Experimental teaching of harmonic conjugate and circular point in projective geometry

Abstract

Harmonic conjugate is one of the most important and basic theories in projective geometry. According to the theory of projective transformation, the vanishing point's coordinates can be obtained, after that, combined with the corollary of Laguerre theorem: the infinity points of which the two mutually perpendicular lines and their circular points harmonic conjugate, the image coordinates of the circular points can be obtained. In the experiment, using the positive tri-prism as the experimental object and according to the same primal and the cross-ratio invariance, the coordinates of infinity points can be obtained. Then, according to the corollary of Laguerre theorem, the corresponding coordinates of the circular points can be got. Established the equations which constraint on the intrinsic parameters, then the intrinsic parameters can be solved. So, this theory application can be verified in computer vision.