Thu, Apr 27, 2006 -- Final Exam
The final will be given on Monday, May 8 from 4:40 pm to 6:30 pm in our classroom, PSA 303.

The final will be cumulative, but will emphasize material from the second half of the course.

In general, be very familiar with the homework questions and the midterm. Expect modifications to these questions that will assess how well you understand the material to an extent that you can apply it in a slightly different setting.

From the first half: In general, be very familiar with the homework questions and the midterm. Expect modifications to these questions that will assess how well you understand the material to an extent that you can apply it in a slightly different setting.

From the first half:
From the second half:
1. H (Δ) = Σ βi - π = 2π - Σ αi Same proof for spherical and hyperbolic
2. K Area(Δ) = Σ βi - π = 2π - Σ αi Different proofs for spherical and hyperbolic
3. H (Δ) = K Area(Δ) Just combining 1 & 2 above
4. K Area(Γ ) = [ Σ βi - (n - 2) π] = 2π - Σ αi Applying 2 to several triangles combined
5. K area(M) = 2π ( v - e + f ) = 2π χ(M) Using 4 to prove
6. χ(M) = Σ indices of zeroes of a vector field Partial proof by constructing a special vector field that works
7. χ(M) = Σ (2π - cone angles) Not a rigorous proof, but know how to combine 3 & 5 with the idea of squeezing curvature to cone points.
Wed, Apr 05, 2006 -- Hyperbolic Sketchpad
Here is a link to the Java-based hyperbolic geometry sketchpad, NonEuclid.

http://cs.unm.edu/~joel/NonEuclid/

Mon, Mar 06, 2006 -- Midterm
March 9 or 10
To be taken in the Mathematics Department Testing Center.
Thursday:
   Open: 8:00am - 8:00pm
   Last exam given out: 6:30pm
   Last exams collected: 7:55pm
Friday:
   Open: 8:00am - 3:00pm
   Last exam given out: 1:30pm
   Last exams collected: 2:55pm

You will need your ASU Sun Card!

Topics to be covered:
  1. Isometries
    1. General definition
    2. Isometries of the plane, sphere, cone, cylinder
  2. Symmetries
    1. General definition
    2. Symetries of geodesics on the plane and sphere
  3. Geodesics
    1. General definitions (from symmetries, from differential geometry, intuitive versions of no-turning conditions and shortest distance)
    2. Characterization of geodesics on the plane, sphere, cone, and cylinder
  4. Angles
    1. Definitions (measure, geometric shape, dynamic)
    2. Definition of congruence
    3. Vertical angle theorem and proof
  5. Covariant Derivative
    1. Intrinsic and Extrinsic objects and operations
    2. Computation of the covariant (intrinsic) derivative
    3. Relation to geodesics (no turning condition)
    4. Spherical and Cylindrical coordinates
  6. Congruence Theorems
    1. Angles: VAT, ITT
    2. Triangles: SAS, SSS, ASS, SAA, & AAA
    3. Determine whether they are true on the plane and sphere
    4. Proof on the plane
    5. Counterexamples for false cases
  7. Approaches to geometry
    1. Basics of different approaches
      1. Axiomatic/Synthetic Geometry (Euclid, Hilbert) (undefined terms, axioms or postulates, rules of logic, definitions, theorems, proofs)
      2. Analytic/Coordinate Geometry (Descartes)
      3. Riemannian Geometry (space, metric, covariant derivative)
      4. Transformation Geometry / Erlangen Program (Klein) (spaces, isometries, invariants)
    2. Assumptions
      1. Identifying your assumptions in proofs and definitions
      2. Locating assumption in the various branches of geometry

External Links

NonEuclid: a Java-based hyperbolic geometry sketchpad

Curved Spaces: Interactive 3D program that allows you to explore different flat, spherical, and hyperbolic spaces.

The Geometry of Rotation: A powerpoint file for the shape of SO3


Homework 10 Comments
Homework 10 Comments (98.174 Kb)
Download this document for some notes on how to think about problems 2 and 3 from homework 10.


Hilbert's Axioms
I. Axioms of incidence

Postulate I.1. For any two points A, B, there exists a line a that contains each of the points A, B

Postulate I.2. For any two points A, B there exists no more than one line containing both A and B

Postulate I.3. There exist at least two points on any given line. There exist at least three points that do not lie on a given line.

Postulate I.4. For a set of three points {A, B, C} that do not lie on the same line, there exists a plane α that contains each of the points in the set. For every plane there exists at least one point which it contains.

Postulate I.5. For a set of three points {A, B, C} that do not all lie on the same line, there exists only one plane that contains each of the points in the set.

Postulate I.6. If two points {A, B} of a line, a, lie in a plane, α, then every point in a lies in α

Postulate I.7. If two planes {α, β} have a point A in common, then they have at least one other point, B, in common

Postulate I.8. There exists at least four points which do not lie in a plane

II. Axioms of order

Postulate II.1. If a point B lies between points A and C, then the points {A, B, C} are three points on the same line and B lies between C and A

Postulate II.2. Given two points {A, C}, a point B exists on the line AC such that C lies between A and B

Postulate II.3. Given any three points {A, B, C} of a line, one and only one of the points is between the other two

Postulate II.4. Given three points {A, B, C} that do not lie on a line and given a line, a, that lies in the plane ABC but does not contain any of the points A, B, C: if the line a passes through a point of the segment AB, then it also passes through a point in the segment AC or through a point in the segment BC

III. Axioms of congruence

Postulate III.1. Given two points {A, B} on a line a and given a point A' on a or another line a', there exists a point B' on a side of the line a' such that ABA'B' are congruent

Postulate III.2. Given segments A'B' and A"B" such that both are congruent to the same segment AB, then A'B' A"B"

Postulate III.3. Given a line a with segments AB and BC such that the point B is the only intersection of the two points and on the same line or a line a' with segments A'B' and B'C' such that the point B' is the only intersection: if ABA'B' and BCB'C' then ACA'C'

Postulate III.4. If ABC is an angle and B'C' is a ray, then there is one and only one ray B'A' on each side of the line B'C' such that A'B'C'ABC

Corollary: Every angle is congruent to itself

Postulate III.5. Given two triangles ABC and A'B'C' with congruences such that ABA'B', ACA'C' and BACB'A'C' then ABCA'B'C'.

IV. Axiom of parallels

Postulate IV.1. Given a line a and a point A not on a, there is at most one line in the plane that contains a and A that passes through A and does not intersect a

V. Axioms of continuity

Postulate V.1 (Archimedes axiom). Given segments AB and CD, there exists a number n such that n copies of CD constructed contiguously from A along the ray AB will pass beyond the point B

Postulate V.2 (line completeness). There exists no extension of a set of points on a line with order and congruence relations that would preserve the relations existing among the original elements as well as preserving line order and congruence, i.e., Axioms I-III and V.1.