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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) f (x, y) = a11x + a12y + b1 a31x + a32y + b3 , a21x + a22y + b2 a31x + a32y + b3

• ˜

f (˜ x) = P˜ x, ˜ x = [x0 : x1 : x2], P ∈ R3×3, det(P) = 0

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

Recognizing curves up to certain transformations.

Projectivities (projective equivalence) f (x, y) = a11x + a12y + b1 a31x + a32y + b3 , a21x + a22y + b2 a31x + a32y + b3

• ˜

f (˜ x) = P˜ x, ˜ x = [x0 : x1 : x2], P ∈ R3×3, det(P) = 0

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) = (αx ∓ βy + b1, βx ± αy + b2), α2 + β2 = 1 f (x) = Qx + b, QTQ = 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(αx ∓ βy) + b1, λ(βx ± αy) + b2), α2 + β2 = 1, λ = 0 f (x) = λQx + b, QTQ = 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) = (a11x + a12y + b1, a21x + a22y + b2) f (x) = Ax + b, A ∈ R2×2, det(A) = 0

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

Projectivities.

Projectivities (projective equivalence) f (x, y) = a11x + a12y + b1 a31x + a32y + b3 , a21x + a22y + b2 a31x + a32y + b3

• ˜

f (˜ x) = P˜ x, ˜ x = [x0 : x1 : x2], P ∈ R3×3, det(P) = 0

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) f (x, y) = a11x + a12y + b1 a31x + a32y + b3 , a21x + a22y + b2 a31x + a32y + b3

• ˜

f (˜ x) = P˜ x, ˜ x = [x0 : x1 : x2], P ∈ R3×3, det(P) = 0 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. x(t) = a(1 − t2) 1 + t2 , 2bt 1 + t2

• .

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) E

ξ

− → W, where W can be written as y 2 = x3 + 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

P Q P ⊕ Q R

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

P Q P ⊕ Q R

Ramification points: P ⊕ P = 2P = O. Flex points: Q ⊕ Q ⊕ Q = 3Q = O. Aligned points: P ⊕ Q ⊕ R = O.

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

Elliptic curves

P Q P ⊕ Q R

Translation map by (fixed) P: τP(Q) = P ⊕ Q; if P = (α, β), τP(x, y) =

• −

y − β x − α 2 − x − α, y − β x − α − y − β x − α 2 − x − 2α

• + β
• Juan G. Alc´

azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

Projective equivalences between elliptic curves

Theorem Let E1, E2 ⊂ R2 be two elliptic curves, with Weierstrass forms W1, W2 ∈ R2, such that there exists a projectivity g mapping E1 to E2. Then there exists a birational transformation ϕg of R2, associated with g, mapping W1 onto W2, making the following diagram commutative: E1

g

• ξ1
• E2

ξ2

• W1

ϕg

W2 (1) In particular, for a generic point (x, y) ∈ E1 we have ξ2 ◦ g = ϕg ◦ ξ1 (2)

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

Projective equivalences between elliptic curves

Assuming Wi ≡ y 2 = x3 + rix + si, Theorem (A., Hermoso) The birational transformation ϕg satisfies that ϕg = τP ◦ φ, with P ∈ W2, φ = (a2x, a3y) and a = 0 is a real root of gcd(r2 − r1a4, a6s1 − s2).

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

Projective equivalences between elliptic curves

For cubic elliptic curves Ei, the Ei and the Wi are projectively equivalent! E1

g

• ξ1
• E2

ξ2

• W1

ϕg

W2 (3) with ϕg = ξ2 ◦ g ◦ ξ−1

1

(4) So ϕg = τP ◦ φ must be a projectivity, and also τP (restricted to W2)!!

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 ∈ W2 aligned τP(Q ⊕ R ⊕ S) = τP(O) = (P ⊕ Q) ⊕ (P ⊕ R) ⊕ (P ⊕ S) = 3P = O Theorem (A., Hermoso) The projectivities g : E1 → E2, with Ei a cubic elliptic curve, are the mappings g = ξ−1

2

• ϕg ◦ ξ1,

with ϕg = τP ◦ φ, where P = O (i.e. τP is the identity) or P is a flex point of W2, and φ = (a2x, a3y), with a = 0 a real root of gcd(r2 − r1a4, a6s1 − s2).

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) H

ξ

− → W, where W can be written as y 2 = h(x), with h(x) square-free and of degree 2κ + 1 or 2κ + 2.

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

Projective equivalences between hyperelliptic curves

Theorem Let H1, H2 ⊂ R2 be two hyperelliptic curves, with Weierstrass forms W1, W2 ∈ R2, such that there exists a projectivity g mapping H1 to H2. Then there exists a birational transformation ϕg of R2, associated with g, mapping W1 onto W2, making the following diagram commutative: H1

g

• ξ1
• H2

ξ2

• W1

ϕg

W2 (5) In particular, for a generic point (x, y) ∈ H1 we have ξ2 ◦ g = ϕg ◦ ξ1 (6)

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

Projective equivalences between hyperelliptic curves

Theorem (A., Hermoso) Let W1, W2 ⊂ R2 be two hyperelliptic curves in Weierstrass form of genus κ ≥ 2. Any real birational mapping ϕg between W1, W2 has the form ϕg(x, y) = ax + b cx + d , ey (cx + d)κ+1

• ,

(7) with ad − bc = 0, and a, b, c, d, e ∈ R, e = 0.

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

More...

Space elliptic and hyperelliptic curves?? (e.g. intersections between a quadric and a ruled surface)

Picture from Intersecting Quadrics: An Efficient and Exact Implementation, S. Lazard, L. Pe˜ naranda, S. Petitjean Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

More...

More on similarities, affine and projective equivalences: Alcazar J.G., Lavicka M., Vrsek J. (2019) Symmetries and similarities of planar algebraic curves using harmonic polynomials, Journal of Computational and Applied Mathematics Vol. 357, pp. 302–318. Bizzarri M., Lavicka M., Vrsek J. (2018), Computing projective equivalences of special algebraic varieties, ArXiv 1806.05827. Hauer M., J¨ uttler B., Schicho J. (2018), Projective and affine symmetries and equivalences of rational and polynomial surfaces, Journal of Computational and Applied Mathematics Vol. 349, pp. 424–437.

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves

The end

Thanks for your attention!!

Juan G. Alc´ azar, Carlos Hermoso Projective equivalences of elliptic and hyperelliptic planar curves