Peculiar shape surface/volume

[Shadow]

Hey all,

 

I'm curious as to how to calculate the surface/volume of a shape I though of the other day. Since I don't know if what it's called (heck, I don't even know if it exists) and I'm not even close to being good enough with Photoshop to make a picture, I'll have to describe it, which may prove a little tricky. So, here we go.

 

It's three dimensional. Imagine you have a circle with center [math]S_1[/math] of radius [math]r_1[/math] in space. Now imagine on either side of it, an n-sided convex regular polygon with a center [math]S_2[/math] and diameter (that is, the distance between the center and the place where two edges meet) [math]r_2[/math]. The distance between [math]S_1[/math] and [math]S_2[/math] is [math]d[/math]. Now here comes the complicated part; connect the circle with the polygons in such a way that the shape will transform from a circle to the polygon the closer you get to the polygon. The circle will sort of blend into a polygon. I can't think of a better way to explain. If it helps you, if the polygons were circles instead, you'd get a "double" right circular cone with it's tip missing.

 

Anyway, in the rare case you understood the above, how would you calculate the surface/volume? I'm sure you'd need calculus, but that's about as far as I got. That's not to say that at any time I kidded myself into thinking I could solve this on my own.

 

Cheers,

 

Gabe

 

PS.: I know I only use one center for two polygons, but the same thing applies to both of them, they're just on opposite sides of the circle.

[D H]

Since you "you'd get a double right circular cone with it's tip missing" (better: a right frustum of a right circular cone), I am going to assume that

 

a. The planes containing the circle and the polygon are parallel.

b. The line connecting the centers of the circle and polygon is perpendicular to the circle and polygon.

 

You haven't adequately defined the shape. So, let me take a hack at defining it for you. The definition is constructive.

 

0. Label the vertices of your n-gon.

 

1. Circumscribe a regular n-gon around the circle, label these vertices, and orient this new polygon so that corresponding vertices of the original n-gon and the circumscribed n-gon "line up" (i.e., lines connecting corresponding vertices are either parallel to one another or meet at a point). Note that this new n-gon will have a radius of [math]r_n = r_1/\cos(\pi/n)[/math].

 

2. Develop a function [math]f(x)[/math] with [math]f(0)=0[/math], [math]f(d)=1[/math], and [math]0<f(x)<1[/math] for all intermediate values of x. The linear mapping [math]f(x)=x/d[/math] will do nicely.

 

3. Construct n-gons parallel to and co-aligned with the two end n-gons along the line segment connecting the two end n-gons. These n-gons will change in size from that of the original n-gon to that of the circumscribed n-gon according to the distance [math]x[/math] between the centers of the original and intermediate n-gons: [math]r_p(x) = r_2 - f(x)\cdot(r_2-r_n)[/math].

 

4. For each of these intermediate n-gons, construct a circle with radius of [math]r_c(x) = f(x)*r_1[/math] inside the intermediate n-gon. This circle will have a radius of [math]r_c(x) = f(x)*r_1[/math]. Use this circle to round the vertices of the intermediate n-gon.

 

6. Note that the original n-gon is not changed and the new n-gon becomes the circle. At intermediate points you will get ever more rounded polygons the closer you get to the circle.

 

 

With this, you should be able to compute the perimeter of each rounded n-gon as a function of distance from the original n-gon. From there calculating the surface area or volume is simple calculus.