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Universal Hyperbolic Geometry Summary of N Universal Hyperbolic Geometry Summary of N. J. Wildberger's online lecture series http://www.youtube.com/watch?v=EvP8VtyhzXs Martin Pitt ([email protected]) 1 Notation Conics: intersections of a (double-sided) cone with a plane: point, circle, elipse, parabola, hyperbola (consists of two separate parts) Null conic: Single fixed given circle or other conic; symbol C Points: points have lower-case variables (a, p); points on C have lower-case greek variables (α, η); the product of two points ab is the line that goes through a and b algebraic coordinate notation: a = [x] (one-dimensional), a = [x; y] (two-dimensional) Lines: lines have upper-case variables (A, B); tangents on C have upper-case greek variables (Γ); the product of two lines AB is their intersection, i. e. a point algebraic coordinate notation: A := (a : b : c) representing ax + by = c 1 2 Polarity http://www.youtube.com/watch?v=AjVM5Q-pvjw Pole: The pole a of a line A is the inversion point of A's closest point to C . a = A? Polar: The line A is the polar of the pole point a. A = a? Construction of polar of a: • Find null points α, β; γ; δ on C so that (αβ)(γδ) = a (i. e. select two secants of C which pass through a) • Define b := (αδ)(βγ) and c := (αγ)(βδ), i. e. the two inter- sections of the other diagonals of the quadrangle αβγδ • Then A = a? = bc. Also, b? = ac and c? = ab (three- fold symmetry of polars and poles produced by the three intersections of diagonals) Construction of pole of A: Choose any two points b; c 2 A. Then a = (b?c?). Polar Independence Theorem: The polar A = a? does not depend on the choice of lines through a, i. e. the choice of α, β; γ; δ. Polar Duality Theorem: For any two points a and b: a 2 b? , b 2 a? Construction of polar of null point γ: • Chose any two secants A, B that pass through γ • Construct their poles A?, B? • Polar Γ = γ? is the tangent (A?B?) 2 3 Harmonic conjugates http://www.youtube.com/watch?v=t7oXlrcPBb4 Four collinear points a; b; c; d are a harmonic range if a and c are harmonic conjugates to b, d, i. e. if they divide bc internally and externally by the same ratio: ab~ cb~ = − ad~ cd~ In that case, b and d are then harmonic conjugates to a and c: ba~ da~ = − bc~ dc~ Note: ab~ measures displacements on a linear scale, not distances. (affine geometry only, no units) Harmonic Ranges Theorem: The image of a harmonic range un- der a projection from a point onto another line is another harmonic range. ! harmonic ranges are not dependent on the choice of a scale (affine geometry), but are really part of projective ge- ometry If a; b; c; d are a harmonic range, and p a point not on the line abcd, then the four lines ap; bp; cp; dp are a harmonic pencil. By the previous theorem, the intersections of any line through these four lines are harmonic ranges. Harmonic Pole/polar Theorem: For a point a and any secant through a that meets C at two points β, γ there is a point c = (βγ)a?. The points a; β; c; δ are a harmonic range. 3 Harmonic Bisectors Theorem: If C, D are the two bisectors of two non-parallel lines A, B, then A; C; B; D is a harmonic pencil. Harmonic Vectors Theorem: If ~a;~b are linearly independent vec- tors, then the lines spanned by ~a;~b are harmonic conjugates to the lines spanned by ~a +~b;~a −~b Harmonic Quadrangle Theorem: If pqrs is a quadrangle, find the two intersections a = (ps)(qr) and c = (pq)(rs) and then b = (pr)(ac), d = (sq)(ac). Then a; b; c; d are a harmonic range. 4 Cross ratios http://www.youtube.com/watch?v=JJbh0iJ1Agc Cross ratio of four collinear points a; b; c; d: ~ac bc~ a − c b − c R(a; b : c; d) := = = = ad~ bd~ a − d b − d a; b; c; d are a harmonic range if R(a; b : c; d) = −1. Cross-ratio Transformation Theorem: If R(a; b : c; d) = λ then 1 R(b; a : c; d) = R(a; b : d; c) = λ and R(a; c : b; d) = R(d; b : c; a) = 1 − λ Four points determine 24 possible cross ratios, but only 6 1 1 will generally be different (permutations of λ, λ , 1−λ, 1−λ , λ−1 λ λ , λ−1 ). 4 Cross-ratio Theorem: The cross ratio is invariant under projection from a point p to another line L: With a0 := (pa)L, b0 := (pb)L, c0 := (pc)L, d0 := (pd)L: R(a; b : c; d) = R(a0; b0 : c0; d0) Therefore the cross-ratio can be transferred to lines: R(pa; pb; pc; pd) := R(a; b; c; d) Chasles Theorem: If α, β; γ; δ are fixed points on a null conic C and η a fifth point on C , then R(αη; βη : γη; δη) is independent of the choice of η. 5 Introduction to hyperbolic geometry http://www.youtube.com/watch?v=UXQas-B5ObQ 5.1 Definitions Hyperbolic geometry : Geometry on a projective plane and a sin- gle fixed given circle C ; only tool is a straightedge Duality: Terminology of pole and polar get replaced by simply duality: the dual of a point a is the line a?, and vice versa; points and lines are completely dual concepts Line perpendicularity: A ? B , B? 2 A , A? 2 B (A is p. to B if A passes through the dual of B) Point perpendicularity: a ? b , a 2 b? , b 2 a? (a is p. to b if a lies on the dual of b) 5 Quadrance between points: q(a1; a2) := R(a1; b2 : a2; b1) with ? ? b1 := (a1a2)a1 and b2 := (a1a2)a2 Spread between lines: S(A1;A2) := R(A1;B2 : A2;B1) with B1 := ? ? (A1A2)A1 and B2 := (A1A2)A2 5.2 Basic theorems ? ? Quadrance{Spread duality: q(a1; a2) = S(a1 ; a2 ) Pythagoras: If a1a3 ? a2a3, and q1 = q(a2; a3), q2 = q(a1; a3), q3 = q(a1; a2): q3 = q1 + q2 − q1q2 If A1A3 ? A2A3, and S1 = S(A2;A3), S2 = S(A1;A3), S3 = S(A1;A2): S3 = S1 + S2 − S1S2 Triple Spread/Quadrance: 2 2 2 2 If a1; a2; a3 are collinear: (q1 + q2 + q3) = 2(q1 + q2 + q3 ) + 4q1q2q3 2 2 2 If A1;A2;A3 are concurrent: (S1 + S2 + S3) = 2(S1 + S2 + 2 S3 ) + 4S1S2S3 Spread Law: For a triangle: S1 = S2 = S3 q1 q2 q3 2 Cross law: (q1q2S3 −(q1 +q2 +q3)+2) = 4(1−q1)(1−q2)(1−q3) Relation to Beltrami-Klein model: 2 For points a1; a2 inside C : q(a1; a2) = − sinh d(a1; a2) 2 For lines A1;A2 inside C : S(A1;A2) = sin \(A1;A2) 6 6 Calculations with Cartesian coordinates http://www.youtube.com/watch?v=YDGUnGGkaTs, http://www.youtube. com/watch?v=XomxP2pxYnw ? Point/line duality: a = [x0; y0] , a = (x0 : y0 : 1) h 2 i Point on null circle: Parameterized by t 2 : e(t) := 1−t ; 2t Q 1+t2 1+t2 This reaches any point except [−1; 0], which corresponds to e(1) Line through points: [x1; y1][x2; y2] = (y1 − y2 : x2 − x1 : x2y1 − x1y2) Line through null points: e(t1)e(t2) = (1 − t1t2 : t1 + t2 : 1 + t1t2) Line relation to C : The line (a : b : c) • is tangent to C , a2 + b2 = c2 • meets C at two points , a2 + b2 − c2 is a square • does not meet C , a2 +b2 −c2 is a non-square (negative or no rational root) Quadrance: If a1 = [x1; y1] and a2 = [x2; y2], then q(a1; a2) = 2 (x1x2+y1y2−1) 1 − 2 2 2 2 (x1+y1 −1)(x2+y2 −1) Spread: If L1 = (l1 : m1 : n1) and L2 = (l2 : m2 : n2), then 2 (l1l2+m1m2−n1n2) S(L1;L2) = 1 − 2 2 2 2 2 2 (l1+m1−n1)(l2+m2−n2) Point perpendicularity: [x1; y1] ? [x2; y2] , x1x2 + y1y2 − 1 = 0 Line perpendicularity: (l1 : m1 : n1) ? (l2 : m2 : n2) , l1l2 + m1m2 − n1n2 = 0 7 7 Projective homogeneous coordinates http://www.youtube.com/watch?v=gzalbNLcwGI, http://www.youtube. com/watch?v=KhxVy75NetE In affine geometry, lines from a 3D object through the projec- tive plane are parallel, and projections maintain linear spacing. In projective geometry, they meet in two points (\observer" or "horizon"), and projections distort linear spacing. In one-dimensional geometry, the projective vector space is 2 an one-dimensional subspace of two-dimensional V , i. e. a line through [0; 0]). This is specified by a proportion [x : y] (projective point); canonically, either [x : 1], or [x : 0] for the special case of the subspace being the x axis (representing \point at 1") 2 In two-dimensional geometry, the projective plane P is de- 3 scribed with a three-dimensional vector space V , projective points a = [x : y : z] (lines through the origin) and projec- tive lines A = (l : m : n) , lx + my − nz = 0 (planes through the origin). In particular, we introduce a viewing plane yz=1. Then any projective point [x : y : z]; z 6= 0 meets the viewing plane at x y [ z : z : 1]. Any projective point [x : y : 0] corresponds to the two-dimensional equivalent [x : y], representing the \point at 1 2 in the direction [x : y]".
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