Questions tagged [mathematics]
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160
questions
2
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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?
1
vote
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 ...
0
votes
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 ...
0
votes
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
votes
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.
...
0
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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
votes
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
votes
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
vote
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 ...
0
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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
votes
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 ...
1
vote
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
votes
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
votes
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
vote
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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
vote
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 ...
0
votes
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 ...
1
vote
1answer
54 views
Finding geodesics on a mesh?
What are some state of the art methods/algorithms to find geodesics on arbitrary manifold meshes?
1
vote
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
votes
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
votes
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 ...
0
votes
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 ...
1
vote
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
vote
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 ...
0
votes
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")
0
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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, \...
1
vote
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 ...
0
votes
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 ...
1
vote
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):
...
