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

<p><a href="https://i.sstatic.net/CWujH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/CWujH.png" alt="enter image description here"></a></p>

<p>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 $2$.</p>

<p>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.</p>

CC BY-SA 3.0
2016-03-07T08:23:15.437Z
2016-03-07T10:48:23.870Z
2016-03-07T10:48:23.870Z
Martin Ender
<p>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).</p>

<p>Can I replace this operation by Uniform or Non-uniform Scaling + Translate? </p>

<p>Will resulting surfaces be mathematically identical in these cases?</p>

<p>For example, I want to transform this surface (side view):</p>

<p><a href="https://i.sstatic.net/JuAlC.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JuAlC.png" alt="enter image description here"></a></p>

<transformations><mathematics>
Moving each point of a surface in direction of corresponding normal
Ibraim Ganiev
# Moving each point of a surface in direction of corresponding normal

(Asked by: username_0 on 2016-03-07)

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](https://i.sstatic.net/JuAlC.png)](https://i.sstatic.net/JuAlC.png)

**Tags:** transformations, mathematics


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## 1 answers

### Answer by username_1 on 2016-03-07. Score: 7 | Accepted answer

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

[![enter image description here](https://i.sstatic.net/CWujH.png)](https://i.sstatic.net/CWujH.png)

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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