点绕 z 轴的旋转 [英] Rotation of a point about the z-axis

问题描述

我在 3D 空间中有 3 个向量.我们称它们为 xaxis、yaxis 和 zaxis.这些向量以 3D 空间中某处的任意 point 为中心.我有兴趣将 xaxis 和 yaxis 向量绕 zaxis 向量旋转一定度数 θ.

I have 3 vectors in 3D space. Let's call them xaxis, yaxis, and zaxis. These vectors are centered about an arbitrary point somewhere in 3D space. I am interested in rotating the xaxis and yaxis vectors about the zaxis vector a number of degrees θ.

对于以下具有任意值且不重要的代码:

For the following code with values being arbitrary and unimportant:

double xaxis[3], yaxis[3], zaxis[3], point[3], theta;

我将如何将 xaxis 和 yaxis 绕 zaxis 旋转 theta 度?

How would I go about rotating xaxis and yaxis about the zaxis by theta degrees?

未来注意:这些尝试不起作用.请参阅我在 BlueRaja-DannyPflughoeft 的帮助下找到的正确解决方案的答案

我尝试基于矩阵的旋转:

My attempt at matrix-based rotation:

double rx[3][3];
double ry[3][3];
double rz[3][3];
double r[3][3];

rx[0][0] = 1;
rx[0][1] = 0;
rx[0][2] = 0;

rx[1][0] = 0;
rx[1][1] = cos(theta);
rx[1][2] = sin(theta);

rx[2][0] = 0;
rx[2][1] = -1.0 * sin(theta);
rx[2][2] = cos(theta);

ry[0][0] = cos(theta);
ry[0][1] = 0;
ry[0][2] = -1.0 * sin(theta);

ry[1][0] = 0;
ry[1][1] = 1;
ry[1][2] = 0;

ry[2][0] = sin(theta);
ry[2][1] = 0;
ry[2][2] = cos(theta);
//No rotation wanted on the zaxis
rz[0][0] = cos(0);
rz[0][1] = sin(0);
rz[0][2] = 0;

rz[1][0] = -1.0 * sin(0);
rz[1][1] = cos(0);
rz[1][2] = 0;

rz[2][0] = 0;
rz[2][1] = 0;
rz[2][2] = 1;

vtkMath::Multiply3x3(rx, ry, r); //Multiplies rx by ry and stores into r
vtkMath::Multiply3x3(r, rz, r); //Multiplies r by rz and stores into r

vtkMath::Multiply3x3(r, xaxis, xaxis);//multiplies a 3x3 by a 3x1
vtkMath::Multiply3x3(r, yaxis, yaxis);//multiplies a 3x3 by a 3x1

此尝试仅在飞机处于 x-y 平面时有效:

This attempt only worked when the plane was in the x-y plane:

double x, y;
x = xaxis[0];
y = xaxis[1];
xaxis[0] = x * cos(theta) - y * sin(theta);
xaxis[1] = x * sin(theta) + y * cos(theta);

x = yaxis[0];
y = yaxis[1];
yaxis[0] = x * cos(theta) - y * sin(theta);
yaxis[1] = x * sin(theta) + y * cos(theta);

使用 BlueRaja-DannyPflughoeft 给出的轴角方法:

Using the axis-angle approach given by BlueRaja-DannyPflughoeft:

double c = cos(theta);
double s = sin(theta);
double C = 1.0 - c;

double Q[3][3];
Q[0][0] = xaxis[0] * xaxis[0] * C + c;
Q[0][1] = xaxis[1] * xaxis[0] * C + xaxis[2] * s;
Q[0][2] = xaxis[2] * xaxis[0] * C - xaxis[1] * s;

Q[1][0] = xaxis[1] * xaxis[0] * C - xaxis[2] * s;
Q[1][1] = xaxis[1] * xaxis[1] * C + c;
Q[1][2] = xaxis[2] * xaxis[1] * C + xaxis[0] * s;

Q[2][0] = xaxis[1] * xaxis[2] * C + xaxis[1] * s;
Q[2][1] = xaxis[2] * xaxis[1] * C - xaxis[0] * s;
Q[2][2] = xaxis[2] * xaxis[2] * C + c;

double x = Q[2][1] - Q[1][2], y = Q[0][2] - Q[2][0], z = Q[1][0] - Q[0][1];
double r = sqrt(x * x + y * y + z * z);

//xaxis[0] /= r;
//xaxis[1] /= r;
//xaxis[2] /= r;

xaxis[0] = x;// ?
xaxis[1] = y;
xaxis[2] = z;

推荐答案

感谢 BlueRaja - Danny Pflughoeft:

Thanks to BlueRaja - Danny Pflughoeft:

double c = cos(theta);
double s = sin(theta);
double C = 1.0 - c;

double Q[3][3];
Q[0][0] = zaxis[0] * zaxis[0] * C + c;
Q[0][1] = zaxis[1] * zaxis[0] * C + zaxis[2] * s;
Q[0][2] = zaxis[2] * zaxis[0] * C - zaxis[1] * s;

Q[1][0] = zaxis[1] * zaxis[0] * C - zaxis[2] * s;
Q[1][1] = zaxis[1] * zaxis[1] * C + c;
Q[1][2] = zaxis[2] * zaxis[1] * C + zaxis[0] * s;

Q[2][0] = zaxis[0] * zaxis[2] * C + zaxis[1] * s;
Q[2][1] = zaxis[2] * zaxis[1] * C - zaxis[0] * s;
Q[2][2] = zaxis[2] * zaxis[2] * C + c;

xaxis[0] = xaxis[0] * Q[0][0] + xaxis[0] * Q[0][1] + xaxis[0] * Q[0][2];
xaxis[1] = xaxis[1] * Q[1][0] + xaxis[1] * Q[1][1] + xaxis[1] * Q[1][2];
xaxis[2] = xaxis[2] * Q[2][0] + xaxis[2] * Q[2][1] + xaxis[2] * Q[2][2]; // Multiply a 3x3 by 3x1 and store it as the new rotated axis

yaxis[0] = yaxis[0] * Q[0][0] + yaxis[0] * Q[0][1] + yaxis[0] * Q[0][2];
yaxis[1] = yaxis[1] * Q[1][0] + yaxis[1] * Q[1][1] + yaxis[1] * Q[1][2];
yaxis[2] = yaxis[2] * Q[2][0] + yaxis[2] * Q[2][1] + yaxis[2] * Q[2][2]; // Multiply a 3x3 by 3x1 and store it as the new rotated axis

这篇关于点绕 z 轴的旋转的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！