Class Log

Thursday 4/30/15: Review.

Wednesday 4/29/15: Classification of orientation preserving isometries of the hyperbolic plane H in terms of fixed points: A hyperbolic rotation has one fixed point in H, a horolation has no fixed points in H and one fixed point on the boundary of H (the x-axis), a hyperbolic translation has no fixed points in H and two fixed points on the boundary of H.

Monday 4/27/15: Classification of compositions of 2 hyperbolic reflections in hyperbolic lines L₁ and L₂. Case (1): L₁ and L₂ intersect in H, then the composite is a hyperbolic rotation about the intersection point through twice the angle between the lines. Case (2): L₁ and L₂ intersect on the boundary of H (the x-axis), then the composite T is called a horolation, and there is an isometry f such that U=fTf^-1 is a Euclidean translation U(z)=z+b parallel to the x-axis. Case (3) L₁ and L₂ do not intersect in H or on its boundary, then the composite is called a hyperbolic translation and there is an isometry f such that U=fTf^-1 is a Euclidean scaling U(z)=az (where a is real and positive).

Friday 4/24/15: Every hyperbolic isometry is the composite of at most 3 reflections.

Wednesday 4/22/15: Hyperbolic reflections: the reflection in a hyperbolic line given by a vertical line is the ordinary Euclidean reflection, the reflection in a hyperbolic line given by a semicircle is inversion in the semicircle. Algebraic formulas for all hyperbolic isometries: a hyperbolic isometry f is given by either (1) f(z)=(az+b)/(cz+d) where a,b,c,d are real and ad-bc>0, or (2) f=gh where h is reflection in the y-axis and g is as in (1). Case (1) are the orientation preserving isometries and case (2) are the orientation reversing isometries.

Monday 4/20/15: No class (Patriots' day).

Friday 4/17/15: The area of a hyperbolic triangle T equals pi minus the sum of the angles. Explanation of this and related facts in the plane and on the sphere in terms of the Gauss Bonnet theorem. Description of a Gaussian curvature of a surface at a point in terms of the circumference of small circles drawn centered at the point. Reflections in the hyperbolic plane.

Wednesday 4/15/15: Hyperbolic lines are defined as vertical lines or semicircles with center on the x-axis in H (these give the shortest paths between points as shown earlier). There is a unique hyperbolic line between two points P and Q in H. There is a unique hyperbolic line through a point P in H with tangent direction v (a vector in R²) at the point P. However, Euclid's parallel postulate fails in the hyperbolic plane: if L is a hyperbolic line and P is a point not on L there are many hyperbolic lines M passing through P and not meeting L.

Monday 4/13/15: The Mobius transformation f(z)=(az+b)/(cz+d) for a,b,c,d real and ad-bc > 0 defines an isometry of H. The shortest path between two points P and Q in H, not lying on a vertical line, is the semicircle with center on the x-axis passing through P and Q.

Friday 4/10/15: Hyperbolic geometry. The upper half plane model H for the hyperbolic plane. Hyperbolic length of curves γ : [a,b] -> H, γ(t)=x(t)+iy(t) defined as the integral of (√(x'(t)^2+y'(t)^2))/y(t) from a to b. The shortest path in H between two points P=(a,b) and Q=(a,c) on a vertical line is the segment of the line connecting the points, and has hyperbolic length ln(c/b) for b < c.

Wednesday 4/8/15: Given 3 distinct points a,b,c in the extended complex plane and another 3 distinct points a',b',c' there is a unique Mobius transformation f such that f(a)=a', f(b)=b', and f(c)=c'. The cross ratio of 4 points a,b,c,d is defined by CR(a,b,c,d)=((c-a)/(c-b))/((d-a)/(d-b)). The cross ratio is preserved by Mobius transformations. The cross ratio is a real number precisely when the 4 points lie on a circle or line.

Monday 4/6/15: Mobius transformations preserve angles and send circles and lines to circles and lines. The image f(C) of a circle or line C under a Mobius transformation f is a line precisely when f sends some point of C to ∞.

Friday 4/3/15: Linear fractional transformations of the extended complex plane CU{∞} (also known as Mobius transformations). Examples: f(z)=z+b (translation, f(z)=az (rotation and scaling), f(z)=1/z=z̅/|z|². Inversion g(z)=z/|z|^2 in the circle with center the origin and radius 1. The Mobius transformation f(z)=1/z is induced by rotation of the sphere about the x-axis through angle π via stereographic projection.

Wednesday 4/1/15: If T is the rotation of the sphere defined by a quaternion q=a+bi+cj+dk then, under stereographic projection, T corresponds to the linear fractional transformation of the (extended) complex plane CU{∞} given by f(w)=((a+di)w+(-c+bi))/((c+bi)w+(a-di)).

Monday 3/30/15: Examples of computation of composition of rotations in R³ using quaternions. The group G of rotations fixing the origin in R³ can be identified with the group of unit quaternions q modulo the equivalence relation given by identifying q and -q, by sending q to the rotation T given by T(x)=qxq̅.

Friday 3/27/15: Quaternions and rotations in R³. If q is a unit quaternion (i.e. |q|=1) we can write q=cos(t)+sin(t)v, where v is a unit vector in R³. Then the formula T(x)=qxq̅ defines the isometry of R³ given by rotation with axis the line through the origin in the direction of v, through angle 2t radians counterclockwise. If T₁ and T₂ are the rotations given by quaternions q₁ and q₂, then the composite T₂οT₁ is given by the product q₂q₁.

Wednesday 3/25/15: Basic properties of the quaternions: Multiplication of quaternions is associative (i.e. (AB)C=A(BC) ) but NOT commutative (AB ≠ BA in general). We define the length of q=a+bi+cj+dk as |q|=√a²+b²+c²+d² and the conjugate of q as q̅=a-bi-cj-dk. We have qq̅=q̅q=|q|², q₁.q₂ = q₂.q₁, and |q₁q₂|=|q₁||q₂|.

Monday 3/23/15: Rotations in R² via multiplication of complex numbers. For complex numbers z,w we have |zw|=|z||w| and arg(zw)=arg(z)+arg(w). The formula e^it = cos t + i sin t. The quaternions. Multiplication law given by i²=j²=k²=-1 and ij=-ji=k, jk=-kj=i, ki=-ik=j (note cyclic symmetry). Relation to dot product and cross product of vectors in R³: if we write a quaternion q=a+bi+cj+dk as q=t+v where t=a is a real number and v=bi+cj+dk is a vector in R³, then q₁q₂=(t₁+v₁)(t₂+v₂)=(t₁t₂-v₁.v₂)+(t₁v₂+t₂v₁+v₁ x v₂).

Friday 3/13/15: Stereographic projection and rotations of the sphere. Warmup: 1-dimensional case. Rotation of the circle corresponds to a linear fractional transformation (LFT) f(t)=(at+b)/(ct+d) of the extended line RU{∞} under stereographic projection. Review of LFT's (see also Stillwell p.104).

Wednesday 3/11/15: Stereographic projection preserves angles. One proof using calculus (233) and another using 3d Euclidean geometry.

Monday 3/9/15: Stereographic projection sends circles on the sphere to circles and lines in the plane. (Proof using algebraic formula for F^-1.)

Friday 3/6/15: Algebraic formula for stereographic projection in the 2 dimensional case: F:S² \ {N} --> R², F(x,y,z)=(x,y)/(1-z), F^-1(u,v)=(2u,2v,u²+v²-1)/(u²+v²+1).

Wednesday 3/4/15: Stereographic projection. Algebraic formula for stereographic projection in the 1 dimensional case: F:S¹ \ {N} --> R, F(x,y)=x/(1-y), F^-1(t)=(2t,t²-1)/(t²+1).

Monday 3/2/15: Maps of portions of the sphere cannot exactly preserve distances. The Gall--Peters projection: geometric description, algebraic formulas for the projection and its inverse. The Gall--Peters projection preserves areas.

Friday 2/27/15: The isometries of the sphere are obtained by restricting isometries of R³ fixing the origin. Geometric description of the resulting isometries from the viewpoint of the sphere.

Wednesday 2/25/15: Examples continued: explicit change of basis calculation. Isometries of the sphere.

Monday 2/23/15: Examples computed using method developed on Friday 2/20.

Friday 2/20/15: Geometric description of isometries T of R³ fixing the origin given algebraically using determinant and trace. If T(x)=Ax then det A = +1 if T is a rotation and det A = -1 if T is a reflection or rotary reflection. We have trace(A)=1+2cos(θ) for a rotation through angle θ, and trace(A)=-1+2cos(θ) for a rotary reflection with angle θ (note reflection can be regarded as a rotary reflection with angle θ=0). This allows us to solve for θ (up to a sign). Finally we can determine the direction of the axis of rotation by solving the linear equation Ax=x in the case of a rotation and Ax=-x in the case of a rotary reflection.

Wednesday 2/18/15: Classification of isometries of R² using the algebraic description (continued).

Monday 2/16/15: An isometry T of Rⁿ is given by T(x)=Ax+b where A is an n x n orthogonal matrix and b is a vector. Classification of isometries of R² using the algebraic description.

Friday 2/13/15: Classification of orthogonal matrices for n=2 and 3, and geometric interpretation.

Wednesday 2/11/15: Isometries of Rⁿ. An isometry T fixing the origin is given by T(x)=Ax where A is an n x n orthogonal matrix.

Monday 2/9/15: Snow day.

Friday 2/6/15: The spherical triangle inequality. Proof that spherical lines (great circles) give shortest paths between points on the sphere. Examples of isometries of R²: translation, rotation, reflection, and glide reflection.

Wednesday 2/4/15: Spherical sine rule (continued): sin a / sin α = sin b / sin β = sin c / sin γ .

Monday 2/2/15: Snow day.

Friday 1/30/15: The spherical sine rule.

Wednesday 1/28/15: The spherical cosine rule. Review of cosine rule in R². Review of spherical coordinates from 233. For a spherical triangle ABC on a sphere of radius 1 with angles a,b,c and opposite side lengths α, β, γ we have cos(α)=cos(β)cos(γ)+sin(β)sin(γ)cos(a).

Monday 1/26/15: The sum of the angles of a spherical triangle ABC on a sphere of radius R=1 is given by a+b+c = pi + area(ABC).

Friday 1/23/15: The angle between two spherical lines is equal to the dihedral angle between the corresponding planes or equivalently the angle between their normal vectors. It can be computed using the dot product.

Wednesday 1/21/15. The sphere. A spherical line (or great circle) is the intersection of the sphere with a plane passing through its center. The shortest path between two points on the sphere is given by the spherical line through the points (stated without proof). Two spherical lines intersect in a pair of antipodal points. Given two points on the sphere, there is a unique spherical line passing through the points if the points are not antipodal. (If the points are antipodal, there are infinitely many spherical lines passing through the points.) Computation of the spherical distance between two points using the dot product.