title: Rotate normal vector onto axis plane in 4D
body: <p>The problem I need to solve is to rotate a 4-simplex given in 4D on the hyperplane with normal vector (1, 1, 1, 1) so that I can draw it in 3D. For instance I need to know the rotation for the regular one having vertices e_i (that is the coordinate vectors), and all its sub-simplices after division.</p>
  
  <p>In order to understand the problem, let's go one dimension back. If you have a 3-simplex in 3D on the hyperplane with normal vector (1, 1, 1) like here (<a href="http://upload.wikimedia.org/wikipedia/commons/thumb/3/38/2D-simplex.svg/150px-2D-simplex.svg.png" rel="nofollow">http://upload.wikimedia.org/wikipedia/commons/thumb/3/38/2D-simplex.svg/150px-2D-simplex.svg.png</a>), one can follow the idea of Nosredna to the question </p>
  
  <p><strong><em>Rotate normal vector onto axis plane</em></strong></p>
  
  <p>It works fine in 3D, but in 4D there is no cross products, so I cannot extend this answer to my question. On the other hand using rotation matrices I managed to rotate the simplex around the x axes by -45 degree, then rotating around the y axes by around 35 degree (atan(sqrt(2)/2) using the coordinate rotation matrices (http:// upload.wikimedia.org/math/2/8/5/2851c9dc2031127e6dacfb84b96446d8.png).</p>
  
  <p>I also tried to calculate a rotation matrix from axes rotations like in <a href="http://ken-soft.com/2009/01/08/graph4d-rotation4d-project-to-2d/" rel="nofollow">http://ken-soft.com/2009/01/08/graph4d-rotation4d-project-to-2d/</a> but I could not find out what should be the angles to use. So I used R=rotXU*rotYU*rotZU  with the angles pi/4, -atan(sqrt(2)/2, and -pi/6, which looked good, but somehow the result wasn't ok. </p>
  
  <p>Sorry, I could not put the images directly as I'm a newbie...</p>
  
  <p>Thank you for any answer!</p>