用椭圆曲线揭示三焦点问题新解法，提升精度。
An Elliptic Curve Based Solution to the Perspective-Three-Point Problem
- 转向求解控制点连线方向，而非距离
- 在合理条件下比Lambda Twist更准确
- 揭示了三焦点问题与椭圆曲线的深层联系
本文通过分析控制点间连线相对于相机的方向关系，而非相机到控制点的距离，解决了透视三焦点问题（P3P）。该方法构建了一个高效、精确且相对简单的P3P求解器，与当前先进方法“Lambda Twist”进行了对比。两者均依赖于三次多项式单根的精确计算。在广泛测试的控制点三角形下，该方法在特定合理限制下显著优于Lambda Twist，尽管速度稍慢。然而，本工作的核心价值不在于提出另一种求解器，而在于发现P3P问题与一类特殊椭圆曲线之间存在深刻联系，这类椭圆曲线也用于密码学。为此，提出并解决了一个有趣的球面版古代“滑动”问题，为多个方向的研究提供了新可能。
原文摘要 · Abstract (English)
The Perspective-Three-Point Problem (P3P) is solved by first focusing on determining the directions of the lines through pairs of control points, relative to the camera, rather than the distances from the camera to the control points. The analysis of this produces an efficient, accurate and reasonably simple P3P solver, which is compared with a state-of-the-art P3P solver, "Lambda Twist." Both methods depend on the accurate computation of a single root of a cubic polynomial. They have been implemented and tested for a wide range of control-point triangles, and under certain reasonable restrictions, the new method is noticably more accurate than Lambda Twist, though it is slower. However, the principal value of the present work is not in introducing yet another P3P solver, but lies rather in the discovery of an intimate connection between the P3P problem and a special family of elliptic curves that includes curves utilized in cryptography. This holds the potential for further advances in a number of directions. To make this connection, an interesting spherical analogue of an ancient "sliding" problem is stated and solved.
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