Piecewise-circular curves for geometric modeling
Piecewise-circular curves for geometric modeling
复制标题
DOI：
10.1147/rd.313.0296
复制
发表时间：
1987-05
影响因子：
1.3
通讯作者：
J. Rossignac;A. Requicha
中科院分区：
文献类型：
--
作者：
J. Rossignac;A. Requicha
现代实体建模器必须能够表示广泛的对象类别，并且必须支持实体上的布尔运算。这些操作对于定义实体、检测干涉和建模制造过程非常有用。计算通过布尔运算定义的实体边界需要曲面/曲面和曲线/曲面求交算法。许多现有的建模器使用封闭形式的参数表达式为二次曲面的相交曲线，并计算这些曲线与其他表面的交点通过寻找低次多项式的根。由于涉及环面或更复杂曲面的相交所产生的曲线通常不能以闭合形式表示，建模者通常通过插值位于真实相交点上的点的三次样条来近似这些曲线。三次样条具有二次连续性，但在实体建模计算中处理它们的成本很高。本文以二次连续性为代价，提出了一种用由直线段和圆弧组成的光滑空间曲线插值三维点及相应的单位切向量的方法。这些曲线被指定为PCC（分段圆形曲线），并且具有连续的单位切线。PCC可以用于执行基本几何计算的有效算法中，例如评估从点到曲线或曲线和曲面的交点的最小距离。给出了在实体建模器中生成和处理PCC的公式和算法。我们还表明，PCC是有用的，将环形图元，以及扫描，增长，收缩，混合操作的系统中，模型的自然二次曲面，平面，圆柱体，圆锥体和球体的固体为界。
Modern solid modelers must be able to represent a wide class of objects, and must support Boolean operations on solids. These operations are very useful for defining solids, detecting interferences, and modeling fabrication processes. Computing the boundaries of solids defined through Boolean operations requires algorithms for surface/surface and curve/surface intersection. Many of the currently available modelers use closed-form parametric expressions for the curves of intersection of quadric surfaces, and compute intersections of these curves with other surfaces by finding the roots of low-degree polynomials. Because the curves that result from intersections involving tori or more complex surfaces generally cannot be expressed in closed form, modelers typically approximate these curves by cubic splines that interpolate points lying on the true intersections. Cubic splines exhibit second-degree continuity, but they are expensive to process in solid modeling computations. In this paper, we trade second-degree continuity for computational simplicity, and present a method for interpolating three-dimensional points and associated unit tangent vectors by smooth space curves composed of straight line segments and circular arcs. These curves are designated as PCCs (for piecewise-circular curves) and have continuous unit tangents. PCCs can be used in efficient algorithms for performing fundamental geometric computations, such as the evaluation of the minimal distance from a point to a curve or the intersection of a curve and a surface. Formulae and algorithms are presented for generating and processing PCCs in solid modelers. We also show that PCCs are useful for incorporating toroidal primitives, as well as sweeping, growing, shrinking, and blending operations in systems that model solids bounded by the natural quadric surfaces—planes, cylinders, cones, and spheres.