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Is there a way to generate k points $x_1,\dots,x_k$ such that $\min_{ i \neq j } d(x_i,x_j) $ is as large as possible? The distance between any two points, $d(x_i,x_j)$, is defined as the minimum path from $x_i$ to $x_j$ via the cone's lateral surface.
Is there further a way to calculate the minimum distance based on the number of points and the cone dimensions?

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A cone is a developable surface. If you cut it open along a ray from the apex, you get a flat sector. Place a pattern of points on this unfolding, being aware that there are two images of the cut ray. Something like this:


            ConePts
You could use a hexagonal pattern instead of the rectangular grid I show above. Then it is like packing pennies on the cone.

See also this question: Packing disks on a cone.

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