Question

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,

Answer 1

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

Answer 2

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,