We present a new approach to detecting projective equivalences and symmetries of rational parametric 3D curves. To detect projective equivalences, we first derive two projective differential invariants that are also invariant with respect to the change of parameters called M\"obius transformations. Given two rational curves, we form a system consists of two homogeneous polynomials in four variables using the projective differential invariants. The solution of the system yields the M\"obius transformations, each of which corresponds to a projective equivalence. If the input curves are the same, then our method detects the projective symmetries of the input curve. Our method is substantially faster than methods addressing a similar problem and provides solutions even for the curves with degree up to 24 and coefficients up to 78 digits.
翻译:我们提出一种新的方法来检测合理参数3D曲线的预测等值和对称性。 为了检测预测等值, 我们首先得出两个预测差异, 这些差异与称为 M\'obius 的参数变化有关。 鉴于两个理性曲线, 我们形成一个系统, 由四个变量中的两个同质多元值组成, 使用预测差异值。 这个系统的解决方案产生 M\'obius 的变异, 每一个变异都与预测等值相对应。 如果输入曲线相同, 那么我们的方法将检测输入曲线的预测对称性。 我们的方法比解决类似问题的方法要快得多, 并且提供了解决方案, 甚至对于达到24度的曲线和达到78位数的系数。