Professor Ken Brakke

June 10th, 2011

http://www.susqu.edu/news/37931.asp

Professor Ken Brakke is the author of the software Surface Evolver which is a powerful and easy to use system for
computing surfaces with minimum energies, including surface tension, volume, and so on. This software is a practical tool, because it implements many math quantities defined on a simple mesh structure, and uses evolution (gradient descent) to compute the surface mesh in a reliable way. The software is widely used, and the paper he published introducing this software is cited by hundreds, which would be a kind of miracle for a graphics paper.. I don’t know his university well, or I have never heard of it, Susquehanna University in PA.

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Part of the story

May 31st, 2011

Besides my work, I will write regular posts about the computational art of computer graphics, as a tribute for the people in this practical yet interesting field.

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Reading CVs

May 20th, 2011

It is interesting to read the CV of a PhD student named Keenan Crane (http://users.cms.caltech.edu/~keenan/) .
He has good publication records. By looking at his CV, we could see how many diverse projects he has been involved into, from rendering to geometry processing. He is in the world, actively. Why is it so hard for me to do a project in a natural way? I mean, a way that is approachable, rather than being difficult for every step? Maybe I am taking research to be too mysterious. Maybe I should be more open to the community so that I could get into it more quickly. Or, I should use the resources that are available to me better, and get my way to the community. Anyway, I can not go to the schools and companies as he did. I feel like that we do not have the circumstance for doing the stuff. I think discussion is really important. Peers are helping you grow. Yes, in this flat world, get into the community by opening my heart.

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The useful stuff I met when preparing a paper

May 18th, 2011

1. Rendering is important if you are doing graphics. Offline rendering using 3DS Max and other ray tracers (e.g. POV Ray) enables very beautiful and realistic images that show the power of your modeling method. Rendering is always about Light, Camera, Material, Reflection, and Refraction. Online rendering is also interesting. I guess it could be used to produce videos and demos, and the major techniques are basic phong model augmented with shaders.

2. Do enough work to keep your program clean. This is important when trying to measure running time. It also helps you do experiments efficiently.

3. In a cooperation, be active and know your position, take up the duties you should perform. It’s like a soccer match.

4. Liking what you do is always the right state.

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Chapter 2. Simple Math for Describing Surfaces and Minimal Surfaces

October 26th, 2010

Math for describing surface, minimal surface, or even constant mean curvature surface should be the same everywhere. The following is a typical introduction of math for MS which I copied from a paper.

Let M=\{\textrm{\bf{x}}(u^1,u^2)\in R^3, (u^1,u^2) \in \Omega \subset R^2\} be a piece of regular smooth parametric surface. The tangent plane and the unit normal vector at \textrm{\bf{x}} of M are respectively defined as

T_\textrm{\bf{x}}M = span\{\textrm{\bf{x}}_{u^1},\textrm{\bf{x}}_{u^2}\} and \textrm{\bf{n}} = \frac{\textrm{\bf{x}}_{u^1} \times \textrm{\bf{x}}_{u^2}}{ \| \textrm{\bf{x}}_{u^1} \times \textrm{\bf{x}}_{u^2} \| },

where \textrm{\bf{x}}_{u^1} and \textrm{\bf{x}}_{u^2} are the coordinate tangent vectors. The first fundamental form of M is

I = <\textrm{d\bf{x}},\textrm{d\bf{x}} > = g_{11}du^1du^1 + 2g_{12}du^1du^2 + g_{22}du^2du^2

with the coefficients g_{\alpha\beta} = <\textrm{\bf{x}}_{u^{\alpha}},\textrm{\bf{x}}_{u^{\beta}}>,\alpha,\beta=1,2. (So we can see that in the first fundamental form the g stuff is measure of two basis vectors of the tangential space T_{x}M.) The second fundamental form of M is

II = - <\textrm{d\bf{x}},\textrm{d\bf{n}}> = b_{11}du^1du^1 + 2b_{12}du^1du^2 + b_{22}du^2du^2

with the coefficients b_{\alpha\beta}=<\textrm{\bf{x}}_{u^{\alpha}u^{\beta}}, \textrm{\bf{n}}>, \alpha,\beta=1,2. (Unlike the first fundamental form which measures the basis vectors of the tangential space and therefore the metric which I like saying;-) , the second fundamental form measures how the surface is curved as it is twice the directed distance from the tangent plane at the point and a neighboring point.) Set

g=det[g_{\alpha\beta}], [g^{\alpha\beta}] = [g_{\alpha\beta}]^{-1}, b = \det[b_{\alpha\beta}].

Weingarten transformation is a linear map which is defined on the tangent plane at any point \textrm{\bf{x}} of M:

W := T_{\textrm{\bf{x}}}M \rightarrow T_{\textrm{\bf{x}}}M,

and satisfies

W(\textrm{\bf{x}}_{u^1})=-\textrm{\bf{n}}_{u^1} and W(\textrm{\bf{x}}_{u^2})=-\textrm{\bf{n}}_{u^2}.

The linear map can be represented by a 2\times 2 matrix S = [b_{\alpha\beta}][g^{\alpha\beta}]. (Yes, given basis before transformation which are \textrm{\bf{x}}_{u^1} and \textrm{\bf{x}}_{u^2}, and after transformation which are -\textrm{\bf{n}}_{u^1} and -\textrm{\bf{n}}_{u^2}, the linear map can be written in matrix format.) The eigenvalues k_1 and k_2 of S are the principal curvatures of M (Therefore eigenvalues and eigenvectors are characteristics of linear maps in matrix form). Their arithmetic average is the mean curvature H:

H = \frac{k_1+k_2}{2} := \frac{tr(S)}{2}.

Mean curvature normal \textrm{\bf{H}} = H\textrm{\bf{n}}.

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Chapter 1. What is a minimal surface?

September 5th, 2010

I would like to start the story from the Plateau’s problem. Joseph Antoine Ferdinand Plateau (October 14, 1801 – September 15, 1883) was a Belgian physicist who had studied the phenomena of capillary action and surface tension. He conducted extensive studies of soap films. Given a closed boundary wire frame, suspended soap films will contract surface area under the force of surface tension. Plateau said that for any such boundary, there exist soap films that have minimum surface area. Following are some pictures of soap films (Images are taken from the address http://www.funsci.com/fun3_en/exper2/exper2.htm which explains in detail surface tension.).

Films in a cubic frame Cubic bubble
Membranes on a cubic frame. These membranes do not arrange on the faces of the cube, but they are in contact each other. Membranes on a cubic frame. The cubic central bubble has been placed with a straw.
Films in a tetragonal frame Films between two circular frames
Membranes in a pyramidal frame (tetrahedron). Place a bubble in the center. Membranes between two rings and having a film in common.
Tube-like film Helicoidal film
Tube-shaped membrane between two rings. It has been obtained by breaking the film in common. helical film.

Minimal surfaces got its name by having locally minimal surface area while fitting to some fixed boundary curve. Therefore the above soap films are examples of minimal surfaces. And Plateau’s problem actually asks if for a given boundary there always exist minimal surfaces. In mathematics minimal surfaces are defined as surfaces with zero mean curvature. So why zero mean curvature implies that the surface area is locally minimal? Mean curvature k = {k_1 + k_2}/{2}.

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Hello world!

September 4th, 2010

This blog is setup to record or organize materials I know about minimal surfaces. I have little experience of writing a technical log which is also intended to be read by people other than me (group members). Clarity may be the first aim. This is not a technical paper, so we should have an easy time when reading it. But this may lead to some imprecise discounts, which I will try to amend by briefly including formal statements. Hope you find it helpful and enjoy reading the stuff. Thank you!

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