Paths of algebraic hyperbolic curves

Cubic algebraic hyperbolic (AH) Bézier curves and AH spline curves are defined with a positive parameter α in the space spanned by {1, t , sinh t , cosh t }. Modifying the value of α yields a family of AH Bézier or spline curves with the family parameter α . For a fixed point on the original curve, it will move on a defined curve called “path of AH curve” (AH Bézier and AH spline curves) when α changes. We describe the geometric effects of the paths and give a method to specify a curve passing through a given point.