Mapping points on a cylinder to points on a cone.

asked 2021-04-20 21:04:07 +0200

Buk gravatar image

updated 2021-04-20 23:27:29 +0200

slelievre gravatar image

My question has two parts.

1) How to map (not project) 3D coordinates from the surface of a cylinder to the surface of a sphere.

In this image the red squares are projections, the blue square is a (one of many possible) mapping. Contrasting projection with mapping

2) How to do this when the cylinder and cone are not concentric; and when their axes are not in the same plane.

By way of better explanation, this view represents a cutting tool (brown; middle left) rotating around its axis (black double-dashed; right) cutting the sides (orange; upper left) into the workpiece blank (green; left). (The yellow shadow shows the resulting gear after 30 cuts and indexing.)

Close up view showing the problem

This one shows the same thing from the top and side to emphasise the non-orthogonal relationship between the cutting plane and the workpiece. Top and front views establishing context

Given that I have a set of 3D points that represent the boundary of the cutting tool, I want to mathematically sweep those points around the axis through the workpiece and then map them into the coordinate space of the workpiece.

I've found a paper (12MB pdf that I believe develops the equations to do this, but my math education was long ago and never covered the Differential geometry used in that paper; and is therefore not strong enough to understand enough to be sure.

I would like help to verify that it does what I think it does; and if so, in working out which of the equations therein I need to program to apply the required transformation(s).

I realise that this problem is ill-defined mathematically as presented; and it is asking a lot for people to read a 16 page paper; but any help offered would be most gratefully received.

(I'm a retired engineer and a competent programmer and this project is a purely personal one with no commercial affiliations.)

Thanks Buk.

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