So let assume that I have some convex smooth and unclosed surface. I'm moving each point of it in a normal direction by some constant factor (This factor is same for all points on surface).

Can I replace this operation by Uniform or Non-uniform Scaling + Translate?

Will resulting surfaces be mathematically identical in these cases?

For example, I want to transform this surface (side view):

enter image description here


1 Answer 1


No this cannot be modelled by (non-uniform) scaling. It's fairly easy to construct a counterexample:

enter image description here

The issue is that the amount a section of the curve/surface grows depends on its curvature, not its orientation in space. Notice here that the circular arc grows uniformly in all directions (by a factor of $3/2$) whereas the length of the horizontal segments remains unchanged at a length of $2$.

Of course, if your surface is not only convex but also has constant curvature, then it's just a circular arc, and for circles your transformation is equivalent to uniform scaling. You can probably also construct curves of varying curvature where your transformation happens to correspond to non-uniform scaling, but for general convex surfaces, that's not the case.


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