Inscribed angle

Abstract: The Coulomb law of the Mohr theory of failure predicts a constant dihedral angle between conjugate failure planes regardless of the magnitude of the stress difference required for failure. The maximum principal stress bisects this angle. Experimental studies show that this angle actually becomes smaller as confining pressure is decreased. For low pressures, Nadai, enlarging on a suggestion by Leon, proposed a curved Mohr envelope which would intersect the normal stress axis at a right angle. If so, there could then be a critical stress circle tangent to the envelope only at the apex. The single plane of failure, lying parallel to the maximum principal stress, is a tension joint in the usual terminology (extension fracture). A slight increase in stress difference will move the critical circle off the apex, and it will then intersect the envelope at two points; the resulting conjugate fractures will enclose a small dihedral angle. Within a small range of increasing stress differences, the angle will increas...

Abstract: A novel vision-based method for the circle pose determination is addressed. This method is based on two particular projected chords of a circle image. The first one is the projection of a circle chord which subtends the largest apex angle of the viewing cone for the circle image and the second one is the projection of a circle diameter whose back-projection plane bisects the above largest apex angle. This method is conceptually simple, since the circle center is the center point of the diameter chord and the circle orientation is given by the cross product of these two (directed) chords. We present theorems on the geometry of both the viewing cone and a re-projected circle image which are essential to the pose determination. We then give a pose determination method which is applicable under all viewing conditions. Experimental results illustrate the good performance of the method, when compared with other existing methods.