计算与 3D 中的第三个向量垂直的两个向量 [英] Computing two vectors that are perpendicular to third vector in 3D

问题描述

计算与第三个向量 (X) 垂直且彼此垂直的两个向量的最佳(最快)方法是什么?

What is the best (fastest) way to compute two vectors that are perpendicular to the third vector(X) and also perpendicular to each other?

这就是我现在计算这个向量的方式:

This is how am I computing this vectors right now:

// HELPER - unit vector that is NOT parallel to X
x_axis = normalize(X);
y_axis = crossProduct(x_axis, HELPER);
z_axis = crossProduct(x_axis, y_axis);

我知道对此有无数种解决方案，我不在乎哪一种是我的解决方案.

I know there is infinite number of solutions to this, and I don't care which one will be my solution.

这个问题的背后是什么:我需要构造变换矩阵，在那里我知道 X 轴(矩阵中的第一列)应该指向哪个方向.我需要计算 Y 和 Z 轴(第二和第三列).众所周知，所有的轴都必须相互垂直.

What is behind this question: I need to construct transformation matrix, where I know which direction should X axis (first column in matrix) be pointing. I need to calculate Y and Z axis (second and third column). As we know, all axes must be perpendicular to each other.

推荐答案

我所做的，前提是 X<>0 或 Y<>0 是

What I have done, provided that X<>0 or Y<>0 is

1. A = [-Y, X, 0]
2. B = [-X*Z, -Y*Z, X*X+Y*Y]

然后对向量进行归一化.

and then normalize the vectors.

[ X,Y,Z]·[-Y,X,0] = -X*Y+Y*X = 0
[ X,Y,Z]·[-X*Z,-Y*Z,X*X+Y*Y] = -X*X*Z-Y*Y*Z+Z*(X*X+Y*Y) = 0
[-Y,X,0]·[-X*Z,-Y*Z,X*X+Y*Y] = Y*X*Z+X*Y*Z = 0

这称为向量的零空间.

如果 X=0 和 Y=0 那么 A=[1,0,0], B=[0,1,0].

If X=0 and Y=0 then A=[1,0,0], B=[0,1,0].

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