The discussion focuses on calculating the mean ray length from the apex to points on or within the base circumference of an oblique circular cone with altitude h and base radius R. The probability measure is specified as uniform across the base area A, leading to the probability expression P(dA) = 1/A x r x dr x dw, where r is the radius from the center to the specified point and w is the azimuthal angle. The apex ray length L(h,R,r,w) is derived using the law of cosines and involves integrating over the specified ranges for r and w to obtain mean and variance. The participants conclude that the integrations are complex and likely do not yield a closed form, suggesting numerical integration as a practical solution.
PREREQUISITESMathematicians, physicists, and engineers involved in geometric modeling, particularly those working with oblique circular cones and numerical integration techniques.
For this question to meaningful, one must specify a probability measure. It is not clear to me how such a measure should be specified in this case. There seems to be several choices...erielb said:Consider an oblique circular cone of altitude h, base radius R, with apex directly above a point on the base circumference. What is the mean length (& variance) for the set of all rays from the apex to points on or within the base circumference?
Yes, very diffucult integrations indeed. I would guess that this cannot be expressed in closed form (unless thare are some special functions which show up here). If the problem comes from a practical situation, I would recommend numerical integration.erielb said:My difficulty is with the integrations of the resulting expressions.