椭圆旋转矩阵？ [英] Elliptical rotation matrix?

问题描述

在C ++中，我们可以绕任意轴旋转一个点：

In C++ we can rotate a point about an arbitrary axis:

  void radRotateAxis( float a, float b, float c, float theta )
  {
    float newX = (
      x*( a*a*(1-cos(theta)) + cos(theta)   ) +
      y*( a*b*(1-cos(theta)) - c*sin(theta) ) +
      z*( a*c*(1-cos(theta)) + b*sin(theta) ) );

    float newY = (
      x*( a*b*(1-cos(theta)) + c*sin(theta) ) +
      y*( b*b*(1-cos(theta)) + cos(theta)   ) +
      z*( b*c*(1-cos(theta)) - a*sin(theta) )  );

    float newZ = (
      x*( a*c*(1-cos(theta)) - b*sin(theta) ) +
      y*( b*c*(1-cos(theta)) + a*sin(theta) ) +
      z*( c*c*(1-cos(theta)) + cos(theta)   )  );

    x = newX ;
    y = newY ;
    z = newZ ;
  }

但是当我们走路theta 0 - > 2PI这绕着绕你旋转的轴的单位圆点。

But as we walk theta 0 -> 2PI this takes the point around a "unit circle" around the axis you're rotating about

我们如何让它像theta 0 - > 2PI的结果是关于一个椭圆的宽度a，高度b？

How can we make it so as theta 0 -> 2PI the results are about an ellipse of width a, height b?

我不想在转换后将变换矩阵应用到点关于轴 - 我正在寻找的是一个椭圆的旋转矩阵，如果有人知道这样的事情！

I do not want to apply transformation matrices to the points after rotating them about the axis - what I'm looking for is an "elliptical" rotation matrix, if anyone knows of such a thing!

推荐答案

定义矩阵A以将目标椭圆缩放到合适的单位圆。

Define a matrix A to scale the target ellipse to a suitable unit circle.

然后，A，旋转矩阵和A的逆的组合是您的椭圆旋转矩阵。

Then the composition of A, the rotation matrix, and the inverse of A is your elliptical rotation matrix.

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