11. I am interested in plane curves only. 12. The second generalized curvature is called "'torsion "'and measures the deviance of from being a plane curve . 13. I realize the formula only holds if the plane curve does not cross the axis of rotation. 14. Consider the affine plane curve defined by the equation y ^ 2 = x ^ 5 + 1. 15. It starts with a plane curve , and repeatedly applies birational transformations to the plane to improve the curve. 16. For example, the disjoint union of a Sierpinski carpet and a circle is also a universal plane curve . 17. The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. 18. Thus for " g " = 3 the canonical curves ( non-hyperelliptic case ) are quartic plane curves . 19. The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. 20. The affine focal set of a plane curve will cusp points ( semi-cubical palabara | semi-cubical parabolae ).
