Projective Equivalences of Rational Algebraic Space Curves Using Differential Invariants

Gözütok U., Çoban H. A. , Sağıroğlu Y.

Curves and Surfaces 2022, Bordeaux, France, 20 - 24 June 2022, vol.1, pp.92

  • Publication Type: Conference Paper / Summary Text
  • Volume: 1
  • City: Bordeaux
  • Country: France
  • Page Numbers: pp.92
  • Karadeniz Technical University Affiliated: Yes


We present an alternative method to existing algorithms [1, 2] for computing the projective equivalences

between two rational space curves. The method is inspired in [3], where torsion and curvature, two classical

and well-known differential invariants of space curves, are used to compute the similarities between two given

space rational curves. In our case, we produce two projective curvature-like invariants κ1 and κ2, that can be

used to characterize the existence of projective equivalences. In more detail, given two rational curves C1 and

C2 properly parametrized by p, q, we prove that C1,C2 are projectively equivalent if and only if

κ1(p) = κ1(q)(φ), κ2(p) = κ2(q)(φ),

where φ is a M¨obius transformation. Then we can detect projective equivalence by checking whether or not

the gcd of the polynomials involved in the above equation has a M¨obius-like factor. In the affirmative case φ

is obtained, and from here the projective equivalence itself can be easily computed. After finding the gcd, we

can also use polynomial system solving; but in that case the system is substantially smaller compared to other

approaches, which leads to better timings. The method has been implemented in Maple. A full description of

the algorithm and the ideas behind it can be found in [4].