Calculate vertical enclosing pitch angles of a box in a coordinate system given center position c, orientation (as a quaternion) q and dimensions l, w, h of the box

Question

Given a coordinate system (which could sit at the center of a laser sensor) and a box with dimensions l, w, h and position of center c⃗ (cx, cy, cz) and its orientation as a quaternion q (qx, qy, qz, qw) in that coordinate system I want to find the minimal enclosing pitch angles (from the perspective of the coordinate system) of a box βmax and βmin

Illustration of the problem

https://i.stack.imgur.com/HfdHl.png

(that means the pitch angles of pitch planes that touch the highest projected point and the lowest projected point - from the perspective of the coordinate system in the laser sensor).

This question seems to be very tricky, because how do I know which is the "highest and lowest point" from the perspective of the coordinate system? It obviously depends on the orientation and and the dimensions of the box: it could be the edge of the box or just the corner, but I have no mathematical way of expressing it.

What sort of mathematics is involved here? Can you guys point me to books and topics I need to study in order to solve this on my own? Is this Projective Geometry?

Answer 1

This is a good question. You can linear programming to determine the "highest and lowest" points.