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

• erielb
In summary, the conversation discusses the mean length and variance for the set of all rays from the apex of an oblique circular cone to points on or within the base circumference. However, in order to calculate these values, a probability measure must be specified, which is not clear in this case. Assuming uniform probability for all points in the base area, the probability of an apex ray of length L through a given point is determined. The length of the planar ray is then established using the law of cosines, and the apex ray length is expressed as the square root of the hypotenuse of a right triangle formed from the length, altitude, and planar ray. Integrations are needed to find the mean and variance, but these may
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?

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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?
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...

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.

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erielb said:
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.

1 person

1. What is the definition of "Mean Ray Length" in the context of an oblique circular cone?

In geometry, the "mean ray length" of an oblique circular cone refers to the average distance from the apex (tip) of the cone to the base, along the surface of the cone. This measure is important in calculating the overall length or surface area of the cone.

2. How is the mean ray length calculated for an oblique circular cone?

To calculate the mean ray length for an oblique circular cone, you need to first find the distance from the apex to the base along the slanted height of the cone. This can be calculated using trigonometric functions. Then, you can use this distance along with the radius of the cone and the height of the cone to find the mean ray length using a formula.

3. What is the difference between "mean ray length" and "slant height" of an oblique circular cone?

The mean ray length and slant height of an oblique circular cone are two different measures. The slant height refers to the distance from the apex to any point on the base, measured along a straight line that is perpendicular to the base. The mean ray length, on the other hand, refers to the average distance from the apex to the base, measured along the surface of the cone.

4. How does the angle of the cone affect the mean ray length?

The angle of the cone does not directly affect the mean ray length. However, it does affect the slant height, which in turn affects the mean ray length. A cone with a smaller angle will have a longer slant height and a larger mean ray length, while a cone with a larger angle will have a shorter slant height and a smaller mean ray length.

5. What is the practical application of understanding the mean ray length of an oblique circular cone?

Understanding the mean ray length of an oblique circular cone is important in various applications, such as in architecture and engineering. It helps in accurately calculating the surface area and volume of the cone, which is essential in designing structures such as buildings, bridges, and tunnels. It is also used in optics and acoustics for determining the path of light and sound waves through a cone-shaped object.

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