Projective equivalences of elliptic and hyperelliptic planar curves Juan G. Alc´ azar , Carlos Hermoso Universidad de Alcal´ a, Alcal´ a de Henares, Madrid (Spain) CCMA 2019 Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Recognizing curves up to certain transformations. Projectivities ( projective equivalence ) � a 11 x + a 12 y + b 1 , a 21 x + a 22 y + b 2 � f ( x , y ) = a 31 x + a 32 y + b 3 a 31 x + a 32 y + b 3 ˜ x = [ x 0 : x 1 : x 2 ] , P ∈ R 3 × 3 , det( P ) � = 0 f (˜ x ) = P ˜ x , ˜ Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Recognizing curves up to certain transformations. Projectivities ( projective equivalence ) � a 11 x + a 12 y + b 1 , a 21 x + a 22 y + b 2 � f ( x , y ) = a 31 x + a 32 y + b 3 a 31 x + a 32 y + b 3 ˜ x = [ x 0 : x 1 : x 2 ] , P ∈ R 3 × 3 , det( P ) � = 0 f (˜ x ) = P ˜ x , ˜ Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Recognizing curves up to certain transformations. Rigid motions ( congruence ) [including symmetries of a curve] f ( x , y ) = ( α x ∓ β y + b 1 , β x ± α y + b 2 ) , α 2 + β 2 = 1 f ( x ) = Qx + b , Q T Q = I Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Recognizing curves up to certain transformations. Similarities ( similaritity ) f ( x , y ) = ( λ ( α x ∓ β y ) + b 1 , λ ( β x ± α y ) + b 2 ) , α 2 + β 2 = 1 , λ � = 0 f ( x ) = λ Qx + b , Q T Q = I , λ � = 0 Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Recognizing curves up to certain transformations. Affinities ( affine equivalence ) f ( x , y ) = ( a 11 x + a 12 y + b 1 , a 21 x + a 22 y + b 2 ) f ( x ) = Ax + b , A ∈ R 2 × 2 , det( A ) � = 0 Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Projectivities. Projectivities ( projective equivalence ) � a 11 x + a 12 y + b 1 , a 21 x + a 22 y + b 2 � f ( x , y ) = a 31 x + a 32 y + b 3 a 31 x + a 32 y + b 3 ˜ x = [ x 0 : x 1 : x 2 ] , P ∈ R 3 × 3 , det( P ) � = 0 f (˜ x ) = P ˜ x , ˜ Pictures from 2D projective transformations (homographies) , C. Gava, G. Bleser, Computer Vision: Algorithms and Applications , R. Szeliski Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Projectivities. Projectivities ( projective equivalence ) � a 11 x + a 12 y + b 1 � , a 21 x + a 22 y + b 2 f ( x , y ) = a 31 x + a 32 y + b 3 a 31 x + a 32 y + b 3 ˜ x = [ x 0 : x 1 : x 2 ] , P ∈ R 3 × 3 , det( P ) � = 0 f (˜ x ) = P ˜ x , ˜ Obs.: projectivities are collineations Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
What curves do we want to study? In CAGD, the preferred type of algebraic curve to study are rational curves , e.g. � a (1 − t 2 ) 2 bt � x ( t ) = , . 1 + t 2 1 + t 2 Alc´ azar J.G., Hermoso C., Muntingh G. (2014), Detecting similarity of Rational Plane Curves , Journal of Computational and Applied Mathematics vol. 269, pp. 1-13 Hauer M., J¨ uttler B. (2018), Projective and affine symmetries and equivalences of rational curves in arbitrary dimension , Journal of Symbolic Computation Vol. 87, pp. 68–86. Problem essentially solved for rational curves. Other algebraic curves? Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
What curves do we want to study? Elliptic and Hyperelliptic curves Parametrizable by square-roots of rational funcions. Non-rational offsets of rational curves, and certain bisectors (line / rat. curve, circle / rat. curve), are either elliptic or hyperelliptic curves. Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
What curves do we want to study? The ellipse is rational, but the offsets to the ellipse, in general are not (in general, offsets to rational curves are not rational). Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
What curves do we want to study? Bisector curves of rational curves are not necessarily rational, either. Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Elliptic curves Elliptic curve: curves of genus 1 (genus 0 means rational) birationally equivalent to a nonsingular cubic curve (its Weierstrass form ) ξ E �− → W , where W can be written as y 2 = x 3 + rx + s Nonsingular cubic curves have a very rich structure! Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Elliptic curves Group law in a Weierstrass curve R Q P P ⊕ Q Conmmutative law ( abelian varieties ); the neutral element is O = [0 : 1 : 0] . Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Elliptic curves R Q P P ⊕ Q Ramification points: P ⊕ P = 2 P = O . Flex points: Q ⊕ Q ⊕ Q = 3 Q = O . Aligned points: P ⊕ Q ⊕ R = O . Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Elliptic curves R Q P P ⊕ Q Translation map by (fixed) P : τ P ( Q ) = P ⊕ Q ; if P = ( α, β ), � � 2 � � � 2 � � � y − β � y − β � y − β τ P ( x , y ) = − − x − α, − − x − 2 α + β x − α x − α x − α Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
� � � Projective equivalences between elliptic curves Theorem Let E 1 , E 2 ⊂ R 2 be two elliptic curves, with Weierstrass forms W 1 , W 2 ∈ R 2 , such that there exists a projectivity g mapping E 1 to E 2 . Then there exists a birational transformation ϕ g of R 2 , associated with g, mapping W 1 onto W 2 , making the following diagram commutative: g (1) E 1 E 2 ξ 1 ξ 2 � W 2 W 1 ϕ g In particular, for a generic point ( x , y ) ∈ E 1 we have ξ 2 ◦ g = ϕ g ◦ ξ 1 (2) Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Projective equivalences between elliptic curves Assuming W i ≡ y 2 = x 3 + r i x + s i , Theorem ( A. , Hermoso) The birational transformation ϕ g satisfies that ϕ g = τ P ◦ φ, with P ∈ W 2 , φ = ( a 2 x , a 3 y ) and a � = 0 is a real root of gcd ( r 2 − r 1 a 4 , a 6 s 1 − s 2 ) . Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
� � � Projective equivalences between elliptic curves For cubic elliptic curves E i , the E i and the W i are projectively equivalent! g E 1 E 2 (3) ξ 1 ξ 2 � W 2 W 1 ϕ g with ϕ g = ξ 2 ◦ g ◦ ξ − 1 (4) 1 So ϕ g = τ P ◦ φ must be a projectivity, and also τ P (restricted to W 2 )!! Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Projective equivalences between elliptic curves Cubic elliptic curves: since projectivities are collineations, for Q , R , S ∈ W 2 aligned τ P ( Q ⊕ R ⊕ S ) = τ P ( O ) = ( P ⊕ Q ) ⊕ ( P ⊕ R ) ⊕ ( P ⊕ S ) = 3 P = O Theorem ( A. , Hermoso) The projectivities g : E 1 → E 2 , with E i a cubic elliptic curve, are the mappings g = ξ − 1 ◦ ϕ g ◦ ξ 1 , 2 with ϕ g = τ P ◦ φ , where P = O (i.e. τ P is the identity) or P is a flex point of W 2 , and φ = ( a 2 x , a 3 y ) , with a � = 0 a real root of gcd ( r 2 − r 1 a 4 , a 6 s 1 − s 2 ) . Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Projective equivalences between elliptic curves For non-cubic elliptic curves, P = ( α, β ) must be included as an unknown in the computation (polynomial system solving, instead of linear system solving). Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
Hyperelliptic curves Hyperlliptic curve: curves of genus κ ≥ 2 birationally equivalent to a (singular) curve (its Weierstrass form ) ξ H �− → W , where W can be written as y 2 = h ( x ) , with h ( x ) square-free and of degree 2 κ + 1 or 2 κ + 2. Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
� � � Projective equivalences between hyperelliptic curves Theorem Let H 1 , H 2 ⊂ R 2 be two hyperelliptic curves, with Weierstrass forms W 1 , W 2 ∈ R 2 , such that there exists a projectivity g mapping H 1 to H 2 . Then there exists a birational transformation ϕ g of R 2 , associated with g, mapping W 1 onto W 2 , making the following diagram commutative: g H 1 H 2 (5) ξ 1 ξ 2 � W 2 W 1 ϕ g In particular, for a generic point ( x , y ) ∈ H 1 we have ξ 2 ◦ g = ϕ g ◦ ξ 1 (6) Juan G. Alc´ azar , Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves
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