# Find a rotary reflection in 3D space between two given tetrahedra

My problem is the following:

Given two congruent tetrahedra (same side lengths but not equally oriented), I have to find the symmetry plane, the axis and the angle of a rotary reflection that goes from the first one to the second.

A rotary reflection is a combination of a reflection and a rotation with rotation axis ortogonal to the reflection plane.

As I don't have any coordinates of the tetrahedra, I have to find those three things using geometrical arguments and show them with help of a geometry software (geogebra), but I am not able to find the correct reflction plane.

• What is the order of operations: reflexion followed by rotation or the converse ? Jan 6, 2017 at 20:08
• Hint: For a pure reflection, the symmetry plane passes through the bisectors of the segments connecting corresponding vertices. Rotating about an axis orthogonal to this plane moves each point in a plane parallel to the plane of symmetry. This doesn’t change a point’s distance from the symmetry plane, so the symmetry plane will still pass through the new bisectors after rotation.
– amd
Jan 7, 2017 at 7:43
• The order of the operations doesn't matter (but I think it's easier to find the correct reflexion first). @amd, I am sorry but I think I don't get the hint, with two corresponding vertices you mean A and A' (If I name the two of them ABCD and A'B'C'D')? (thanks for your help anyways, I am thinking about it). Jan 9, 2017 at 8:52