I am implementing a conical filter for Gupta-Sproull anti-aliased line algorithm.
Given a cone with the total volume of 1 and a radius of 1. Find the subvolume of the intersection of a line. The line has a size of 1 as well. I need the formula so I can generate a table of values. I could find the total volume of the area under the cone. But then I need to cutout the subvolume of where the line intersects with the cone. I need a formula for this subvolume in terms of height so I can plug in the perpendicular distance from the center (this is what changes depending on where the line is drawn) and output the height.
The height will determine the intensity of the pixel because the height decreases as the pixel moves farther away from the center.
So I guess I first need to find the area of that intersection of the line and the circle base….then I can find the volume. How do I find this area since the edges of the circle are arced shaped? I suppose I need to follow the edge of the circle and find out the change in slope of the line between the edges of the line but I do not know how to do it.
The line is flat and only intersects at the base. It marks or cuts out the subsection of the cone.
Best Answer
It looks like you cut the base of the cone with a chord. Then you want to know what the volume of the portion of the cone that lies above one of the base partitions is (doesn't really matter which if you know the volume of the cone).
If the height of an arbitrary point on the upper surface of the cone is h and the maximum of the upper surface of the cone is $H$ then:
$h=f(\sqrt{x^{2}+y^{2}})=H-a\sqrt{x^{2}+y^{2}}$
where a is the gradient at which the cone descends. When $\sqrt{x^{2}+y^{2}} = R$ , where $R$ is the radius of the base of the cone, h equals zero. So, $a = \frac{H}{R}$ , and:
$h = H\left(1 - \frac{\sqrt{x^{2}+y^{2}}}{R}\right)$
The problem should be isotropic so nothing about the line should matter except the minimum distance between the line and the center of the cone's base ($d$ in your second diagram above). To get the volume of the smaller partition integrate:
$\int\limits_{-\sqrt{R^{2}-d^{2}}}^\sqrt{R^{2}-d^{2}} \int\limits_d^\sqrt{R^{2}-x^{2}} H(1-\frac{\sqrt{x^{2}+y^{2}}}{R}) \,dy \,dx$
Sorry about the ugly notation. It's my first post and I don't know how to mark equations up here (figured my answer would get stale if I looked it up first).
Edit: AH! $\LaTeX$!