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We present a new approach using differential invariants to detect projectiveequivalences and symmetries between two rational parametric $3D$ curvesproperly parametrized. In order to do this, we introduce two differentialinvariants that commute with M\"obius transformations, which are thetransformations in the parameter space associated with the projectiveequivalences between the curves. The M\"obius transformations are found byfirst computing the gcd of two polynomials built from the differentialinvariants, and then searching for the M\"obius-like factors of this gcd. Theprojective equivalences themselves are easily computed from the M\"obiustransformations. In particular, and unlike previous approaches, we avoidsolving big polynomial systems. The algorithm has been implemented in Maple,and evidences of its efficiency as well as a comparison with previousapproaches are given.

A new method to detect projective equivalences and symmetries of rational $3D$ curves