找到点最小化从一组N行的距离 [英] Find the point minimizing the distance from a set of N lines

问题描述

由于多（N）在三维空间的线条，找点距离最小为所有行。

Given Multiple (N) lines in 3d space, find the point minimizing the distance to all lines.

1. 在给定一个行[AX + B]，并点[P]之间的最短距离将在垂直线[AX + B] - [P]我可以EX preSS最小平方距离为平方和线的距离，例如。 （[AX + B] - [P]）^ 2 + ... +（[AX + B] N- [P]）^ 2
2. 由于线垂直，我可以在该行的术语使用点积EX preSS [P]

我已经用最小二乘估计点距离最小考虑的问题是，标准的最小二乘将接近最佳拟合线/曲线给出一组点，我需要的是相反的，给定一组行估计最佳拟合点。

I have considered using Least Squares for estimating the point minimizing the distance, the problem is that the standard least squares will approximate the best fitting line/curve given a set of points, What I need is the opposite, given a set of lines estimate the best fitting point.

应如何加以处理？

推荐答案

从维基百科，我们读到这条线之间的平方距离 a'x + B = 0 键，点 P 是 （A'P + B）^ 2 /（A'A）。因此，我们可以看到，最小化平方距离的总和的点是一个加权线性回归问题与一种观察每一行。回归模型具有以下属性：

From wikipedia, we read that the squared distance between line a'x + b = 0 and point p is (a'p+b)^2 / (a'a). We can therefore see that the point that minimizes the sum of squared distances is a weighted linear regression problem with one observation for each line. The regression model has the following properties:

• 的样本数据 A 每行 AX + B = 0
• 样品结果 -b 每行 AX + B = 0
• 样品重量 1 /（A'A）每行 AX + B = 0

• Sample data a for each line ax+b=0
• Sample outcome -b for each line ax+b=0
• Sample weight 1/(a'a) for each line ax+b=0

您应该能够解决这个问题，任何标准的统计软件。

You should be able to solve this problem with any standard statistical software.

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