Omni 3D project
300R: Mathematics of 3D image synthesis
A tetrahedron problem
Office: SciCtr 434
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
Oliver Knill | Department of Mathematics | Harvard Summer School | Harvard University
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