基于三次Bezier分段构造g2 -光滑复合曲线

V. Korotkiy
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引用次数: 5

摘要

自20世纪60年代中期以来,在技术设计中使用的形成复合g2 -光滑(两连续可微)曲线的理论和实践,仍然没有以任何方式反映在技术大学的课程或俄罗斯工程和计算机图形学教科书中。同时,这类曲线也被广泛用于各种几何物体和物理过程的建模。本文讨论了构造一条复合g2 -光滑曲线的问题,该曲线经过给定的点,并在这些点处接触预定的直线。为了解决这个问题,我们使用了三次贝塞尔线段。构造光滑复合曲线的主要问题是保证线段连接处曲率的连续性。本文表明,对于参数化三次曲线,该问题可简化为求解一个二次方程。编写了一个软件模块,可以构造平面g2 -平滑曲线,通过预定点,并在这些点与预定直线相切。曲线的形状(其分段的“完整性”)由用户在程序模块的对话模式中进行调整。解决了构造三次曲线平滑连接非连通贝塞尔段的问题。提出了一种构造具有给定切线和边界点曲率的Bezier线段的算法。讨论了三次Bezier线段的一些性质。特别是,对于平行切线的情况,线段末端的曲率仅由一个控制点的位置决定(定理1)。当贝塞尔线段末端的曲率等于零(定理2)时,考虑这种情况。使用贝塞尔线段执行三点物理样条的近似。近似误差小于2%,与处理实验数据的误差相当。提出了一种空间g2 -光滑曲线的建模方法,该曲线通过空间中预先设置的点,并在这些点上接触空间中任意方向的直线。这篇文章具有教育性质,旨在深入学习计算几何和计算机图形学的基础知识。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Constructing a G2-Smooth Compound Curve Based on Cubic Bezier Segments
The theory and practice of forming composite G2-smooth (two-continuously differentiable) curves, used in technical design since the mid-60s of the 20th century, is still not reflected in any way in the curriculum of technical universities or in Russian textbooks in engineering and computer graphics. Meanwhile, such curves are used in modeling a wide variety of geometric objects and physical processes. The article deals with the problem of constructing a composite G2-smooth curve passing through given points and touching at these points pre-specified straight lines. To solve the problem, cubic Bezier segments are used. The main problem in constructing a smooth compound curve is to ensure the continuity of curvature at the joints of the segments. The article shows that for parametrized cubic curves, this problem is reduced to solving a quadratic equation. A software module has been compiled that allows one to construct a plane G2-smooth curve passing through predetermined points and tangent at these points with predetermined straight lines. The shape of the curve (“completeness” of its segments) is adjusted by the user in the dialog mode of the program module. Solved the problem of constructing a cubic curve smoothly connecting unconnected Bezier segments. An algorithm for constructing a Bezier segment with given tangents and given curvature at its boundary points is proposed. Some properties of the cubic Bezier segment are considered. In particular, it was shown that for the case of parallel tangents, the curvature at the end of a segment is determined by the position of only one control point (Theorem 1). Cases are considered when the curvature at the ends of the Bezier segment is equal to zero (Theorem 2). An approximation of a three-point physical spline is performed using Bezier segments. The approximation error was less than 2%, which is comparable to the error in processing the experimental data. A method is proposed for modeling a spatial G2-smooth curve passing through points set in advance in space and touching at these points arbitrarily oriented lines in space. The article is of an educational nature and is intended for an in-depth study of the basics of computational geometry and computer graphics.
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