Strophoids, a family of cubic curves with remarkable properties Hellmuth Stachel [email protected] — http://www.geometrie.tuwien.ac.at/stachel 6th International Conference on Engineering Graphics and Design University Transilvania, June 11–13, Brasov/Romania Table of contents 1. Definition of Strophoids 2. Associated Points 3. Strophoids as a Geometric Locus June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 1/29 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). June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 2/29 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 . June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 3/29 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. June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 4/29 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 June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 5/29 1. Definition of Strophoids ′ m F P The product of the polarity and Fp e the inversion in K is the pedal T transformation t 7→ T w.r.t. N. asymptote t Polar to H is the parabola P. H G′ T g Theorem: The strophoid S is F y G the pedal curve of the parabola P with respect to N. S The parabola’s directrix m is K parallel to the asymptote of S. F is the midpoint between N and N x the parabola’s focus Fp. June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 6/29 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: ′ N F Q and Q are called x t associated points of S. m asymptote ′ g Q and Q are associated iff the 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. June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 7/29 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. June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 8/29 replacements 2. Associated Points X′β′ =Xβ β • β has a center Z Z X such that X,Xβ,Z are ′ ′ collinear. Y =Y β t1 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. June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 9/29 2. Associated Points P S G T On the equicevian cubic , 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′ June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 10/29 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. June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 11/29 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 June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 12/29 2. Associated Points P m F g asymptote X1 Theorem: Given the non- ′ A S collinear points A, A and N, X the locus of points X such t2 that the line XN bisects the ′ ′ N A angle between XA and XA , is a strophoid with node N t1 and associated points A, A′. The respectively second angle bisectors are tangent to the parabola P. June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 13/29 3. Strophoids as a Geometric Locus P m C2 Q′ C1 M C3 Theorem: The strophoid S is the Q locus of focal points (Q,Q′) of conics A′ N which contact line AN at A and A line A′N at A′. a′ a The axes of these conics are tangent F to the parabola P S N June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 14/29 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 . T G1 t T α α On the other hand, the F1 F2 tangent t at T bisects C 2a t the angle between T F1 and T F2. g1 June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 15/29 3. Strophoids as a Geometric Locus S N T The points of contact of T ′ tangents drawn from a fixed point N to confocal F conics as well as the A A′ foot points of normals through N lie on a strophoid. P June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 16/29 3. Strophoids as a Geometric Locus z C The curve V of intersection between the sphere (radius 2r) and the vertical right cylinder V (radius r) is called Viviani’s F c window. N c X X c Central projections with V y N c center C ∈V and a horizontal x image planes map V onto a strophoid. F June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 17/29 3. Strophoids as a Geometric Locus ′′ V Z ′′ V′′′ X X′′′ X N′′ N′′′ V λ N′ V′ ′ 45◦ X x y N λ V lies also on a cone of revolution and on a torus. Points of V have equal geographic longitude and latitude. June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 18/29 3. Strophoids as a Geometric Locus B A′ Two particular examples of flexible octahedra C′ C′ where two faces are omitted. Both have an B′ A axial symmetry (types 1 B and 2) A C A′ Below: Nets of the two octahedra. B′ C ′ C ′ ′ ′ A A A B C A B ′ ′ ′ ′ C A B C ′ B ′ C A C June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 19/29 3. Strophoids as a Geometric Locus B′ t2 According to R. Bricard there are C′ t1 3 types of flexible octahedra (four- ′ A M sided double-pyramids).