Martin Flashman's Courses - Math 371 Spring, '09

Geometry Notes
The New Course: Geometric Structures for the Visual

[Work in Progress DRAFT VERSION Based on noted from 07]

Blue sections indicate tentative plans for those dates.

1-21-09 Introduction
1-23-09 Continue discussion of what is "geometry"? 
Start on Euclid- Definitions, Postulates, and Prop 1.
1-26 Euclid- Definitions, Postulates, and Prop 1.
1-28 Pythagorean plus... Dissections ?
Dissections- equidecomposeable polygons
Assign viewing of  looking at Euclid 

2-2 Begin Constructions and the real number line.
 2-4 M&I's Euclidean Geometry

2-6 More on Equidecomposeable polygons
2-9 Construction of rational numbers. Constructions and  The real number line.- Continuity 2-11  Inversion and Orthogonal Circles
2-13  More on Inversion. and Continuity.  Similar Triangles
2-16  Odds and ends, The real number line.- Continuity. Coordinate based proofs.
.Isometries:Classification of Isometries & More :)
 Isometries. Coordinates and Classification
Proof of classification result for plane  isometries
  More Isometries

Recognizing Isometries
  Similarity & Proportion (Euclid V) More on Proportion and Measurement   More on Similarity and transformations.
  Inversion and Beginning to See The Infinite.
The Affine Line and Homogeneous Coordinates.
Homogeneous Coordinates
More on seeing the infinite.
The Affine Line and Homogeneous Coordinates.

No class Spring Break!
 No class Spring Break!
 No class Spring Break!
3-23 More on The Affine Line and Homogeous coordinates and the Affine Plane
3-25 Homogeneous Coordinates and the Affine Plane


3-27 Introduction to projective geometry with homogeneous coordinates.Finite geometries!

3-30 Axioms
Connecting Axioms to Models.
Z2 and  Finite Projective Geometry.
Models for Affine and Projective Geometries.

4-1 Introduction to Desargues' Theorem- a result of projective geometry
Axioms for Synthetic Projective Geometry (see M&I)

4-3  homogeneous functions and coordinates: the circle and parabola in the affine and projective plane from equations. A beginning to conics.

4-6 Examples of proofs in synthetic projective geometry.

4-8 Desargues' Theorem-
4-10 Quiz 2 Proof of Desargues' Theorem in the Plane
Introduction to duality.

4-13 A look at duality and some applications. 

4-15 Sections and Perspectively related Figures
Some key configurations. space duality, perspective reconsidered.
Space Duality and polyhedra
Sections in Space
Perspectivies as transformations.
The complete Quadrilateral.
Inversion Video?
4-20 More on Perspectivities and "mapping figures"
Conics. Introduction to Pascal's Theorem
4-22 Start Isometries with Homog. Coord.
4-24 More on Matrix Projective Transformations?
Harmonics: uniqueness and coordinates for Projective Geometry.

4-27 More on Harmonics.
Projectivities. Projective relations
Projective Line transformations: Synthetic Projectivities;
4 -29 Harmonics Theroems
5-1 Quiz #3
5-4 Projective Conics Video?
Pascal and Brianchon's theorem.
Proof of Brianchon's theorem
Planar transformations and Matrices
An Inversion Excursion?

[Side Trip] Moving line segments:


            Read  the definitions in M&I section 1.1
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 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.

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2-16/18 Odds and ends...!
    Convexity: Another brief side trip into the world of convex figures.
[ cos(t)
] [ 1
] [ cos(t) sin(t)
cos2(t)-sin2(t) 2cos(t)sin(t)

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

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=h2. Notice this says that h = sqrt(ab).


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Application of this construction: Choosing a = 1, this proportion becomes h = sqrt(b)

Interlude (not covered in Lecture): Using Vectors in a geometric structure:
The segment connecting midpoints of the sides of a triangle proposition: A Vector Proof

S(x , y) = (5x -12 ,5y - 8).



                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:


Not covered in class:Another example of vector methods in geometry:
Proposition: Show that the perpendicular bisectors of any triangle meet in a single point [which is equidistant from the vertices]

                    View video: Central perspectivities VIDEO4206
We consider these statements in algebraic models for these planes.

 Sections and Perspectively related figures in a projective plane.

More on  projectivities.