8
$\begingroup$

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

$\endgroup$
7
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.