Second Order Nurbs Interpolation of Real Affine and Projective Plane Curves

A method is presented to interpolate real affine plane curves with second degree NURBS (Non Uniform Rational Bezier Splines). The curve into object is initially partitioned into an arbitrary number of arcs. Then each arc is approximated with a conic section passing thru its extremes, thru a third intermediate point and being tangent to the arc at the extremes. Each arc is then parameterized by a second degree NURBS whose coefficients are computed in order to exactly fit the conic. Finally, the resulting NURBS arcs are joined together to form a unique global curve. The method is capable to manage also non-simple (i.e. self-intersecting), non-regular (i.e. piece wise differentiable) curves. A generalization of the method is also presented, capable to manage curve arcs containing improper points. This procedure, fully deploying NURBS interpolation capabilities, is theoretically set in the framework of real projective geometry.