Strophoids – cubic curves with remarkable properties Hellmuth Stachel [email protected] — http://www.geometrie.tuwien.ac.at/stachel Slovak–Czech Conference on Geometry and Graphics = 24th Symposium on Computer Geometry SCG’2015 = 35th Conference on Geometry and Graphics, Sept. 14–18, Terchová, Slovakia Table of contents 1. Definition of Strophoids 2. Associated Points 3. Strophoids as a Geometric Locus September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 1/34 replacements 1. Definition of Strophoids F ′ Definition: An irreducible cubic is called circular if it passes through the absolute circle-points. asymptote A circular cubic is called strophoid if it has a double point (= node) with ′ G orthogonal tangents. F g A strophoid without an axis of y G symmetry is called oblique, other- wise right. S S : (x 2 + y 2)(ax + by) − xy =0 with a, b ∈ R, (a, b) 6= (0, 0). In x N fact, S intersects the line at infinity at (0 : 1 : ±i) and (0 : b : −a). September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 2/34 1. Definition of Strophoids F ′ The line through N with inclination angle ϕ intersects S in the point asymptote 2 2 ′ sϕ c ϕ s ϕ cϕ G X = , . a cϕ + b sϕ a cϕ + b sϕ F g y G X This yields a parametrization of S. S ϕ = ±45◦ gives the points G, G′. ϕ The tangents at the absolute circle- N x points intersect in the focus F . September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 3/34 1. Definition of Strophoids F ′ The polar equation of S is 1 Te S : r = . a + b asymptote sin ϕ cos ϕ H G′ The inversion in the circle K transforms S into the curve H T g F y with the polar equation G a b H: r = + . S sin ϕ cos ϕ K This is an equilateral hyperbola which satisfies N x H: (x − b)(y − a) = ab. September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 4/34 1. Definition of Strophoids G H has two axes of P2 symmetry =⇒ the inverse curve S is self-invers w.r.t. two circles through N N with centers G, G′. F m S is the envelope of circles centered on S confocal parabolas ′ G P1 P1 and P2. P September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 5/34 ′ 1. Definition of Strophoids F ′ P m The product of the polarity and Fp the inversion in K is the pedal Te transformation t 7→ T w.r.t. N. asymptote t Polar to H is the parabola P. H G′ Theorem: The strophoid S is T g F y the pedal curve of the parabola G P with respect to N. S The parabola’s directrix m is K parallel to the asymptote of S. F (on the tangent at the vertex N x of P) is the midpoint between N and the parabola’s focus Fp. September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 6/34 1. Definition of Strophoids W. Wunderlich: Ebene Kinematik, p. 115 as a particular pedal curve of a parabola, the strophoid is a particular trajectory during a blau symmetric rolling of parabolas September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 7/34 1. Definition of Strophoids P Tangents t of the parabola P G y T intersect S beside the pedal point T in two real or conjugate complex Q points Q and Q′. F1 Definition: Q and Q′ are called N F x t associated points of S. m asymptote Q and Q′ are associated iff the g ′ F lines QN and Q′N are harmonic ′ G S Q′ w.r.t. the tangents at A. For given t, the points Q and Q′ lie on a circle centered on g. September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 8/34 replacements 2. Associated Points Projective properties of X′β′ =Xβ β Z cubics with a node: X There is a 1-1 cor- ′ ′ t1 Y =Y β S respondance between S N and lines through N, X′ except N corresponds to t2 t1 and t2. β′ Y β =Y The involution α which Z′ fixes t1, t2 determines ′ Xβ′ =X′β pairs X,X of associated points. Involutions which exchange t1 and t2 determine involutions β on S with N 7→ N and several properties, e.g., there exists an ‘associated’ involution β′ = α ◦ β = β ◦ α. September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 9/34 replacements 2. Associated Points X′β′ =Xβ β • β has a center Z Z X such that X,Xβ,Z are collinear. ′ ′ t1 Y =Y β S • The centers Z of β and N Z′ of β′ are associated. X′ ′ ′ t2 • The lines Z X, Z Xβ β′ Y β =Y correspond in an involu- ′ Z′ tion which fixes Z N and the line through the fixed Xβ′ =X′β points Y,Y ′ of β. • For associated points, the diagonal points XY ∩ X′Y ′ and XY ′ ∩ X′Y are again on S. The tangents at corresponding points intersect on S. September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 10/34 2. Associated Points Theorem: A B On each irreducible cubic S a R commutative group can be defined with an arbitrary chosen point O as A+B neutral element. S O Conversely, point B is uniquely defined by A and A + B. Addition of points September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 11/34 2. Associated Points R Theorem: On each irreducible cubic S a B A commutative group can be defined P B+C with an arbitrary chosen point O as neutral element. C A+B Q A+B+C At cubics with a node the group is O isomorphic to (R \ {0}, · ). Pairs S of associated points differ only by their sign. N corresponds to zero. associative law September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 12/34 2. Associated Points P G T On the equicevian cubic S, the y following pairs of points are Q associated: F1 • Q,Q′, N F x t • the absolute circle-points, m asymptote • The focal point F and the point ′ g F at infinity, F ′ ′ ′ • G, G on the line g ⊥ NF . G S Q′ September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 13/34 2. Associated Points Theorem: • For each pair (Q,Q′) of associated points, the lines NQ, NQ′ are symmetric w.r.t. the bisectors t1, t2 of <) BNC . • The midpoint of associated points Q,Q′ lies on the median m = NF ′. • The tangents of S at associated points meet each other at the point T ′ ∈ S associated to the pedal point T on t = QQ′. • For each point P ∈ S , the lines PQ and PQ′ are symmetric w.r.t. PN. Other consequences: For each pair (Q,Q′) of associated points (as Laguerre-points) the represented two complex conjugate points are again associated points of S. September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 14/34 2. Associated Points The right-angle involution at N induces an involution β on S. P G Corresponding points P, Pβ are collinear mit F ; their midpoint is on m. N F m Given m, N ∈ m and F∈ / m, asymptote the strophoid S is the locus of F ′ intersection points between circles g through N and centered on m with G′ P β diameter lines through F . September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 15/34 2. Associated Points X We recall: Theorem: Given three e S N aligned points A, A′ and N, A N A′ the locus of points X such that the line XN bisects the ′ S angle between XA and XA , is the Apollonian circle. The second angle bisector passes through the point N harmonic to N w.r.t. A, A′. e September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 16/34 2. Associated Points P m Theorem: Given the non- F g asymptote ′ X1 collinear points A, A and N, the locus of points X such A S that the line XN bisects the ′ X angle between XA and XA , t2 is a strophoid with node N ′ ′ N A and associated points A, A . This holds also when A is at t1 infinity. The respectively second angle bisectors are tangent to the parabola P. When N is at infinity, then the locus is an equilateral hyperbola. September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 17/34 3. Strophoids as a Geometric Locus P m C2 Q′ C1 Theorem: The strophoid S is the locus of focal points (Q,Q′) of conics M C3 Q N which contact line AN at A and line A′N at A′. A′ A The axes of these conics are tangent ′ a a to the negative pedal curve, the F parabola P. Therefore, the real focal points are associated — as well as the complex conjugate point. S N September 17, 2015: Slovak–Czech Conference on Geometry and Graphics, Terchová/Slovakia 18/34 3. Strophoids as a Geometric Locus h Let the tangents to C at ′ Q T and T intersect at Q. Then G1 α = <) T F2Q = <) QF2T .