Omni 3D project

300R: Mathematics of 3D image synthesis

A tetrahedron problem

by Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

Fact:
The structure of motion problem for 2 oriented
omnidirectional cameras and two points in space
is solvable uniquely if and only if the 4 points are not coplanar. Mathematical reformulation:
We look for a tetrahedron ABCD in space for which the directions
a=AC,b=AD,c=BC,d=BD are known. Is the tetrahedron uniquely determined
by these directions?
The answer is yes, if the four points are not coplanar, no, if the four
points are coplanar. Proof: If the four points are not coplanar, the plane generated by a,b and the
plane generated by c,d are not parallel and intersect in a line. Lets call it the
hinge line. Chose a point on that
line and call it D. Draw the line in the cirection of c and chose a point B on that line.
Now the rest is determined. The line in the direction d from B hits the hinge line in a
point D. Attach the other directions to get the point A. If the four points are coplanar, one can chose any pai of points A,D, attach the directions o AC,AD at A and BD,CD at D to get a quadrilateral. These quadrilaterals are not all similar. |

Questions and comments to knill@math.harvard.edu