关键词: 曲线造型, 有理二次Bé, zier 曲线, 双曲线, 共轭双曲线, 线性插值

Abstract: Consider the relationship between the control points of the conjugate hyperbolas presented
as rational quadratic Bézier forms. Given a rational quadratic Bézier curve with standard form
presenting a hyperbolic segment, the target is to calculate the control points of the corresponding
segment on its conjugate hyperbola. Firstly, the natural definition of the segment of conjugate
hyperbola is given. Secondly, parameter conversion is used to transform the hyperbola between the
rational quadratic Bézier form and a general parametric form. The transformations correspond to
matrices. Thus the explicit expressions of the control points of the conjugate segments are obtained.
Finally, the geometric meanings of the expressions are shown. Each control point of the conjugate
hyperbola segments can be given by two linear interpolations of the control points of the original
hyperbola segment.

Key words: curve modeling, rational quadratic Bézier curve, hyperbola, conjugate hyperbola, linear
interpolation