您如何确定一个点位于线段上的其他两个点之间?

2022-01-14 00:00:00 python geometry math

问题描述

假设您有一个二维平面，上面有 2 个点(称为 a 和 b)，每个点用一个 x 整数和一个 y 整数表示.

Let's say you have a two dimensional plane with 2 points (called a and b) on it represented by an x integer and a y integer for each point.

如何确定另一个点 c 是否在 a 和 b 定义的线段上?

How can you determine if another point c is on the line segment defined by a and b?

我最常使用 python，但任何语言的示例都会有所帮助.

I use python most, but examples in any language would be helpful.

解决方案

检查 (ba) 和 (ca) 的 叉积 是否为 0，正如 Darius Bacon 所说，告诉您积分是否a、b 和 c 对齐.

Check if the cross product of (b-a) and (c-a) is 0, as tells Darius Bacon, tells you if the points a, b and c are aligned.

但是，由于您想知道 c 是否介于 a 和 b 之间，您还必须检查 (ba) 和 (ca) 的 点积 是否正 并且小于 a 和 b 之间距离的平方.

But, as you want to know if c is between a and b, you also have to check that the dot product of (b-a) and (c-a) is positive and is less than the square of the distance between a and b.

在未优化的伪代码中:

def isBetween(a, b, c):
    crossproduct = (c.y - a.y) * (b.x - a.x) - (c.x - a.x) * (b.y - a.y)

    # compare versus epsilon for floating point values, or != 0 if using integers
    if abs(crossproduct) > epsilon:
        return False

    dotproduct = (c.x - a.x) * (b.x - a.x) + (c.y - a.y)*(b.y - a.y)
    if dotproduct < 0:
        return False

    squaredlengthba = (b.x - a.x)*(b.x - a.x) + (b.y - a.y)*(b.y - a.y)
    if dotproduct > squaredlengthba:
        return False

    return True

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