Plane detector: information provided is convex hull and pose (transl + rot). Are the "extents" the scaling?

Question

While calibrating a system that includes the automatic detection of planes from the SLAM output, the information about planes provided is:

1 - (x,y) pairs representing the plane's convex hull

2- the "pose", defined as a 3D translation and a quaternion.

3- the "extents", two real valued numbers.

The question is how to recover the 3D position of this plane from this data.

Based on observed data, the convex hull appears to be scaled from -1 to 1 in both dimensions, so my best guess would be that the "extents" are each dimension's scale. I haven't found the word "extent" particularly tied to the concept of spatial scale in Computer Graphics, so any help would be appreciated.

The proposed algorithm for reconstruction is:

• Build the interior of the convex hull in a 2-dim plane (e.g. $(x,y,0)$) bounded by $[-1,1]^3$

• Scale it by $(extent_0, extent_1,1)$

• Rotate it according to the quaternion, and translate it according to the 3D vec.

Answer 1

Extents is a somewhat archaic term for geometric length, or more properly a set of boundaries that define the space between them. I also haven't seen it used in Computer Graphics literature, but it shows up in Geometry.

Your plan seems sound, but take care of doing the rotation and translation in the correct order because they are not commutative. It's not clear to me here what the correct order ought to be, but there will be a right and wrong way.