How to check/decide/segment whether a point is inside a 3d cone?

Question

How do I know which points are inside the cone, for example for the following bunny data?

library("onion")
library("rgl")

#bunny data
data(bunny)

#cone
pts <- cbind(c(0, 0), c(0, 0.23), c(0, 0))# the centers
radii <- c(0.0, 0.15)
cone <- cylinder3d(pts, radii, sides = 20)

#visualize
points3d(bunny, col="pink", alpha = 0.3)
shade3d(cone, col = "lightblue", alpha = 0.9)

Answer 1

This is more of a geometry question than a programming question. For any given point in the x, y, z space, the points where x^2 + z^2 < (0.15/0.23 * y)^2 will lie inside the cone you have defined, so you can do:

data(bunny)

pts <- cbind(c(0, 0), c(0, 0.23), c(0, 0))
radii <- c(0.0, 0.15)
cone <- cylinder3d(pts, radii, sides = 100)

cone_bunny <- bunny[bunny[,1]^2 + bunny[,3]^2 < (0.15/0.23 * bunny[,2])^2,]

points3d(cone_bunny, col="pink", alpha = 0.5)
shade3d(cone, col = "lightblue", alpha = 0.3)

I'm not sure if you were looking for a more general solution for intersecting 3D shapes - that is far more difficult, but many simple geometric intersections can be calculated from straightforward algebra.

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Addendum

It was pointed out in the comments that you may not want an actual cone, but rather a 20-sided pyramid as shown in your example. This is also amenable to a mathematical solution:

cone <- cylinder3d(pts, radii, sides = 20) 

lims <- (0.15/0.23 * bunny[,2])^2/cos(atan2(bunny[,3], bunny[,1]) %% 
         (2*pi)/20 - pi/20)

cone_bunny <- bunny[bunny[,1]^2 + bunny[,3]^2 < lims,]

points3d(cone_bunny, col="pink", alpha = 0.5)
shade3d(cone, col = "lightblue", alpha = 0.3)