title: Precise, linear interpolation between two 3D planes
body: <p>I have two planes <code>P1</code> and <code>P2</code> specified as normal-distance, or 4D vectors.  I also have a parameter <code>t</code>. I need to construct a derived plane <code>P</code> which is coincident with <code>P1</code> for <code>t = 0</code>, coincident with <code>P2</code> for <code>t = 1</code>, and in-between for <code>t = (0, 1)</code>. In the end <code>P</code> should also be represented as a 4D-vector.</p>
  
  <p>Intuitively <code>P</code> should be constructed such that <code>P1</code>, <code>P2</code> and <code>P</code> intersect at the same line, and the angle P1&angle;P = (P1&angle;P2)<em>t</em>.</p>
  
  <p>Is there an efficient math for this? Can this be solved precisely?</p>
  
  <h3>Clarification</h3>
  
  <p>Linear interpolation of the angle between planes is actually a requirement. In other words the normal N<sub>P</sub> must linearly rotate between N<sub>P1</sub> and N<sub>P2</sub> through the shortest arc.</p>