Abstract: The aim of the paper is to do further study on the geometric properties of linear hyperbolic Bézier-like curves. A given hyperbola with standard equation is changed into the linear hyperbolic Bézier-like form by using geometric and parametric transformations. Comparing the new-form hyperbola with any linear hyperbolic Bézier-like curve, some equations are gotten. By solving the equations, a conclusion that any non-degenerate linear hyperbolic Bézier-like curve is a segment of a hyperbola is drawn. The explicit expressions of the geometric elements of the hyperbola, such as the center, the vertices of the real and imaginary axes, and the foci, in terms of the control points of the linear hyperbolic Bézier-like curve, are also obtained. The theoretical analysis and examples show that all the above-mentioned geometric elements of the hyperbola can be given by the bilinear interpolation forms of the control points of the linear hyperbolic Bézier-like curve.

Key words: computer aided geometric design, linear hyperbolic Bézier-like curve, hyperbola, explicit expression, bilinear interpolation

摘要： 为进一步研究线性双曲拟Bézier曲线的几何性质，从标准的双曲线方程出发，使用几何变换与参数变换，将其化为线性双曲拟Bézier曲线的形式；经过与任意线性双曲拟Bézier曲线相比较，得到方程组以求解，从而得出非退化的线性双曲拟Bézier曲线必为双曲线的结论，并给出该双曲线的中心、实、虚轴顶点和焦点这些几何元素关于控制顶点的显式表达；通过理论和实例表明，提出的双曲线的几何元素均可由线性双曲拟Bézier曲线控制顶点的双线性插值得到。

关键词: 计算机辅助几何设计, 线性双曲拟Bé, zier曲线, 双曲线, 显式表达, 双线性插值