MATH 461 - Affine and Projective Geometry

The class meets TuTh 1:00 - 2:15pm in LGRT 141.

Professor: Jenia Tevelev

Office: LGRT 1235E. Office hours are TuW 2:30-4:00

E-mail: tevelev(at)math.umass.edu

Textbook: The four pillars of geometry by John Stillwell. We will cover pretty much the whole book except for the last section. Take lecture notes because I will often give examples and proofs that are not contained in the book.

As in any math class, it is very important to understand everything written in the textbook and covered in lectures. Read slowly and take your time understanding all definitions, examples, and proofs. Don't take anything on faith. Use as a motto a famous phrase of Alexandre Grothendieck: "I never once doubted that I would eventually succeed in getting to the bottom of things." Every time you read a statement of a theorem or lemma, pause and think about it before reading the proof. Do you know definitions of all concepts mentioned in the statement? What makes this statement non-trivial? Have you seen similar statements before? How would you prove it? When you read the proof, try to isolate its main idea. Remember that each problem is unique but there aren't that many different proof techniques. A trick used in the proof of a theorem can often be re-used in the proof of a homework problem. Reading a mathematical textbook is not always easy. Sometimes you will be under impression that two consecutive sentences are not logically connected. If this happens, the author probably wants you to make this logical connection yourself.

The homework will be posted on the class website. It will be collected each week on Thursday. Most of the problems will require writing proofs. Write them in an essay style (like proofs in the book) using complete English sentences. Check that everything you are saying is rigorously justified. You will make the grader's life easier by providing as many details as possible.

There will be two in-class midterms (October 5 and November 9) and occasional 15-minute quizzes (I will e-mail the class before every quiz).

The final grade will be 20% midterm 1, 20% midterm 2, 20% final exam, 30% HW, 10% quizzes. Two worst HW grades will be dropped. All grades will be posted on Moodle.

HW assignments:

   HW due on             Sections to read              Homework problems       Comments   
September 14 Chapter 1 + lecture notes. 1.2.3, 1.3.3, 1.3.5, 1.4.3, 1.4.4
I. Given a circle, construct the center of this circle
II. Given a circle C and points A and B, construct a point on C equidistant from A and B. Is there only one such point?
III. Construct a right isosceles triangle circumscribed around a given circle.
IV. Given a circle and a line, construct a tangent line to the circle parallel to the given line.
V. Given a circle, construct an inscribed rectangle such that the angle between its diagonals is 30 degrees.
Lecture slides.
Euclid's Elements: An interactive web-based version of the famous "Elements".
Homework solutions.
September 21 Chapter 2 + lecture notes. 2.2.2, 2.3.1, 2.3.2, 2.3.3, 2.5.2, 2.7.5
I. Let A be a point outside a circle. Draw lines AX, AY tangent to the circle at X and Y. Prove that AX=AY.
II. Let ABCD be a quadrilateral such that angles ACB and ADB are equal. Show that all vertices of this quadrilateral lie on a circle.
III. Prove that every triangle has an inscribed circle and that the center of that circle is the intersection point of all three angle bisectors of angles of the triangle.
IV. Prove that every triangle has a circumscribed circle and that the center of that circle is the intersection point of all three perpendicular bisectors of sides of the triangle.
Homework solutions.
October 3 Lecture notes Some of the 15 problems in this worksheet are simple and some could be a bit tricky. Choose and solve 6 problems from the worksheet. (So different students can solve different subsets). You are welcome to try all problems but you are not allowed to submit more than 6. All of them will be graded. Quiz on September 28. Review of Euclidean geometry on October 3. Midterm on October 5. Solutions for midterm I.
October 12 Chapter 3 (3.1-3.5). 3.2.4, 3.2.5, 3.2.6, 3.4.1, 3.4.2, 3.4.3, 3.5.1, 3.5.2
I. A quadrilateral ABCD has vertices given by intersection points of lines y=kx+b, y=kx-b, y=mx+b and y=mx-b. Find coordinates of the intersection point of diagonals of the quadrilateral ABCD.
II. The line y=kx+k+1 (with k>0) intersects coordinate axes at points A and B. Find the smallest possible area of the triangle ABO (where O is the origin).
Some homework solutions.
October 19 3.6, 3.7, 3.8 and lecture notes. 3.6.1, 3.6.2, 3.6.3, 3.7.1, 3.7.2, 3.7.3
I. Prove that every isometry of the plane takes circles to circles.
II. Suppose circles S1 and S2 of the same radius R intersect at points A and B. Let M and N be intersection points of S1 and S2 with the perpendicular bisector of the segment AB. Suppose, in addition, that M and N lie on the same side of the line AB. Prove: AB2+MN2=4R2.
III. Suppose a quadrilateral has an axis of symmetry. Show that either this quadrilateral is a trapezoid or the axis of symmetry is a diagonal of the quadrilateral.
IV. Construct an equilateral triangle with vertices on three given parallel lines.
Some homework solutions.
October 26 Chapter 4 + lecture notes. 4.1.3, 4.1.4, 4.3.3, 4.3.5
I. Let points A1, B1, C1 be midpoints of sides BC, AC and AB of a triangle ABC. Let O be an arbitrary point. Prove equality of vectors OA1+ OB1+ OC1= OA+ OB+ OC.
II. Let ABCD be a parallelogram and take an arbitrary point M. Draw lines through points A, B, C, D parallel to lines MC, MD, MA and MB, respectively. Show that these lines all pass through the same point.
III. In a quadrilateral ABCD, consider segments PQ and RS connecting midpoints of opposite sides of ABCD. Show that PQ and RS intersect and the point of intersection X divides each of them in half.
IV. Continuing with the previous problem, let Y and Z be midpoints of diagonals of ABCD. Prove that points X, Y, Z are collinear.
V. Use vectors and inner product to show that diagonals of a parallelogram are perpendicular if and only if it is a rhombus.
VI. Prove that lines AB and CD are perpendicular if and only if AC2−AD2=BC2−BD2.
Quiz on October 24.
Some homework solutions.
November 2 7.1, 7.2 and lecture notes. 4.5.3, 7.1.1, 7.2.3, 7.2.6.
An affine transformation is a composition of a linear transformation followed by a translation (by a vector). In problems I and II, show that given isometries are affine transformations by expressing them in the form u → Lu+c, where L is a 2x2 matrix of a linear transformation and c is a vector of a translation.
I. Rotation by 30 degrees around the point (1,1).
II. Reflection in the line 2x+3y=6.
III. Let ABC and A'B'C' be arbitrary triangles given by triples of non-collinear points. Show that there exists a unique affine transformation which takes ABC to A'B'C'. Thus in affine geometry all triangles are "congruent"!
IV. Show that every affine transformation T take the center of mass of a system of points P1, ..., Pn (with masses m1, ..., mn) to the center of mass of points T(P1), ..., T(Pn) (with masses m1, ..., mn).
V. Let AA' be an angle bisector of a triangle ABC (with A' on the side BC). Show that A'B/A'C=AB/AC. Use this and Ceva's theorem to conclude that three angle bisectors intersect in one point.
VI. Let ABC be an arbitrary triangle and draw two lines through each vertex which cut an opposite side in three equal parts. These six lines form a hexagon. Show that diagonals of this hexagon connecting pairs of opposite vertices intersect in one point.
Lecture slides.
Homework solutions.
November 9 No reading assignment No HW due but check out these midterm II sample problems. Review of analytic geometry on November 7. Midterm on November 9. Solutions for midterm II.
November 16 5.1-5.3 and lecture notes. 5.1.1, 5.1.2, 5.1.3, 5.2.1, 5.2.2
I. Let R(a) be a matrix of rotation by ao around the origin. Explain geometrically why R(a+b) is equal to R(a)R(b) and R(2a) is equal to R(a)R(a). Next compute this matrix produce explicitly. One of the trig identities obtained this way is sin(2a)=2sin(a)cos(a). What are the other three identities?
II. Recall that complex numbers can be drawn on the plane: a number x+iy is represented by the point (x,y). Show that addition of complex numbers corresponds to addition of vectors.
III. Complex numbers are multiplied as follows: (x+iy)(z+iw)=(xz-yw)+i(yz+xw). Writing down x=rcos(a) and y=rsin(a) in polar coordinates and similarly for z and w, use part I to explain geometric meaning of multiplying complex numbers.
IV. Let G be a finite group of isometries of the plane which contains n reflections. Show that G also contains n rotations and that angles between consecutive reflection lines are equal. This finishes the proof of Leonardo's theorem.
V. Let G be a group of isometries of the plane which only contains translations by an integer multiple of a fixed vector u. Show that, in addition to these translations, G can only contain reflections and glide reflections in a unique line L parallel to u, reflections in lines perpendicular to u, and rotations by 180o around points which all lie on the same line L.
Lecture slides - I.
Lecture slides -II.
Homework solutions.
November 30 5.4-5.9 and lecture notes. 5.3.1, 5.3.2, 5.4.1, 5.4.2, 5.4.3,
I. In Desargues', Pappus' and Pascal's theorems, one needs to show that three points are collinear. Explain why we can assume that two of these points lie on the horizon. State carefully theorems in Euclidean geometry obtained this way. They should end like this: if the lines l and l' are parallel and the lines m and m' are parallel then the lines n and n' are also parallel.
II. Use problem I to prove the Pappus' theorem.
III. Use "moving the line to infinity" trick from problem I to prove the following theorem. Consider all possible quadrilaterals ABCD such that its sides AB and CD lie on given lines L1 and L2 while sides BC and AD intersect in a given point P. Show that there exists a line L passing through the intersection point of l1 and l2 such that the intersection point of diagonals AC and BD always lies on L.
IV. The figure shows a light source, a kite, the line where the plane of the kite intersects the plane of the ground, and the shadow of one corner of the kite. Draw the remainder of the shadow.
V. Consider the set of "points" of RP2 with homogeneous coordinates: (1:-1:1), (2:-1:1), (3:-1:1), (4:-1:1), ..., (n:-1:1), ... (a) Find the "line" of RP2 which contains all these points. (b) These "points" converge to what "point" as n goes to infinity?
Homework solutions.
December 12 6.1, 6.2, 6.3, 7.3 and lecture notes. 5.6.3, 5.6.4, 5.7.1, 5.7.2, 5.7.3, 5.8.3, 6.1.1, 6.1.2, 6.3.1
I. Show that a projective transformation of RP2 which preserves the line at infinity is an affine transformation of R2.
II. Suppose A,B,C,D are collinear points and T is a projective transformation of RP2. Show that T(A), T(B), T(C), T(D) are also collinear and their cross-ratio is the same as the cross-ratio of A,B,C,D.
III. Suppose a projective transformation T of RP2 sends the circle x2+y2=1 to itself and preserves its center. (a) Show that T is an affine transformation (Hint: see what happens to tangent lines to the circle and use problem I). (b) Show that T is an isometry.
Final Exam Review on December 12
Notes of all lectures taken by Gabrielle Koch (74 pages).
Last lecture (6 pages).
Last homework solutions.