Richard Baker

" Informally, the Laplacian Δf(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f(p)." Please explain this sentence from the Wikipedia entry for laplacian. https://en.wikipedia.org/wiki/Laplace_operator

Wikipedia article for laplacian says that it is how much the average value of the function deviates from the value of the function but this doesn't makes sense to me. Would it be the average value surrounding the point minus the value at the point divided by the area of the surface or the volume confined within? A while ago I realized that the laplacian equals the curl of the contour line of the function, but I forgot my logic. I know that the laplacian is the sum of second-order partial derivatives, but I would like to know what it means geometrically. Thank you.

Thank you Thanks a lot. Busy thinking on this.

After tinkering with the numbers I realized the barycentric interpolation is inefficient. I divide the surface into quads and then I interpolated the lighting function along the gradient of the lighting function. Gradient in this case means the sum of the first order partial derivatives, not the gradient in paint terminology, and I am using Phong-Blinn as my lighting function. But the question remains: What is a good measure of color change per distance or area?

pixel by pixel on my ti inspire tx2 Hybrid between ray-tracing and rasterization. I need a good threshold formula for the computer to use in deciding when to split triangles. Thank you.

I have a collection of triangles on the screen and the color at each point on each triangle is determined by an algorithm and therefore cannot be approximated by barycentric interpolation. I want to render this triangle quickly and accurately. Here is my attempt at a solution: I have the program calculate the screen area of the tringle using a cross-product. Then it calculates the color difference between each point using the Pythagorean theorem treating rgb as coordinates i.e. square root of (red2 -red1)2 +(green2-green1)2+ (blue2-blue1)2 What I need to figure out next is what function I should use as a function of color difference and area. I want the user to define a specific threshold with units. I want the user to define the threshold. The user-defined threshold should be in units of number of color changes per area. If the function is below a certain threshold it will perform barycentric interpolation. If it is above the threshold it will subdivide the triangle in two more triangles and do the test again. It performs this recursively. Thank you for any help you can provide.

Thank you all. One last question. Relating to the case of the hyperbolic paraboloid. I have a paraboloid in the form of a*x2-b*y2= c*z. I have a very messy parametrization for the hyperbolic paraboloid that has many if...then conditions. Is there a simpler way to do parameterizations in terms of u and v?

Those look good. I will check out prices and send him a late birthday present. Thank you for the suggestions.

Suppose I had a triangular mesh that formed a Tyrannosaurus Rex, for example. Would it be possible to divide that model into quadric surfaces or cubic surfaces? I am concerned about the problem of the double hyperboloid or analogous surface of the third degree. That would split up the model and make the t. rex disconnected.

If I added cubic terms and made it a cubic surface, would there be the same limitations? If I added cubic terms and try to match it up with sixteen points, would there be the same limitations?

I will copy and mail what is in those links. Thank you. While he is waiting, he has another question: I have nine points in 3d space and would like to find with the same formula previously discussed the coefficients a-j. I applied Kramer's rule but determinant equals 0. Am I missing something?

Lol-- check your inbox. He has a mini-computer in the form of a TI calculator. Thank you. Will copy and paste this. We may return.

I don't know how relevant the current area of exploration is to this stated interest. But I will relay this. Difficult, but no-one where he is has can answer these questions. And neither can I. So this is the tenth-best solution.

Neither, independent study, preparing for college. He is interested in ray-tracing. He apologizes-- he wrote the formula wrong and I have a hearing loss. This is by telephone. ax2 + bx + cy2 + dy + ez2 + fz + gxy + hxz + iyz + j =0 x,y,z are all functions of u and v He thanks you. So do I.

How do you parametrize an implicit surface? The surface in question is quadratic in terms of x, y, and z. Ie., 0 = ax squared + bx + cy squared + dy + ez + fz + gxy + hxz + uyz +j. I want general parametrization for all arbitrary constants from a-j. Answer should probably involve some combination of trigonometric functions of uv. I am asking this question for my son who does not have internet access. I will copy, paste, print, and send the answer to him. My mathematics background is far short of being able to understand this. Thank you.