About quadrilateral (geometrical problem)

[pizzicato]

Hello,
I have the following problem in geometry:
Given a rectangle A, B, C, D consisting of segments AB, BC, CD, DA.
At one time I apply vertical displacements at the vertices A, B, C and D perpendicular to the plane of the original rectangle.
I have a new quadrilateral A1, B1, C1, D1 in space.
Is it possible to have a formula allowing me to calculate the new diagonals A1, B1 and C1, D1 based solely on the following data:
- The initial distances AB0, BC0, CD0, DA0
- Angles (A1B1, AB), (B1C1, BC) (C1D1, CD), (D1A1, DA).

Thanks a lot.

 

[andrewkirk]

I think the easiest way to solve this will be 3D Cartesian geometry - ie using coordinates.

Without loss of generality you can set A=(0,0,0), B=(AB0,0,0), C= (AB0,BC0,0), D=(0,BC0,0), and A1=A. The last one is valid because shifting the whole set of points A1...D1 up or down by the same amount doesn't change the distances between any of those points, including the 'diagonals'.

Then just use trigonometry in triangle ABB1 with the length AB0 and angle (A1B1,AB) to get the height (z coordinate) of B1. You can then work around the other points C1, D1, getting their z coordinates. Then you just use the Cartesian formula for distance between points to get the 'diagonals'.

BTW I would call the new shape a tetrahedron, rather than a quadrilateral, because it is unlikely to be planar. ALso, the inputs CD0 and DA0 are unnecessary since we are given that the original shape is a rectangle.