在从 3d 空间中的另一个点投影的线上查找点的 3d 坐标 [英] Find 3d coordinates of a point on a line projected from another point in 3d space

问题描述

使用 Swift、ARTKit/SceneKit

Working in Swift, ARTKit / SceneKit

我在 3d 中有一条线 AB，我有 A 点和 B 点的 xyz 坐标.我也有一个 C 点，我也知道它的 xyz 坐标.

I have a line AB in 3d and I have xyz coordinates of both points A and B. I also have a point C and I know its xyz coordinates too.

现在，我想在AB线上找出D点的xyz坐标；假设 CD 垂直于 AB.

Now, I want to find out the xyz coordinates of point D on line AB; given that CD is perpendicular to AB.

在 Swift 中执行此操作的简单方法是什么.

What would be a simple way to do it in Swift.

推荐答案

用标量 t 参数化 AB 行:

Parameterize the line AB with a scalar t:

P(t) = A + (B - A) * t`

点D = P(t)使得CD垂直于AB，即它们的点积为零:

The point D = P(t) is such that CD is perpendicular to AB, i.e. their dot product is zero:

dot(C - D, B - A) = 0

dot(C - A - (B - A) * t, B - A) = 0

dot(C - A, B - A) = t * dot(B - A, B - A)

// Substitute value of t

-->  D = A + (B - A) * dot(C - A, B - A) / dot(B - A, B - A)

Swift 代码:

var BmA = B - A
var CmA = C - A
var t = dot(CmA, BmA) / dot(BmA, BmA)
var D = A + BmA * t;

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