# Directional Distortion of a Surface

Hi everyone,

I am facing a math road block.

I have two surfaces (3D) described by two functions $f_1$ and $f_2$ (known). I would like to create some sort of directional distortion along the loading direction. See the image below. (Original, full-resolution, rotated version here.)

[The upside-down handwritten text says Distorted "forward" and Distorted "backward".—JOR]

It "protrudes" along the loading direction and becomes "flatter" in the opposite direction. Can that be done? I have the feeling that some kind of interpolation between the shapes of the original surfaces $f_1$ and $f_2$ can do the job, but the interpolation must be done directionally, i.e., it will depend on the loading.

I am not sure how this distortion can be formulated mathematically, and I would love to have your suggestions. Many Thanks,

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@Denise: I took the liberty of including your image, rotated and scaled. Although it seems no rotation is ideal. Apologies if I distorted your intention. – Joseph O'Rourke Aug 23 '11 at 1:28
Thanks a lot Joseph, I really appreciate it! – Denise Aug 23 '11 at 1:31
Could you clarify "along the loading direction"? This loading direction is independent of $f_1$ and $f_2$? It seems your question is not yet precisely specified. What is the input? $f_1$, $f_2$, the loading direction $v$, some magnitude of $v$, ... ? – Joseph O'Rourke Aug 23 '11 at 1:47
Correct, we have a closed-form formula for both f1 and f2. A deformation along a specific vector u is applied (know as well). Can the resulting surface be expressed as a combination of f1 and f2? Also, the resulting surface must remain convex and its area should not vary enormously (meaning if one face is protruding, the opposite one should flatten) – Denise Aug 23 '11 at 2:45
– Will Jagy Aug 23 '11 at 2:52

## 2 Answers

Here is a model to think about. Tweak as needed. I am assuming for sake of my understanding that there is some symmetry about the $xy$ plane, namely there are surfaces $f_1$ and $f_2$ with $0 < f_1 < f_2$ at almost all points in the $xy$ plane (more formally, the numerical relation $0 < f_1(x,y) < f_2(x,y)$ holds for all $(x,y)$ in an open set in the $xy$ plane) and there are also surfaces $-f_1$ and $-f_2$ and what is desired is two surfaces $f_3$ and $f_4$ that satisfy among other conditions $f_1 < f_3 < f_2$ and $-f_1 < f_4 < 0$.

Try a straw model where the volume between $f_1$ and $-f_1$ is made up of parallel straws packed together in the desired direction $u$. If you want volume preservation, push on the straws in the direction u; push so that neighboring straws move close to the same amount. Mathematically, this is reparameterizing the surfaces involved so that the $z$ axis is replaced by the $u$ axis, and then adding a distortion $d$ to the reoriented $f_1$ and $-f_1$.

For an area preserving distortion, try something similar, except shrink or stretch the straws as needed. Instead of $d$ added to both reparameterized surfaces $f_1$ and $-f_1$, you will need to compute the difference in surface areas between $f_1$ and $d+f_1$, and borrow that difference from $-f_1$ somehow. As a start, if an area element gets increased by $b$%, find $c$ to shrink the area element on the other end of the straw so that the net change in the sum of the two areas is zero.

Another model is the soap film model, where $f_1$ is like a soap film with a variety of pressures acting on it. However, this leads to minimal surfaces and/or odd metrics, and is way out of my comfort zone. Perhaps a differential geometer can tell you what an appropriate transformation would be for this kind of model.

Gerhard "Ask Me About System Design" Paseman, 2011.08.22

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EDIT (David White): This is really a comment to Gerhard's answer, not an answer in and of itself.

Thank you for your input Gerhard. A small clarification: what is desired is not two surfaces $f_3$ and $f_4$, but just one function that would undergo directional distortion and that would be some interpolation of $f_1$ and $f_2$ (to ensure tangentiality as $f_1$ grows closer to $f_2$).

Regarding the straw model. Are you using some discrete description (particles)? Where each point would undergo a different displacement $u$? If so, we can generate a surface $f_3$ but we loose the closed-form formula for it, am I correct?

Thanks again,

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I would be happy to discuss this by email, as I think the answer-comment form may not be proper for me to relate my understandings and misunderstandings. You can click on my name after this comment to get my user page and directions for contacting me. Briefly, the straw model is both a recipe for doing a discrete version and a tool for an alternative view. If closed form is important to you, you might investigate one-parameter families f1(p)+ g(t,p)(f2(p) - f1(p)), where f1 and f2 are rewritten using u as the vertical. Gerhard "Ask Me About System Design" Paseman, 2011.08.23 – Gerhard Paseman Aug 23 '11 at 18:28
Also, because of the symmetry of the picture, I am rephrasing it as I view it. If f2 were a sphere and f1 an ellipsoid inside the sphere, with u a vector like (1,0,7); I would reparameterize the system so that I could express the surfaces in a new coordinate system where u was the vertical axis, and I would split the surfaces in half so that I would compute the half where f1 was being pushed toward f2, and consider separately the case where f1 was being pushed away from f2. (I like surfaces that are graphs of functions.) Gerhard "Yes, Call Me Old School" Paseman, 2011.08.23 – Gerhard Paseman Aug 23 '11 at 18:42
Thanks for your help Gerhard – Jeremy Aug 24 '11 at 2:51
Happy to help you Jeremy. Also happy to help Denise too. Point of information, you get better service on MathOverflow if you behave like a single entity. If you prefer to be known as 'Denise and Jeremy', I suppose that is acceptable too. Good luck with your approximation. Gerhard "The Many Sides of Me" Paseman, 2011.08.23 – Gerhard Paseman Aug 24 '11 at 3:12