提出几何方法快速计算空间点到轴对称二次曲面的最短距离。
A geometrical approach to determine the proximity of a point to an axisymmetric quadric in space
- 将三维距离问题转化为二维解析，利用圆锥曲线几何特性求解。
- 针对抛物线、椭圆/双曲线分2至3类情况处理，提升精度与效率。
- 代码实现简单，比商业库Bullet更快，适合工程应用。
本文提出一种将一般二次曲面分类为轴对称二次曲面（AQ）的方法，并解决空间中给定点到AQ的最近距离问题。该三维距离问题被转化为二维情形，这一降维方法在现有文献中尚未出现。作者基于圆锥曲线的几何性质（如子正弦、半长轴、离心率、斜率、半径）提出新的二维求解方法。此外，根据点的位置，将二维问题进一步细分为两类（抛物线）和三类（椭圆/双曲线），这是作者所知的新颖处理方式。所提方法适用于如C语言等通用编程语言实现，实测速度优于商业库Bullet。
原文摘要 · Abstract (English)
This paper presents the classification of a general quadric into an axisymmetric quadric (AQ) and the solution to the problem of the proximity of a given point to an AQ. The problem of proximity in $R^3$ is reduced to the same in $R^2$, which is not found in the literature. A new method to solve the problem in $R^2$ is used based on the geometrical properties of the conics, such as sub-normal, length of the semi-major axis, eccentricity, slope and radius. Furthermore, the problem in $R^2$ is categorised into two and three more sub-cases for parabola and ellipse/hyperbola, respectively, depending on the location of the point, which is a novel approach as per the authors' knowledge. The proposed method is suitable for implementation in a common programming language, such as C and proved to be faster than a commercial library, namely, Bullet.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。