  # Richard Baker

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1. He is eager to be able to interact directly. I will see what can be done to make communication easier. Glad you responded. I will be gone for a good part of today.
2. The way I was taught to conceptualize it on scratchapixel.com the focus point is behind the screen plane so it was probably awkward wording on my part because some people conceptualize the focus point in front of the screen plane. My question still stands; in the case of the interpolating quadric with two intersection points how do I determine which intersection point is a valid interpolation of the implicit surface. Thanks.
3. I am using bi-quadratic interpolation but the coefficients of the interpolating quadric surface are weighted according to the position in screen-space. I know it seems unorthodox but it seems like a real viable method. If anybody has ever ray-traced a sphere (a quadric surface) they would know that no meshes are required; you simply find the ray-sphere intersection using the quadratic formula. My solution didn't work for weeding out the false hits. What I currently do is calculate the local Lipschitz constant and divide the value of f at the surface of the quadric where the intersection of the ray occurs. Then subtract c, and divide that difference by the Lipschitz constant. This doesn't work because it is overzealous; it eliminates a lot of true hits. Thank you for showing interest in my problem. Specific language is difficult, my scribe is not perfect .
4. OK. I got it figured out. Would be easy with a larger computer, but I am limited to my TI super-calculator.
5. Let me rephrase my question: I think the problem is the hyperboloid of two sheets. How do I determine whether the interpolant quadric surface is on the same sheet as the implicit surface I am interpolating?
6. I have an implicit surface given by s(x,y,z)=c where c is a constant. I take a square let's say 16 pixels high by 16 pixels wide Each of the four corners of the square, I do this: Send a ray passing through the pixel into the scene Use Harts sphere-tracing algorithm to determine the point on the ray that is closest to the level set f=c Now instead of applying Phong Blinn I just calculate the first order partial derivatives fx, fy and fz but I also calculate the second order partial derivatives fxx, fyy, fzz, fxy, fyz and fzx. I use theses derivatives to calculate the coefficients of a quadric surface that is essentially a three-variable Taylor series truncated to the first ten terms. Now, for all the pixels inside the square 256 pixels I calculate the weighted average bilinearly interpolated values of the coefficients. The point for doing all this is that quadric surfaces are easy to ray-trace and an analytic solution exists using the quadratic formula. This is way faster than using Moller-Trombore on thousands of triangles. But there are two solutions to the quadratic formula and sometimes the other solution pops up as a false hit resulting in random specks of surface in space. My question is how do I weed out these false hits? Thank you PS I have already implemented third-order approximation and used Cardano's formula.
7. (PDF) Sphere Tracing: A Geometric Method for the Antialiased Ray Tracing of Implicit Surfaces (researchgate.net) On page 5-113 of this article, it says that the distance to the implicit blend is greater than 2/3 times f(x) times the sum of radiuses but it also says the Lipschitz constant of a single contributing sphere equals 3/(2r). It also says that the Lipschitz constant of a sum is bounded by the sum of the Lipschitz constants. Is this an error? The inverse of the sum of the Lipschitz constants does not equal the sum of the inverse of the Lipschitz constants. My question is what is the Lipschitz constant of the level set of a metablob? Thank you.
8. Correction on third line: outermost cone working inward. Also the radius of each concentric circle is tan of θ . Is that correct?
9. I am working on a TI Inspire video game, in which the player is in a dark room holding a flashlight. The flashlight is located at the player's eye and is pointed in the look direction. I render the light by coloring in the conic sections formed by the intersection of each wall, floor, or ceiling and concentric cones of light starting from the innermost cone working outward. As it turns out whenever the conic section formed is an ellipse, it appears to transform into a perfect circle located at the dead center of the viewport after performing the perspective rotations and z divide. This appears to always be the case regardless of the relative position and orientation of the wall. floor, or ceiling. I am not certain if this is the case for hyperbolic conic sections. My question is what shape will the conic section appear to have when the conic section is a hyperbola? Thank you.
10. Thanks for the response. No ray-tracing this time, actually. Right now I am working with a platform that has limited resources. And I decided that for fast, live graphics processing it would be advantageous to render concentric polygons in the shape of level sets of light intensity starting with the outermost level sets and working my way inwards drawing over the inner part of each polygon not unlike the painter's algorithm. In theory these polygons approximate the ellipse, parabola, or hyperbola. The light from the spotlight forms a cone. The surface the spotlight shines upon is a plane, the intersection of a cone and a plane is one of the three conic sections. Hence, what I need here is the formula for each curve. I think the eccentricity equals cosine (angle of spotlight) divided by (unit vector from origin to center of spotlight dot product the normal vector of the surface). But I need to find the focus and the semi-lattice rectum (pardon my Latin, LOL). I have to cut corners here for increased performance so for now I am ignoring the effects of distance and dot product of the ray vector and unit normal. If it is possible to split or deform these conic sections so that it would approximate the effects that would be great. But right now I am having trouble finding the proper shape parameters for the conic sections.
11. I decided that the effects of the distance and the cosine of the angle are negligible so I can just trace out a conic section but I am having trouble for the hyperbolic and parabolic cases. In particular, what is the maximum and the minimum angle for the hyperbola or parabola in terms of the variables already stated? Thank you.
12. Hello again. I am trying to determine the level sets of light intensity due to a spotlight cast on a flat surface. I am only taking into account the diffused lighting and no specular and no reflections for simplicity. I figure the light intensity due to the spotlight equals the following: (normal vector to the plane) dot product (unit vector from origin to the point in question) times (ray intensity function) times (ray dispersion function). The ray intensity function is a function of the angle between the center of the spotlight to the point in question. The ray dispersion function is a function of the distance from the origin to the point in question. For example a point light that emits light equally in all directions the ray intensity function is a constant and the ray dispersion function follows the inverse square law. What I want to do is find rif and rdf such that the level sets of light intensity are an elementary function such as an ellipse or a hyperbola so that I can construct the shape of the spotlight. Thank you
13. Bump. Instead of interpolating the normal factors I could interpolate a function of the normal vectors such as diffuse and specular lighting which is still a function of second-order partial derivatives but the question is still the same. If I have a property that is a function of the u, v parameters of a surface, and I also have the rate of change of the function I am trying to interpolate then how do I factor in the rate of change at the vertices of a triangle as well as the properties of the triangle at the vertices?
14. Bump. Please help poster doesn't have access to internet. Tried to interpolate u and v values then . - multiplied the change in u and v by the gradient of the normal vector with respect to u and v. But it didn't work.
15. I already know that if I have a triangle with vertices v0, v1, v2 in property p0, p1, p2 then the property linearly interpolated at location l = a*p0 + b*p1+(1-a-b)*p2 a=(area of triangle formed by the points v1, v2 and l) / (area of the triangle formed by points v0, v1, v2) b= (area of triangle formed by the points v0, v2, and l) / (area of the triangle formed by points v0, v1, v2) My question is how do I generalize this to a higher order interpolation method? In my problem the property p = the normal vector to a surface and I have already calculated the Gaussian curvature of the surface to decide whether to subdivide the triangle into more triangles, but I don't want the second-order partial derivatives to go to waste.
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