Mathematics 138
What you should know how to do for the final exam

I will use this list to construct the final exam. New items start with E7.  About 2/3 of the final will be devoted to them. 
Topics that were thoroughly covered on the midterm will not reappear on the final.

A. Finite affine geometry
1. State, recognize, or explain the axioms for an affine plane.
2. Draw or apply diagrams to show how the axioms of Desargues and Pappus apply to a specified "affine faculty senate," especially to proving key results about addition or multiplication.
3. Prove the following theorems about affine planes, given either a diagram to illustrate the proof, or a proof that requires a diagram, or a proof with key steps omitted:
a. The result of addition or multiplication does not depend on the choice of "auxiliary instructor" in the special case where the two auxiliary instructors are on the same committee as the appropriate identity.
b. Addition or multiplication is associative.
c. Addition or multiplication is commutative.
d. The equation AC = E or A+C = E can be solved uniquely for A or C, given the other, as long as division by 0 is not required.
4. Given a list of instructors and committees for an "affine senate" and coordinates for the members of two intersecting committees, determine
a. the equation satisfied by a specified committee.
b. the committee that satisfies a specified linear equation.
c. the coordinates of a specified instructor.
d. the instructor with specified coordinates.

B. Incidence properties of Cayley-Klein geometries
1. Fill in all or part of the 3x3 grid enumerating the nine possible Cayley-Klein geometries in terms of whether the measurement of distance and angle is elliptic, parabolic, or hyperbolic and assigning a name to each geometry.  For each geometry, given a line m and a point P not on m, specify how many points on m cannot be joined to P by a line and how many lines through P do not intersect m.
2. Given a model of one of the nine Cayley-Klein plane geometries, use incidence properties (number of lines through a given point parallel to a given line and number of points on a line that cannot be joined to a specified point) to determine the nature of the geometry and give it a name. (You are not responsible for knowing the details of the models that were listed in the handout.)
3. Given the definition of distance or measure of angle in a geometry, identify it as elliptic, parabolic, or hyperbolic.
4. Explain or apply the connection between the elliptic, parabolic, or hyperbolic measurement of angle and the measurement of the area of an appropriate portion of a circle, triangle, or hyperbola.
5. State and apply the definition of cross-ratio for the segments determined by four points on a line or the angles determined by four rays with a common vertex, and use the law of sines to prove that cross-ratio is the same for any line cutting four rays with a common vertex.
6. Use the law of sines to prove theorems of Euclidean geometry such as the theorems of Menelaus and Ceva. (You are not responsible for memorizing the statements of these specific theorems.)

C. Isometries of the Euclidean plane
1. Write or identify the 2x2 matrix for a specified reflection about a line through the origin or rotation or rotation about the origin.
2. Write the 3x3 matrix that represents a specified reflection, translation, rotation, or glide reflection.
3. Given a Euclidean isometry, represent it as the product of three or fewer reflections, and prove or apply theorems that are used to accomplish this, especially
  a. The product of three reflections in lines through a single point equals a reflection in a suitably chosen line though that point.
  b. The product of three reflections in parallel lines equals a reflection in a suitably chosen line parallel to the given line.
  c. The product of two half-turns is a translation.
  d. A glide reflection followed by a reflection equals a translation or a rotation.
4. Given an isometry that is specified by a 3x3 matrix or verbally as a composition of isometries, determine its fixed points and fixed lines
5. Construct an isometry with specified fixed points or lines, either by writing down a 3x3 matrix or by giving a verbal description of the isometry.
6. Given a matrix or a sequence of isometries that specifies a glide reflection, determine the axis of the glide reflection.
7. Construct or identify the direction vector for the bisector of a specified angle in the Euclidean plane.
8. Construct graphically the result of performing successive Euclidean isometries (by expressing them cleverly as products of reflections).

D. Affine transformations of the Euclidean plane.
1. Prove that an affine transformation that fixes three noncollinear points must be the identity and show that there is a unique affine transformation that carries three specified noncollinear points into three other specified noncollinear points.
2. Given two triangles in the plane, construct the affine transformation that maps one into the other.
3. Prove that any affine transformation is a collineation.
4. Carry out steps of the proof that any collineation of the Euclidean plane is an affine transformation, and indicate how this proof can fail to apply to other affine planes.
5. Determine what isometries are required in order to make specified triangles congruent in the Euclidean plane.

E. Galilean geometry
1. Interpret the "points" and "lines" of Galilean geometry in terms of events and uniform motions along a line, and provide a physical interpretation of the isometries of Galilean geometry.
2. Write down or identify the 3x3 matrices that represent specified Galilean isometries.
3. Determine ordinary or special distances or angles for points or lines in the Galilean plane.
4. State, prove, and apply the Galilean “triangle equality” and the Galilean Law of Sines.
5. Construct the bisector of the Galilean angle between two lines.
6. Given a theorem of Galilean geometry, formulate its dual theorem.  Given the proof of the theorem, construct a proof for its dual.
7. Carry out graphically the addition of two doublets A and B in three cases:
a.	the end of A is the beginning of B.
b.	both A and B have the same beginning.
c.	the general case.
8. Given a doublet, express it as a linear combination of basis doublets I and O. Represent graphically a doublet xI+hO.
9. State or identify the axioms for Galilean geometry, and explain how a given axiom can be satisfied either by vectors and points or by doublets and lines.
10. Given a theorem about vectors in Galilean geometry, state and prove a dual theorem involving doublets. 
11. Given a parabola in the Galilean plane, characterize it in terms of the directed angle made by the lines joining an arbitrary point on the parabola to the ends of a segment joining two fixed points on the parabola.
12. Write down or identify the matrix for a cyclic rotation that holds a given parabola invariant, or find a parabola that is held fixed by a given cyclic rotation.
13.Given a triangle of Galilean geometry, construct the parabola through its vertices, and interpret the invariant property of this parabola physically in terms of acceleration.

F. Minkowski Geometry
1.	State the axiom that distinguishes Minkowski geometry from Euclidean and Galilean geometry, and show that it divides vectors into timelike, lightlike and spacelike classes.
2.	Define distance and angle in Minkowski geometry, and prove a Cauchy-Schwartz inequality to show that the angle between two forward timelike vectors is well defined.
3.	Write down or identify the matrix for a Lorentz transformation, and show that its eigenvectors are lightlike.
4.	Prove and apply the Law of Cosines and the Law of Sines.
5.	Determine the angle bisector for two intersecting timelike lines and show in what sense a point on this bisector is equidistant from the two lines.
6.	State the properties of “circles of the first kind” and “circles of the second kind.”
7.	Formulate and prove theorems about hyperbolas that are equivalent to the corresponding theorems about circles in Euclidean geometry.
Note: in the interest of fairness, there will not be questions on the final exam that give an advantage to people who have studied special relativity in a physics course.

G. Spherical Geometry
1. State and apply the fundamental identities involving cross products, inner products, and determinants. These involve <u x v, w>, (u x v) x w, 
and <u x v, w x z>
2. Given two non-antipodal points on the unit sphere, find the pole of the great circle that passes through them; given two great circles, find the two antipodal points in which they intersect.
3. Given two points on a great circle, represent the great circle parametrically in the form (cos t)P + (sin t)Q, where either P or Q may be required to be one of the original given points.
4. Given two points on the unit sphere, calculate the distance in radians between them; given two rays of spherical geometry, calculate the angle between them.
5. State, prove, and apply the Law of Sines, the two Laws of Cosines, and the triangle inequality for spherical geometry.
6. From the Law of Sines and/or the Law of Cosines, derive formulas that relate the sides and angles of a right spherical triangle.
7. Describe the all motions of the sphere that are products of three or fewer reflections.
8. By exploiting the fact that a rotation of the sphere can be represented in different ways as the product of reflections, simplify products of motions of the sphere (for example, ones that involve half-turns), and show how to represent any motion as the product of three or fewer reflections.
9. By considering eigenvalues of real orthogonal 3x3 matrices, prove that any isometry of the sphere is represented by an orthogonal 3x3 matrix and can be represented as a product of three or fewer reflections.
10. State, prove, or apply the formula that expresses the area of a spherical triangle in terms of its angles.
11. Construct a midpoint and a perpendicular bisector of a segment in spherical geometry and describe the effect of reflection in the perpendicular bisector. Construct a bisector of the angle between two lines in spherical geometry, and describe the effect of reflection in this bisector.  
12. Prove theorems of spherical geometry that are similar to well-known theorems of Euclidean geometry.

H. Projective geometry
1. State the axioms of projective geometry, and draw diagrams to illustrate the two axioms that are related to the theorems of Desargues and Pappus.
2. Given an "affine senate," show how to add instructors and committees "at infinity" to convert it to a projective senate; given a projective senate, show how to convert it to an affine senate by removing a specified "committee at infinity."
3. Given an assignment of coordinates to the instructors of an affine senate, assign homogenous coordinates to the instructors and committees of the projective senate so that the instructor with affine coordinates (x,y) acquires coordinates (x,y,1).
4. Show that the one-dimensional and two-dimensional subspaces of a three-dimensional vector space over a field satisfy the first three axioms of projective geometry.
5. Use cross products to determine the committee of finite projective geometry on which two specified instructors serve or to determine the instructor who serves on both of two specified committees.
6. Show how to introduce coordinates into a projective plane so that four specified points have coordinates (1,0,0), (0,1,0), (0,0,1) and (1,1,1) respectively, and use such coordinates to assign coordinates to all the points in a diagram for the theorem  of Desargues or the theorem of Pappus.
7. State, prove and apply the "fundamental theorem of projective geometry."
8. State and apply the properties (bilinearity, symmetry, and non-degeneracy) of a polarity that is represented by a nonsingular symmetric matrix B, and prove that the matrix A of a collineation that preserves such a polarity satisfies AtBA = B.
9. Given a polarity of a projective plane for which the equation <v, Bv>=0 has a solution, classify the points of the projective plane as lightlike, spacelike, or timelike.

I. Hyperbolic geometry
1. Draw a diagram of the real projective plane with a conic represented by a circle (the Klein model), and show how any spacelike or timelike point or line can be constructed from lightlike points and lines.
2. Prove and apply theorems about the number of  lightlike points on lightlike, timelike, or spacelike lines.
3. Either algebraically or by constructing tangents in the Klein model, find the pole of a spacelike line or the polar line of a spacelike point.
4. Either algebraically or by carrying out an appropriate construction in the Klein model, find the pole of a timelike line or the polar line of a timelike point.
5. Given a timelike line and one point on the line represented by a spacelike unit vector, find one more spacelike vector and one timelike vector so that the three vectors form an orthonormal basis.
6. Given a spacelike line and one point on the line represented by a timelike unit vector, find one spacelike vector on the line and one other spacelike vector so that the three vectors form an orthonormal basis.  Using two of these basis vectors, express all the timelike points on the line parametrically in terms of hyperbolic functions.
7. Using the Klein model, illustrate how lines of hyperbolic geometry can be intersecting, parallel, or ultraparallel.
8. Prove that two ultraparallel lines have a unique common perpendicular, and construct this common perpendicular either algebraically or by a diagram in the Klein model.

J. Hyperbolic isometries and hyperbolic trigonometry
1. State and apply the definitions of distance and of angle in real hyperbolic geometry in terms of vectors that represent points and lines and the symmetric matrix B that defines the polarity of hyperbolic geometry.
2. Either analytically or by using a diagram of the Klein model, construct the midpoint and the perpendicular bisector of a segment in hyperbolic geometry and describe the effect of reflection in the perpendicular bisector. 
3. Construct the bisector of the angle between two interecting rays in hyperbolic geometry, and describe the effect of reflection in this bisector. 
4. Given two ultraparallel lines of hyperbolic geometry, construct their midline graphically using the Klein model, and describe the effect of reflection in this midline.
5. State the formula for the image of a point of hyperbolic geometry under reflection in the line whose pole is a given unit vector x . Use this formula to prove that any reflection is an isometry and to derive the matrix for reflection in a given line.
6. Using a diagram of the Klein model, determine by geometric construction the image of a given (timelike) point under reflection in a given (spacelike) line.
7. State what isometries result form the product of reflections in two lines of a concurrent, ultraparallel, or parallel pencil.  In simple cases, construct the matrix for such an isometry.
8. State the three-reflections theorems and representation theorems for hyperbolic geometry, and use them to simplify products of isometries.
9.	Apply the two Laws of Cosines and the Law of Sines for a general hyperbolic triangle.
10. From the Law of Sines and/or the Law of Cosines, derive formulas that relate the sides and angles of a hyperbolic right triangle, and derive the formula for the angle of parallelism.
11. State the key facts about area and congruence for triply asymptotic and doubly asymptotic triangles.  Using these, prove Gauss's formula for the area of a hyperbolic triangle.
12. Define and identify the three types of cycles in hyperbolic geometry.
13. Identify or invent statements that can be used in place of Euclid's fifth postulate to distinguish Euclidean geometry from hyperbolic geometry.

Note: so that you will not make sign errors, statements of the Law of Sines and Law of Cosines for several geometries will be supplied on the final examination.  However, these will not be identified.  It is up to you to know which law goes with which geometry.