Mean ray length from apex to base of an oblique circular cone

[erielb]

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?

 

[Erland]

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?

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]

Assuming uniform probability for all points in base area A, the associated probability of an apex ray of length L through a given point in A is P(dA) = 1/A x r x dr x dw where r is the radius from the center of base to the specified point and w is the corresponding central azimuthal angle (reckoned from the diameter constructed from the point where altitude h intercepts the circumference). Establish the length c(R,r,w) of the planar ray from the base of the altitude to the point in question via law of cosines say and express apex ray length L(h,R,r,w) as the SQRT of the hypotenuse of the right triangle formed from L, h, & c. Integrate L(h,R,r,w)x P(dA) over r & w from r==0 to r= R, and w=0 t0 2(pi) respectively for mean and variance accordingly. My difficulty is with the integrations of the resulting expressions.

 

[Erland]

My difficulty is with the integrations of the resulting expressions.

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.