Questions tagged [mathematics]

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2
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1answer
55 views

Deriving formula for perspective correct interpolation

I am trying to derive the formula for perspective correct texture interpolation on my own while implementing my own software rasterizer (projecting an arbitrarily rotated triangle in camera space on ...
3
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1answer
132 views

Cubic Splines - Do Parametric and Explicit representation give different curves?

I asked a similar question here before but since the previous post original question was different, I think it was confusing people. So I've voted to close that and asking the new question here. I'm ...
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1answer
57 views

Help understanding tangent from dot product and max distance from component wise vector multiplication

I am looking through this code and seeing two things which confuse me (well, the whole functions does) in the top functions. First, dir * p where ...
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1answer
88 views

How compute new camera parameters given a velocity vector?

My goal is to update camera parameters given a velocity vector so that the camera points in the direction of the velocity vector. How should one compute the update matrix for the camera parameters?
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0answers
65 views

Fake cubic Hermite spline interpolation with smoothstep

When scaling an image with Bicubic Interpolation, the Cubic Hermite spline interpolation is used. smoothstep is one of the four basis/blend functions of this kind ...
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0answers
36 views

Bottom up (as opposed to top down) text on projective geometric algebra?

I have been reading the SIGGRAPH course notes on projective geometric Algebra, and I get some of what's going on. However a lot of the knowledge needed to understand these PGA notes seems to be based ...
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1answer
37 views

Computing heat diffusion creates weird results

I am trying to model heat diffusion on the surface of a mesh. I annexed the most important bits of theory about this topic as screenshots on the question see the bottom. The crux of the issue is we ...
3
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0answers
38 views

Is this way of transforming QMC samples into barycentric tri coordinates agnostic to mesh-topology?

While I'll try my best to give all relevant info in all possible brevity below, please refer to the spoiler and link at the bottom of the post for the (more lengthy) original description if needed. ...
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2answers
377 views

understanding SDF[Signed Distance Functions] for a torus

This topic would contain a lot of sub questions because I have one for each shape but for today I just want to understand the function for a torus and a capped torus This is the article for all shapes ...
2
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1answer
33 views

Understanding inequality to equality in the Hartmann & Gribbs frustum planes extraction method

I've been reading this article on how to extract the plane equations from the view and projection matrices. And I can understand most of it. However, the only thing that is unclear to me (and that ...
3
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0answers
262 views

How to implement PCSS correctly?

I'm trying to implement PCSS in OpenGL/GLSL, but I have problems understanding the details especially the conversions between the coordinate spaces. The existing implementations are so different from ...
0
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1answer
18 views

Plotting points along a 3D line segment

I'm trying to plot some points along a 3D line segment and could really use some help. In 2D, I've found success getting the angle of the line, the sine & cosine of the angle, and then adding the ...
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0answers
37 views

How do I figure rotation from the difference between two vectors?

I've been playing around with an idea, but I'm struggling with some of the math involved. Now, each face of a mesh has a normal vector, and that the normal map fragment shader modifies that face ...
0
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1answer
35 views

Moving a vertex through the cursor

There is a way to move a selected point in a mesh with a cursor(assuming a camera that doesn;t change between frames). The way I remember the algorithm (but seems to be wrong) is: Unproject the ...
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1answer
49 views

Calculating Material/Texture Placement from Bitmap to 3D Model

everyone! I am currently developing a program that converts a 3D model to 2D isometric pixel art. Part of my program extracts the texture from an FBX file (as FBX files can have textures built in), ...
2
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1answer
44 views

Determining the function of a radial gradient fill with midpoint

Recently I have been trying to create a shader that imitates a shader within Affinity Photo. It is used in the radial gradient fill and essentially you set your endpoint colors and then set the ...
0
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1answer
89 views

Stylized math like rendering?

How can I render plots and graphics that mimic how mathematicians draw diagrams? For example Look at this shirt: The diagrams represent 2D and 3D shapes, however they have characteristics of human ...
1
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1answer
35 views

Subdivision scheme where the faces and edges have weights (not necessarily scalar weights)

Subdivision schemes work by considering the vertices and their connectivity information to calculate averaging weights. However, other than specifying which vertices are connected, and perhaps which ...
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0answers
91 views

Model View Projection Matrix Multiplcation Order

I'm working on a simple software renderer and have a working implementation so far. I'm curious as to why it's actually working since I would expect the multiplication ordering for my world, view and ...
0
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1answer
110 views

Calculate the position and rotation of a quad in 3d space given a 2d projection of that quad from a camera

I am trying to build a VR tracking system with a laptop webcam, and I have succeeded in identifying, and tracking paper markers I put in front of my webcam. For context I am using OpenCV with the ...
2
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1answer
124 views

Determining Rational Quadratic Bezier Curve Weights for Circle

I am trying to create a spherical interpolation with 3 points. I'm currently using Quadratic Bezier Interpolation but have been told I should use Rational Quadratic Bezier Curve in order to get a ...
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0answers
47 views

Explanation of converting a look-at / direction vector to quaternion rotation?

Can someone please explain the math behind converting a direction-vector or lookAt-vector into a quaternion rotation? What I am trying to do? I am trying to create a custom aim-constraint for a DCC (...
0
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1answer
35 views

In place sorting of a half edge DS?

Cross posting from SO because I didn;t know where to put this question. I have an implementation of the half edge and I am trying to sort the edges such that edge n+1 is the pair of edge n. Setup I ...
2
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1answer
95 views

Improve accuracy of mesh gradient?

So to make it simple. I currently use a method I found online to compute the gradient of a scalar field of a mesh. To test how accurate this is, I made a sphere and followed the gradient direction of ...
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0answers
23 views

Finding a flat plane within iPhone depth data

I have real time depth data coming from an iPhone and I want to identify flat objects within the point cloud so that I can highlight the Hard edges in the images. The objects can be at any angle and I'...
3
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1answer
53 views

Mis understanding of the Heat Method

I have been trying to understand a paper in CG for a while, called the Heat Method by Ken Many things have clicked but I don't fully understand it yet. In particular. In the following $u$ is a vector ...
4
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1answer
75 views

Gradient descent (Not ML) on arbitrary meshes

So I am doing a gradient descent like algorithm on the surface of a mesh and I just noticed something: The above is the geodesic gradient (the distance to a single vertex) Look at where the ear ...
1
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2answers
105 views

Discrete Gradient?

I am trying to understand how to get the discrete gradient of a mesh that is being used as the input of some function $f$. In other words for every vertex $v$ there is a scalar quantity $s$ associated ...
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2answers
74 views

Heat Method (Crane et Al) How do we pick u?

The heat method is a very interesting paper for distance computation: https://www.cs.cmu.edu/~kmcrane/Projects/HeatMethod/paperCACM.pdf The idea behind the paper is that, heat travels along the ...
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1answer
57 views

Is the laplacian operator for meshes just the sum of the differences of neighbouring vertices?

I am starting to learn about the laplacian operator $\Delta = \nabla \cdot \nabla\phi(p)$ Which can be described as the divergence of the gradient of a scalar function $\phi$. This is equivalent to ...
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0answers
44 views

Are there algorithms besides Canny edge detection that can create consistent width edges?

Sobel edge detection is pretty much the quintessential way to get edges out of an image. It however suffers from certain quirks. One example is, because it's gradient, based certain surfaces exhibit ...
1
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1answer
78 views

Approximating Geodesics in a half edge DS, how can I refine my mesh to get good approximations

I implemented Djikstra's shortest path algorithm to approximate Geodesics on arbitrary meshes. Djikstra's works, but I noticed a problem inherent to the discretization of my meshes. Consider the ...
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1answer
42 views

Fast and exact Geodesics on meshes, Backtracking confusion

The following is an excerpt from a 2005 paper on geodesics on triangular meshes, taken from section 3.5 In this case $p$ is a point on some arbitrary face in a mesh, $p'$ is a point on one of the 3 ...
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1answer
54 views

Finding geodesics on a mesh?

What are some state of the art methods/algorithms to find geodesics on arbitrary manifold meshes?
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0answers
80 views

Why does my self-written rendering engine make further away objects look larger?

I am writing a very simple rendering engine. I have already made a few tests, but somehow the images it creates look wrong. Objects that are further away from the camera look larger than objects ...
0
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1answer
42 views

Closest sphere on segment

I am trying to get a segment closest point on a surface and display a circle (or sphere in this case) at that position. I actually get the correct position but I can't get a uniform sphere based on ...
2
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1answer
41 views

Interpreting Image Processing Math Equation

I am reading "Image Processing and Analysis" by Chan and Shen, c 2005 SIAM. They introduce some notation I'm not 100% sure how to interpret: $$ u_0(x)=u(x)+n(x), x=(x_1,x_2) \in \Omega $$ They ...
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0answers
44 views

What are some uses for the vector triple product in computer graphics?

I have just learned about the vector triple product (Lagrange formula) from a math for computer graphics textbook, however the book does not mention any use cases for it beyond the fact that it's a ...
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0answers
79 views

What is the projection matrix of reverse (Byzantine) perspective?

I would like to construct a projection matrix for reverse perspective. I'm using OpenGL and tried to modify concepts from this excelent tutorial. I came up with: $$ \begin{bmatrix} 2\frac{(near-M)}{...
1
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1answer
120 views

Prove radiance contributions of deeper paths fade away

When ray tracing, I find it intuitively clear that deeper paths have a lower contribution to the overal picture than shorter paths. This is the main reason why it is generally okay to render a picture ...
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0answers
53 views

Mathematics in Computer Graphics

Can you please recommend some topic for research work related to mathematics in CG? (Maybe something like "Affine transformations in CG")
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1answer
67 views

How do you compute the winding number of a closed poly curve?

Pretty much the title, given a closed curve in 2D, defined by a set of points, and a point. What's the algorithm to calculate the winding number of that curve, point pair?
0
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1answer
71 views

Wrong reflection

Here is a youtube video showing my problem: youtube.com/watch?v=y8J7VhS2pkM I think I'm either doing the texture lookup on the reflection texture wrong, or the reflection or view matrix is somehow ...
3
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0answers
131 views

Procedural Generation of 4D Shapes

I'm curious if anyone has any insight into how one might procedurally generate 4D objects, as showcased in Miegakure (or the developer's other game, 4D Toys ). I built a program a while back to do ...
1
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1answer
232 views

How to handle Half edge boundary edge iteration from vertex?

Ok so let's say we have a portion of a mesh that looks somewhat like this: A triangle fan with one of the triangles removed. If that triangles was not removed, we could get all the edges adjacent to ...
2
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1answer
115 views

How do people come up with subdivision schemes?

Be it chaikin subdivision, loop subdivision, catmull-clark subdivision... How do people come up with the coefficients for an arbitrary subdivision scheme?
4
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1answer
63 views

Understanding distinct vs. not distinct knots in B-splines

In reading the wikipedia page about B-Splines https://en.wikipedia.org/wiki/B-spline, I am getting confused about the concept of knots. If we have a spline of order $n$, with a knot vector ($t_0, \...
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0answers
258 views

Why does Schlick's approximation contain a \$(1-\cos\theta)^5\$ term?

The approximation writes the reflection coefficient as$$R(\theta)=R_0+(1-R_0)(1-\cos\theta)^5,\,R_0=\left(\frac{n_1-n_2}{n_1+n_2}\right)^2.$$Why is the exponent 5? Schlick 1994 introduces this ...
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1answer
248 views

Reading thesis on cone tracing, did the author make a mistake?

I am reading this document that talks about signed distance field rendering. In that document section 6.3, on the topic of cone tracing, it says: Cone tracing is an extension of the sphere tracing ...
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1answer
205 views

Confusion around Lambert's Cosine Law in Ray Tracing in One Weekend

Peter Shirely's Ray Tracing in One Weekend presents the following function to calculate the color of a lambertian material (See chapter's 7 and 8): ...