Martin Flashman's Courses - Math 371 Spring, '07

Geometry Notes
The New Course: Geometric Structures for the Visual

[Work in Progress DRAFT VERSION 3-1-07]

Blue sections indicate tentative plans for those dates.
  1/17 Introduction
1/19 Continue discussion of what is "geometry"? 
Start on Euclid- Definitions, Postulates, and Prop 1.
1/22 Begin The Pythagorean Theorem plus...
1/24 Pythagorean plus... Dissections ?
1/26 Guest Lecture- Here’s looking at Euclid 
1/29 Dissections- equidecomposeable polygons
Begin Constructions and the real number line.
1/31 M&I's Euclidean Geometry
Constructions and  The real number line.- Continuuity
2/2 Breath
 2/5 Construction of rational numbers. 2/7 Inversion and Orthogonal Circles
  2/9. Odds and ends, More on Inversion. Isometries 
2/12 Classification of Isometries & More :)
2/14  Isometries. Coordinates and Classification
2/16  Proof of classification result for plane  isometries
2/19 More Isometries

2/21 Recognizing Isometries
2/23 Symmetry.
2/26 Similarity & Proportion (Euclid V) More on Proportion and Measurement 2/28  More on Similarity and transformations.
3/2 Similarity.
 Inversion and Beginning to See The Infinite.
The Affine Line and Homogeneous Coordinates. 
3/5 The Affine Plane and
Homogeneous Coordinates
More on seeing the infinite.
The Affine Line and Homogeneous Coordinates.
3/10 More on The Affine Line and Homogeneous Coordinates and the Affine Plane 
3/9 Homogeous coordinates and the Affine Plane

3/12 No class Spring Break!
3/17 No class Spring Break!
3/16 No class Spring Break!
Introduction to projective geometry with homogenous coordinates.
Lab time: Video "A non-Euclidean Universe."
 Introduction to Desargues' Theorem- a result of projective geometry.
3/21Another vector geometry Proof.
Connecting Axioms to Models.

3/23 Finite geometries!

3/26 Z2 and  Finite Projective Geometry.
Models for Affine and Projective Geometries.

Axioms for Synthetic Projective Geometry (see M&I)
Examples of proofs in synthetic projective geometry. 
3/30 No class CC Day 
4/2 Quiz 2
More examples of proofs in synthetic projective geometry.

4/4  Desargues' Theorem-
Proof of Desargues' Theorem in the Plane
A look at duality and some applications.
Sections and Perspectively related Figures
Some key configurations. space duality, perspective reconsidered.

4/9 Conics. Introduction to Pascal's Theorem
Video: Orthogonal Projection
Perspectivies as transformations.

4/13  Space Duality and polyhedra
Sections in Space
The complete Quadrilateral.
4/16 Start Isometries with Homog. Coord.
4/18 More on Matrix Projective Transformations.
Harmonics: uniqueness and coordinates for Projective Geometry.  Planar transformations and Matrices
4/23   An Inversion Excursion?
Projectivities. Projective relations
Projective Line transformations: Synthetic Projectivities;
Inversion Video? Projective Conics Video?
4/30 Quiz #3? 5/2

[Side Trip] Moving line segments:

            1-31 and 2-2

            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

    Convexity: Another brief side trip into the world of convex figures.
        The relation of the inversion transformation with respect to a circle and orthogonal circles.
][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

    2-21 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: 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:

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 3/28
We consider these statements in algebraic models for these planes.