二次有理双圆弧曲线及其在CAD中的应用

Rational Bicircular Arcs and Their Application to CAD

阐明了由三点Ｐ０，Ｐ１，Ｐ２确定的双圆弧的连接点可用控制点表示，其轨迹是圆心在Ｐ０Ｐ１中垂线上，半径为的圆弧是（ｌ＝｜Ｐ０Ｐ２｜，θ１，θ２△Ｐ０Ｐ１Ｐ２的底角）。提出了三种满足光顺性要求的选择连接点的方法。证明了用二次有理Ｂｅｚｉｅｒ曲线精确表示圆弧的充要条件是其中ｋ≠０，ω０，ω１，ω２为权因子。进而建立了双圆弧曲线的有理参数方程，即其中Ｆｉ（ｔ）与Ｇｉ（ｔ）是双圆弧基函数。作为应用，最后构造了一类以双圆弧为横向截线的曲面，包括其特殊情形双圆弧锥面和双圆弧柱面。

It is explained that the joint of a bicircular arc determined by three points P0,P1,P2may be represented by control points and its path is a circular arc with radius andits center is on the perpendicular bisector of, where θ1, and θ2 are the bottomangles of the △P0P1P2). Three methods by which the joint of bicircular arc is chosen are provided to satisfy the demand. The necessary and sufficient conditions for representing a circulararc as rational quadric Bener curve are investigatedwhere k≠0, ω0, ω1 and ω2 are weights.Hence the bicircular arc may be represented in a rational parametric equationwhere Fi(t) and Gi(t) are bicircular basic functions and tp is the parament of joint. Lastly as anapplication, surfaces whose cross--sections are bicircular arcs are constructed. Bicircular conicaland bicircular pillar surfaces are contained in these surfaces as particular cases. These algorithms are useful in shape design and numerical control.