parallel lines.

We then considered Euclid's treatment of the side-angle-side congruence [Proposition 4] and how it related to transformations of the plane that preserve lengths and angles. Such a transformation T: plane -> plane, would have 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].

After reviewing briefly the outline of Euclid's argument, we noted the key connection between the congruence of figures in the plane and isometeries: 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 discussed briefly how there were at least four types of isometries of the plane: translation, rotation, reflection and glide reflection.

We watched the video Isometries up to the place where it was demonstrated that 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.
We'll watch the remainder of the film and continue the discussion of isometries next class.

In the second part of class we were in the lab. We worked on constructing a square using Geometer's Sketchpad and then illustrated how GS can do translations, rotations, and reflections. We'll try to look at GS's way to do measurements next time in the lab. This week students should try to reporduce some figure from Book I of Euclid using GS.

• 2/9 For the first part of class we discussed convex figures. A figure F is convex if whenever A and B are point in F, the line segment AB is a subset of F. We looked the half plane determined by a line l and a point P not on the line. We discussed informally why this set is convex using a definition of this half plane as the set of point Q in the plane where the line segment PQ does not meet the line l.  We discussed briefly the problems on convex figures in Problem Set 1. We applied the intersection property [The intersection of convex sets is convex.] to show why the interior of a triangle is convex. We also discussed how to show that the region in the plane where (x,y)  has y>x2 is convex using the tangent lines to the parabola y=x2 and the focus of the parabola to determine a family of half planes whose intersection would be the decribed region.

In the second part of class we discussed the continuity axiom for a euclidean line: Any nonempty family of nested segments will have at least one point in the intersection of the family. We saw how this leads to the result that  any list (possibly infinite) of points in a given segment of a euclidean line will not have every point in that segment on the list. [We also noted that we could make a list of points corresponding to the rational numbers once a unit length had been established.  1/1,1/2,2/1,1/3,2/2,3/1,1/4,2/3,3/2,4/1, ....]

Class concluded with showing more of the isometries video. We stopped after the video had demonstrated that any isometry is the product of at most 3 reflections.

• 2/11 We considered what information would determine an isometry. For an 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'.  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 od 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.

We then watched the remainder of the video on isometries, which classified the product of 3 reflections as either a reflection or a glide reflection. We noted that 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

In the second hour we worked on measurements using GSPad.

• 2/16  We started with a look at problems caused by the diagonal of a square and the issue of finding a unit that would measure both the side and the diagonal. We first looked at the Euclidean algorithm for finding a common segment with which to measure two segments. The procedure appeared to stop in the example on the board, but we showed 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 handed by using simple ratios of whole numbers for measurements.

We turned then to Euclid's (Eudoxus') resolution of the issue in  Book V def'ns 1-5.  We looked at these definitions and noted some key items:
* Ratios exist only between magnitudes of the same type. (Homogeneity)
* For ratios to be equal the magnitudes must be capable of co-measuring.
* Euclid's axioms do not deny the existence of infinitesmals- but will not discuss equality of ratios that use them.
* It is Archimedes axiom that stays that any two segments either one can be used  to measure the other. [No infinitesmals.] We discussed briefly how this might be connected to calculus by looking at the change in the area of a square when the length of a side is changed by an infinitesmal.

We examined the connection between Euclid's definition of proportionality (equal ratios) and real number equality of quotients. We showed for line segments the following two propositions for segments A,B,C,and D with m(A)=a,m(B)=b, m(C)=c, and m(D)=d:

1. If A:B::C:D then a/b=c/d .
2. If a/b=c/d then A:B::C:D

At the end of class we discussed some features of isometries of a line, notation for these using coordinates and ways to visualize these.

• 2/18 We began by looking further at how Euclid uses the theory of proportion in Book VI, Proposition 1and Proposition 2. Turning to the concept of similarity as a transformation we considered a similarity on a line with a center of similarity and a given positive magnification factor. This lead 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 saw 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 or using T(Px)=Px', we find that T is described by x'=2x. 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).

We observed that if we let S(Px)=P(x-3) then ST(Px)=S(T(Px))=S(P2x)=P(2x-3) = T*(Px), so ST=T*.

We'll continue the discussion of transformations using plane coordinates next week and beyond.

We turned our attention to right triangles and noted that if an altitude is constructed in a right triangle with the hypotenuse as the base, the figure that results has 3 similar right triangles. Using similarity of these triangles we saw that there is a proportion of the segments of the hypotenuse AD, DB and the altitude CD given by AD:CD::CD:BD.

We then looked at the concept of inverses of points with respect to a given circle and the construction of inverses. These were connected to the similar triangles just mentioned showing that the constructions in M&I were correct.
Looking at the issues of doing arithmetic with constructions in geometry, we noted that this would allow 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.]

In the lab we discussed how to create a script in GSP and started to look at the use of coordinates in GSP. Next Wednesday we'll do more with coordinates, along with the use of traces and locus to see some further aspects of coordinate geometry.

• 2/23 We went over some problems like those on the last assignment checking that distance was invariant using line transformations. We then went further into the relation of the inversion transformation with respect to a circle and orthogonal circles. In particular we showed that if C2 is orthogonal to  C1  (with center O) and Ais a point on C2 then the ray OP will intersect C2 at the point A' where A and A' are inverses with respect to the circle C1.

We used this fact to 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.  [ First construct the inverse 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 disired circle.] Likewise we discussed constructing an orthogonal circle C2 through two points A and B inside a circle C1, as shown in the sketch below.  [still more... this writing is in progress.]

• 2/25 [vague] We looked at how we could begin to see an infinite (ideal ) point on a line or in the plane... leading to developing figures to represent an affine line and an affine plane.

In lab we looked at measurements, calculations, and some visual features such as locus definition, animation, and hide/show buttons.

• 3/2 [Still sketchy.] Homogeous coordinates for an affine line or plane (in 1 and 2 dimensions) were introduced and related to lines through the origin. <a,b>=<d,e> if and only if  there is some k, not 0, where d=ka and  e=kb. these coordinates correspond to the ordinary point  on a line with coordinate x with <x,1>=<a,b> so x=a/b with b not 0. A point with homogeneos coordinate <a,0> corresponds to the ideal (infinite) point on the affine line.A similar treatment works for planar coordinates, (x,y) corresponds to homogeneous coordinates <x,y,1>=<a,b,c> so x=a/c and y=b/c with c not 0. A point with homogeneous coordinate <a,b,0> corresponds to an ideal point in the affine plane.

Discussion turned to five axioms for a committee structure, and then a corresponding geometric structure where committee members are points and committees are lines.

• 3/4 We continued looking at the axioms for a geometry structure. This led to a seven point geometry. In this geometry it is not possible to have more than seven points and satisfy the axioms. (An 8th and 9th point would be necessary and these would cause a failure of the 5th axiom on any pair of lines havng at one point in common.)

In discussing the issue of whether the 5th axiom could be proven from the other four axioms, we looked at an example of another axiom system, with an axiom N (any pair of lines havng at one point in common) and an axiom P (given a line l and a point P not on that line there is a line m where P is on m and m and l have no common points) which is a version of the parallel postulate (Playfair's - not Euclid's). We gave examples showing that the four axioms and P were possible as well as the four axioms and N were possible. This showed that one can not prove axiom P or N from the other four axioms since P and N are contradictary.

By a similar analysis of the axioms for the seven point geometry we showed that it not possible prove the 5th postulate from the other four. The analysis examines the example of the seven point geometry and notices that by including an 8th and 9th point the resulting geometry would satisfy the other 4 axioms.

We briefly discussed at the end of the first hour how the model we had for affine geometry still satisfies the parallel postulate, since the ideal (infinite) points of affine geometry are not considered as ordinary points of the geometry. However, be removing this distinction between ordinary and ideal points and considering the geometry that results we obtain a geometry in which there are no parallel lines. (A projective plane.) This will be a major focus of discussion for the remainder of the term- especially using the homogeneous coordinates to consider points in this geometry from an analytic/algebraic approach.

The lab time was spent working on sketches showing ways to understand that result of the CAROMS film about the inscribed triangles of minimum perimeter.

• 3/9 Initial discussion considered the problems of perception and how the position of an object in a picture can affect our judgment on its relative size. Turning dirctly to the question of perspective in drawing we looked at how an artist tries to capture the visual reality of perception by drawing figures larger when they are closer to the eye of the beholder. We looked at some Durer drawings showing some mechanical ways to draw accurate perspective figures.

An examination of the problems of transfering an spatial image of a plane to a second plane using the idea of lines of sight we arrived at an understanding of how points in the plane would correspond to lines through a point (the eye).

• 3/11 The beginning of class involved a quiz.

After the quiz:
An overview of what we've considered so far-

 Euclidean Geometry lines/planes Affine geometry lines/planes Finite geometry lines Projective Geometry lines/planes Axioms     Euclid     Hilbert No Axioms Yet A figure indicating  an ideal point or line Axioms 7 points/7lines A figure. No Axioms Yet A figure indicating  all points and lines Parallel lines don't meet Parallel lines meet  at an ideal point. All pairs of lines  have a common point All pairs of lines  have a common point Coordinates      Analytic/Algebraic Ordinary coord's  w/ infinite points Homogeneous Coordinates Homogeneous Coordinates  with cofficients in {0,1}= Z 2 Homogeneous Coordinates with real number coefficients Transformations     Isometries     Siimilarities ? ? ?

In discussing the 7 point geometry we visualized it using vertices of a cube (besides (0,0,0)) with their ordinary coordinates in standard 3 dimensional coordinate geometry and identified the 7 points . This allowed us to identify "lines" using the homogeneous coordinate concepts and their relation to planes in three dimension through (0,0,0). We identified all but one of the lines easily- the last plane has ordinary equation X + Y + Z = 2... but in this arithmetic for {0,1} we have 1+1=0, so 2=0 and the vertices of that satisfy this equation in ordinary coordinates {(1,1,0), (1,0,1),(0,1,1)} form a line as well.

We then discussed using {0,1,2} for homogeneous coordinates connected to the arithmetic given by the tables

 + 0 1 2 * 0 1 2 0 0 1 2 0 0 0 0 1 1 2 0 1 0 1 2 2 2 0 1 2 0 2 1

The homogeneous coordinates for this set identify ordered triples, for example: <1,0,1>=<2,0,2> and <1,0,2>=<2,0,1>.  There are 27 possible ordered triples, and thus 26 when we exclude (0,0,0), and these are each paired by the factor 2 with another triple, so there will be exactly 13 points ( and by the comparable work with lines) and 13 lines in this geometry.
.... some details still to be reported.

The class ended by watching the Open University video, "A Noneuclidean Universe."

• 3/16 and 3/18 were vacation.
• 3/22
• 3/24
• 3/31
• 4/1
• 4/6 Okay I missed reporting on several classes. I hope to get those filled in, but let's go forward from here. Today we looked at the video "Conics" from the BBC which outlined the raditional "Euclidean" views of the conics starting with a cross section of a cone. Two way to characterize the conics were treated in the video -

i....related to 2 foci (ellipse and hyperbola) or 1 focus (parabola),
ii...related to a focus and directrix which was also related to eccentricity.

We turned our attention again to projectivities. We showed how 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.

We then considered 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. We briefly discussed the dual figure which would form a more traditional point conic. [We mentioned 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 deteremined by the derivative: y=x^2-x+3 which is a parabola.]

We turned our consideration to looking at 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: T(x)=x+3 became

 1 3 x x+3 0 1 1 1

.. Reflection R(x)= -x  uses the matrix

 -1 0 0 1

Dilation by a factor of 5  M(x)=5x  uses the matrix

 5 0 0 1

and inversion I(x)=1/x uses

 0 1 1 0

We also considered how composition of these transformations corresponds to matrix multipication and how these transformations interact with the ideal point at infinity using homogeneous coordinates.
More to follow.

• 4/8 The class began with a quiz. We then watched the film on central perspectivities that discussed perspectivites and projectivities in the plane. Any perspectivity between a pencil of points on one line and that one another line can be thought of as a transformation. This transformation is completely determined by the relation of two pairs of

distinct points on two lines. For a projectivity (the composition of a finite number of perspectivities) the result is that a projectivity is completely determined by the correspondence of three points.This result is called the fundamental theorem of projective geometry and is taken as an axiom by M&I.

We also watched the film on projective generation of conics which introduced Pascal's theorem and its converse about hexagons inscribed in a conic and showed how to use this result to construct a conic curve passing through any 5 points. This work was also related to projectivities between pencils of lines. We will be considering this further in the course. After a short break we continued using metric ideas to construct an ellipse  as a locus on sketchpad and discussed how to do a parabola as well. By next Thursday students should construct examples of the three conics on sketchpad using metric ideas.

We also introduced ths concept of 4 points on a line being related harmonically. This will be the chief tool we use to introduce (homogeneous) coordinates into projective geometry.
The idea of a general projective transformation of a projective line using homogeneous coordinates and invertible 2x2 square matrices was also introduced. This showed that all the transformations of the line we had previously discussed were examples of this general type of projective transformation.
These two topic (harmomics and projective transformations) will be a major topic for the next few weeks.

• 4/13 We continued our discussion of four points on a line being  harmonically related. Using the text notation we saw 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 saw H(AB,CD) equiv to H(AB,DC) and H(BA,DC).

We turned our attention to the relation between the "double points" in the figure and the "single points". We proved using the following figure (adapted from Meserve & Izzo) that H(RT,SU) is equivalent to H(SU,RT).
I.e., we show that if H(RT,SU) then H(SU,RT): The new quadrangle used to show this harmonic relation is determined on the figure by the points P3, P4, W and V. [The proof needs only show that WV meets SUat the point T.] Triangle WP1P2 is perspectively related by the line SU to triangle VP3P4. Thus by Desargues' Theorem we have that the triangles are perspectively related by a point. But this point must be T, so the line WV passes through the point T, completing the demonstration that H(SU,RT).

Turning our attention to the construction on the affine line of the point P2 from P0,P1, and Pinf, we noticed that the figure used to construct this point showed that H(P1Pinf,P0 P2).  This led to a discussion of constructing the fourth point, D, on a line given A,B and C already on the line so that H(AB,CD). We used the construction of P2 to show how to construct the point D in general.

The issue then became: was the point constructed from the points A, B and C uniquely determined by the fact that it was in the harmonic relation with A, B, and C? This is the question of the uniqueness of the point D. We proved that in fact the point is D. (The proof followed the argument of Meserve and Izzo. It used Desargues' theorem several times.)

With the existence and uniqueness of the point D established, we turned to some examples of establishing a coordinate system for a projective line by choosing three distinct points to be P0, P1, and Pinf. We constructed P2, P1/2 (in two different ways) and left as an exercise the construction of P3 and P1/3. We'll continue this discussion next class in showing that that with the choice of three points on a projective line we can construct points using harmonics to correspond to all real numbers (as they did in our informal treatment of the affine line).

The class ended with a review of algebraic projective transformations of the projective line P(1) which is characterized as a set of pairs of real numbers <a,b>, not both zero, with <a,b>=<c,d> in the case there is a nonzero real number t so that c=ta and d=td. These transformations correspond to 2x2 matrices with non-zero determinant. These form a group under composition (matrix multiplication) and we looked at one particular example to see how it transformed points on the projective line with coordinates to other points on the projective line. We will see how this transformation is completely determined by the correspondence of three distinct pairs of points. In some cases the transformation will transform an ordinary point to an ordinary point, the ideal point to an ordinary point and an ordinary point to the ideal point, and in some cases the transformation will tranform the ideal point to the ideal point. A transformation that transforms the ideal point to the ideal point is called an affine transformation. The composition of two affine transformations is an affine transformation. The inverse of an affine transformation is an affine transformation and clearly the identity transformation is an affine transformation, so the affine transformations are also a group under the operation of composition (matrix multiplication). We'll continue this discussion next class to examine more about the affine group for P(1).

• 4/13
• 4/15
• 4/20 Okay... just a quick recall of some of what we covered:
1. Now that we've established how to connect homogeneous coordinates to a projective line using harmonics we continued by looking at some more algebraic projective lines- with coefficients of Z2, Z3, and Z5, we saw that these projectuve lines would have 3,4, and 6 points respectively, corresponding to the 2,3,and 5 ordinary points and one ideal point. This lead to
2. A discussion of using the complex numbers, C, and the geometry of CP(1). The ordinary points on this "line" can be visualized as a plane or all the points of a sphere except one- which corresponds to the north pole or the point at infinity. [This was compared to the visualization of RP(1) as a circle.]
3. The disussion turned to algebraic projective transformations of RP(1). When these leave the ideal point fixed they are called affine transformations and form a group under composition. We showed that an affine transformation has a matrix of the form
 a b 0 1

and therefore T(Px)=Pax+b. Thus an affine transformation of the projective line is a dilation/reflection followed by a translation.

4. We looked at the idea of establishing the idea of a distance between ordinary points using absolute value-
d( <a,b>,<c,d>) = |a/b - c/d|.    We showed this is well defined and then discussed what the isometries for this distance would be. After some analysis we saw that this would mean that in the matrix of an isometry |a|=1, so a=1 or a=-1. I.e,  the matrix is
 +/-1 b 0 1

or that T(Px)=P+/-x+b. Thus an isometry of RP(1) is a reflection followed by a translation.

5. We also looked further at Pascal's theorem and its planar dual Brianchon's theorem. Next class we'll prove Brianchon's theorem using an elliptic hyperbaloid.
• 4/22 Today we did more on algebraic projective transformations of CP(1). This meant first examining what it meant for these to be affine, and then the geometry of addition and multiplication of complex numbers. Addition of a constant comex number corresponded to a translation. Multiplication by a constant corresponded to a dilation using the absolute value of the complex number for the factor and a rotation by the angle the corresponding vector mad with the positve real axis. We also saw how affine transformations could preserve the distance between complex numbers  if |a|=1 where a is a complex number, so a can be thought of a point on the unit circle and multiplication by a corresponds to a rotation. So affine transformations of CP(1) are rotations followed by translations of the plane of complex numbers. Notice these are all orientation preserving transformations. so any reflection of the plane is not an isometry in this geometry. We looked briefly at another transformation that was a reflection, namely complex conjugacy, T(a+bi)=a-bi. This is not an isometry in this geometry because it is orientation reversing.

We spent the remainder of the lecture time going over the proof of Brianchon's Theorem using the proof of Hilbert and Cohn-Vossen based on hexagons lying on the surface of an elliptic hyperbaloid (which is a ruled surface).
• 4/27