Class Log

Lecture notes were taken by Gabrielle Koch. References are to the textbook "The four pillars of geometry" by J. Stillwell.

Wednesday 12/12/18. Classification of hyperbolic isometries: identity, hyperbolic reflections, hyperbolic rotations, horolations, hyperbolic translations, and hyperbolic glide reflections. Algebraic formula T(z)=(az+b)/(cz+d) for orientation preserving hyperbolic isometries (where a,b,c,d are real numbers and ad-bc>0). Lecture notes pdf.

Monday 12/10/18. For L a hyperbolic line, if L is a vertical line then hyperbolic reflection in L coincides with Euclidean reflection in L; if L is a semicircle then hyperbolic reflection in L is given by inversion in L. Every hyperbolic isometry is a composite of at most 3 reflections. Lecture notes pdf.

Friday 12/7/18. Hyperbolic lines give shortest paths. Inversion in the unit circle takes circles and lines to circles and lines (and the image of a circle or line C is a circle iff C contains the origin). Inversion preserves angles. Given a hyperbolic line L and a point P there is a unique hyperbolic line M through P perpendicular to L. Lecture notes pdf.

Wednesday 12/5/18. A hyperbolic line is a vertical half line or a semicircle with center on the x-axis in the upper half plane. For any two points P and Q in H there is a unique hyperbolic line through P and Q. Theorem: The shortest path from P to Q is the segment of the hyperbolic line connecting P and Q. The Theorem is proved by reducing to the case of vertical half lines considered earlier using hyperbolic isometries. A hyperbolic isometry is a transformation T:H->H that preserves hyperbolic distances, that is, dH(T(P),T(Q))=dH(P,Q) for all P,Q in H. Examples of hyperbolic isometries: translations parallel to the x axis T(x,y)=(x+a,y); scaling by a factor c > 0 T(x,y)=(cx,cy); reflection in the y-axis T(x,y)=(-x,y) (or reflection in any vertical line); inversion in the unit circle T(x,y)=(x,y)/(x2+y2). The inversion T in the unit circle has fixed locus the unit circle x2+y2=1 and interchanges the inside and outside of the unit circle. The composition T2 equals the identity. We will interpret T:H->H later as the hyperbolic reflection in the hyperbolic line given by the unit semicircle. Lecture notes pdf.

Monday 12/3/18. Hyperbolic geometry. The upper half plane model H. The (hyperbolic) length of a curve in H is defined as the integral of √(x'2+y'2)/y over the domain [a,b] of a parametrization (x(t),y(t)) of the curve. The (hyperbolic) distance dH(P,Q) is the length of the shortest path from P to Q. The shortest path between two points P=(c,a), Q=(c,b) in H lying on a vertical line (x=c) is the segment of the vertical line connecting them, and the distance dH(P,Q)=ln(b/a) (assuming WLOG a < b). Lecture notes pdf.

Friday 11/30/18. Stereographic projection sends spherical circles not passing through the north pole N to circles in the plane. Spherical lines (great circles) on the sphere correspond under stereographic projection to circles or lines in the plane which intersect the unit circle in two antipodal points. Lecture notes pdf.

Wednesday 11/28/18. Proof of formula describing distortion of lengths under stereographic projection. Stereographic projection preserves angles. Stereographic projection sends a spherical circle passing through the north pole N to the line in the plane given by the intersection of the plane containing the spherical circle with the xy-plane. Lecture notes pdf.

Monday 11/26/18. Algebraic formulas for stereographic projection and its inverse: F(x,y,z)=(x,y)/(1-z) and F-1(u,v)=(2u,2v,u2+v2-1)/(u2+v2+1). Relation between the length of a curve on S2 and the length of its image in R2 under stereographic projection via the formula √(x'2+y'2+z'2) = (2/(u2+v2+1))*√(u'2+v'2). Example: computation of length for a line of longitude (a spherical line through the north and south poles) in terms of its stereographic projection (a line throught the origin in the plane). Lecture notes pdf.

Friday 11/16/18. Isometries of S2: A composition of 3 reflections is a rotary reflection. Compositions of rotations. A composition of two rotations of S2 is another rotation; the axis and angle can be computed by drawing a spherical triangle (similar to the case of compositions of rotations in R2 discussed on 10/12/18). Isometries of S2 (or equivalently isometries of R3 fixing the origin) correspond to 3 x 3 orthogonal matrices, and the classification of isometries can be understood in terms of the eigenvalues of the matrix. In particular, the determinant of the matrix is +1 for the identity and rotations and -1 for reflections and rotary reflections. Stereographic projection. Lecture notes pdf.

Wednesday 11/14/18. Classification of isometries of R3 fixing the origin (or equivalently isometries of the sphere S2): identity, reflection, rotation, or rotary reflection. GPS theorem: A point P in S2 is uniquely determined by its distance from 3 points A,B,C not lying on a spherical line. (This uses a Lemma: the perpendicular bisector of a spherical line segment PQ is the locus of points that are equidistant from P and Q). Corollary: A spherical isometry T is determined by T(A),T(B),T(C) for 3 points A,B,C not lying on a spherical line. Theorem: (3 reflections theorem) A spherical isometry is a composite of at most 3 reflections. The classification follows based on the number of reflections --- 0: identity, 1: reflection, 2: rotation (more precisely, the axis of rotation is the intersection of the two planes of reflection and the angle of rotation is twice the dihedral angle between the planes, measured from the first plane to the second), 3: rotary reflection (proof to be completed on Friday). Lecture notes pdf.

Monday 11/12/18. No class (Veteran's day).

Friday 11/9/18. Isometries of R3 fixing the origin: given an algebraic formula T(x)=Ax for T (where A is a 3x3 matrix), we determine a geometric description. First, compute the fixed locus of T: it is either (1) R3 (2) a plane through the origin (3) a line through the origin or (4) the origin. In case (1) T is the identity. In case (2) T is reflection in the plane. In case (3) T is rotation about the line through angle t determined by 2 cos t + 1 = trace(A). In case (4) T is a rotary reflection: a reflection in a plane through the origin followed by a rotation about the line through the origin perpendicular to the plane. To find the axis of rotation solve T(x)=-x. The angle of rotation t is determined by 2 cos t -1 = trace(A). The plane of reflection is the plane through the origin perpendicular to the axis of rotation. Note that the trace of a matrix M is the sum of its diagonal entries, and trace(PMP-1)=trace(M) for an invertible matrix P (this was used to determine the angles of rotation above). Lecture notes pdf.

Wednesday 11/7/18. Algebraic formulas for isometries of R3: Rotations (continued). Lecture notes pdf.

Monday 11/5/18. Algebraic formulas for isometries of R3: Reflections and rotations. Lecture notes pdf.

Friday 11/2/18. The spherical sine rule (continued). Spherical isometries. An isometry of R3 fixing the origin induces a isometry of the sphere. Examples of isometries of the sphere: reflection in a plane through the origin, rotation about a line through the origin, rotary reflection (reflection in a plane throught the origin followed by a rotation about the line through the origin perpendicular to the plane). Every isometry of the sphere is either the identity, a reflection, a rotation, or a rotary reflection (proof later). Lecture notes pdf.

Wednesday 10/31/18. The shortest path on the sphere connecting two points P and Q is the shorter arc of the spherical line through P and Q. The spherical sine rule. Lecture notes pdf.

Monday 10/29/18. The area of a spherical triangle is strictly less that 2 pi, and can be arbitrarily close to 2 pi. The spherical triangle inequality. Lecture notes pdf.

Friday 10/26/18. Spherical cosine rule. Lecture notes pdf.

Wednesday 10/24/18. Proof of angle sum formula for spherical triangle. Review for midterm exam. Lecture notes pdf.

Monday 10/22/18. Spherical geometry. Computing the spherical line through two points using the cross product. Computing the angle between two spherical lines. The angle sum of a spherical triangle ABC equals pi plus the area of the triangle. Lecture notes pdf.

Friday 10/19/18. Spherical geometry. A spherical line is by definition the intersection of the sphere with a plane through its center (also called a great circle). The shortest path between two points on the sphere is given by the shorter arc of the spherical line through the two points (proof later). The angle between two spherical lines meeting at a point is by definition the angle between the tangent lines to the spherical lines at that point, equivalently, the dihedral angle between the corresponding planes, or the angle between the normal vectors to the planes. Lecture notes pdf.

Wednesday 10/17/18. The three reflections theorem: Any isometry of the plane is a composition of at most three reflections. Classification of isometries of the plane (the identity, translations, rotations, reflections, and glide reflections). Lecture notes pdf.

Monday 10/15/18. Two triangles are congruent iff there is an isometry sending one to the other (and moreover the isometry is uniquely determined). Lecture notes pdf.

Friday 10/12/18. Composition of two rotations. GPS theorem: Given 3 points A,B, and C in the plane which are not collinear, a point P in the plane is uniquely determined by its distances from A,B, and C. Corollary: An isometry T is uniquely determined by the images T(A), T(B), and T(C) of 3 non-collinear points A,B, and C. Lecture notes pdf.

Wednesday 10/10/18. Compositions of isometries. The composition of two reflections in parallel lines is a translation. The composition of two reflections in intersecting lines is a rotation about the intersection point of the lines. The composition of a rotation and a translation is a rotation through the same angle about a different point. Lecture notes pdf.

Tuesday 10/9/18. Algebraic formulas for rotations and reflections. Lecture notes pdf.

Friday 10/5/18. 3.6. Isometries of the plane. Examples: Translation, Rotation, Reflection, Glide reflection, Identity. Theorem: Every isometry is one of these types. Lecture notes pdf.

Wednesday 10/3/18. 3.4. Algebra of ruler and compass constructions. A length is constructible iff it can be obtained from 1 by addition, subtraction, multiplication, division, and square roots. Lecture notes pdf.

Monday 10/1/18. 3.1, 3.2, 3.3. Coordinates: Distance. Equation of circle. Equation of line. Slope. Lines are parallel iff slopes are equal. Equation of perpendicular bisector. Lecture notes pdf.

Friday 9/28/18. Center of mass and Orthocenter of a triangle. Lecture notes pdf.

Wednesday 9/26/18. 1.6. Gauss' theorem on constructibility of regular polygons. Lecture notes pdf.

Monday 9/24/18. 2.8. A length is constructible <=> it can be obtained from 1 by addition, subtraction, multiplication, division, and square roots. (We've already proved "<="; we will prove "=>" later using coordinates.) Construction of the regular pentagon. For a regular pentagon of side length 1, the diagonals have length (1+√5)/2, the "golden ratio". Lecture notes pdf.

Friday 9/21/18. 2.8. Sine, cosine, and tangent. Sine rule. Cosine rule. Construction of square roots with ruler and compass. Lecture notes pdf.

Wednesday 9/19/18. 1.4, 2.7. Parallel Pappus' and Desargues' theorems. Angles in a circle. Lecture notes pdf.

Monday 9/17/18. 1.5,1.4,1.3. Similar triangles. Multiplication and division of lengths via ruler and compass. Subdivision of an interval into n equal parts. Impossibility of subdivision of an angle into n equal parts for some values of n (e.g. n=3). Converse of Thales' theorem. Lecture notes pdf.

Friday 9/14/18. 2.3,2.4,2.5,1.3,2.6. Definition of area of polygon in terms of area of a rectangle and behaviour under subdivision. Formulas for area of parallelograms and triangles. Pythagoras' theorem via areas. Thales' theorem. Lecture notes pdf.

Wednesday 9/12/18. 2.1,2.2. If lines L and M cross a given line making interior angles a and b on one side, then L and M are parallel iff a+b=pi. Playfair's axiom: Givena line L and a point P not on L, there is a unique line M through P parallel to L. The angle sum of a triangle equals pi. Parallelograms: Opposite sides have equal lengths, and the diagonals bisect each other. Lecture notes pdf.

Monday 9/10/18. 1.3,2.1. Construction: Bisect a line segment --- the perpendicular bisector. Construction: perpendicular line to a given line passing through a given point. The parallel axiom. Brief overview of hyperbolic geometry (see also Chapter 8). Lecture notes pdf.

Friday 9/7/18. 2.2,1.3. Congruence of triangles. Congruence criteria (SAS,ASA,SSS). Construction: Bisect an angle. Isosceles triangles (two equal angles <=> two equal side lengths). Lecture notes pdf.

Wednesday 9/5/18. 1.1,1.2. Euclidean geometry. Ruler and compass constructions. Motivating problems (constructible lengths, construction of regular polygons). Lecture notes pdf.