In the last post we found a formula for calculating the distance of a point from a plane using the edges of a triangular pyramid which are parallel to the coordinate axes. Determining the edges of the pyramid if the plane and point are known can be found without too much difficulty. We start with a given plane and point in some coordinate system and translate the origin of the coordinate system to that of the point.
In the new coordinate system we can determine the length of the edges as follows.
To verify that the formula works we can work a simple example by selecting an arbitrary plane and a point and then translate the origin to the point. To find two unit vectors in the plane we determine the eigenvectors of the projection operator for e.
The result for the triangular pyramid formula agrees with distance determined by using data from the original coordinate system.