Questions tagged [mathematics]

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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
31 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?
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1answer
56 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 ...
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22 views

GIMP's Additional blending formula

I try to implement GIMP's the Additional blending mode for layers in my application. I've found the formula: E = min((M+I), 255) I think, that if I blend ...
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77 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 ...
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1answer
51 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 ...
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1answer
86 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?
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1answer
46 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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54 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
58 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
87 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): ...
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60 views

GPU Procedural placement - Creating evenly distributed sampling points

I am working on a procedural placement system on the GPU, inspired by the work done by Guerrilla Games in Horizon Zero Dawn: https://www.guerrilla-games.com/read/gpu-based-procedural-placement-in-...
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42 views

When should operations on 4D (homogeneous/projective) vectors use the w-coordinate?

Various operations on 2D vectors naturally generalise to 3D vectors simply by including the $z$-coordinate. The ones that come to mind are: Vector sum and difference Dot product Magnitude (Euclidean ...
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1answer
62 views

Shader or formula that distorts inward

I am looking for a shader (or at least the right terminology) for crushing an image. It should be something very similar to barrel distortion, but instead of the center crushing outward to the edges, ...
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1answer
40 views

Mapping sample to a disk

So I have been reading this wonderful book on ray tracing by Kevin Suffern, Ray tracing from Ground Up. I am reading this book after finishing Peter Shirley's first two book on ray tracing & ...
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2answers
63 views

Visualize the interpolated unit quaternion on the surface of the unit sphere

My major is mechanical, please forgive me for asking questions that may seem trivial to you. First, I'm reading the following paper: Dam, Erik B., Martin Koch, and Martin Lillholm. Quaternions, ...
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70 views

Why does the reflected component in Lambertian path tracing get multiplied by a factor of 2?

I would like to intuitively understand the Path Tracing equation/formula/pseudocode, but I'm not fully getting the last return statement. As per Wikipedia page: <...
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46 views

What library or tool was used to plot these graphs?

I've been wanting to plot graphs that look like these (Taken from this blog post), but I can't figure out what tool, language, or library was used to pull this off.
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119 views

SDF collision detection

I was wondering if there are any techniques for calculating the intersection point/s between two or more distance fields.
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1answer
90 views

Camera Implementation without Limitations

I've developed a simple (Third-Person) Perspective Camera with Orbit controls. However, my implementation doesn't handle the case where if the pitch goes over 90º it flips upside down. From what I've ...
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1answer
56 views

Estimating the position of vertexes in a 3D model

Introduction to my project: We are machine learning scientists and working on a biomedical system for tracking tongue for speech pathology using a technology called EMA (Electromagnetic ...
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1answer
81 views

Want to study computer graphics

I want to study computer graphics but I don't have a strong foundation in mathematics. I am doing a computer science degree that doesn't cover computer vision, image processing or computer graphics, ...
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0answers
14 views

How can I detect if an edge has flipped during the UV unfold process?

I'm writing my own UV Unfold operation (similar to Maya's Optimize UV tool or 3DS Max's Relax Tool). My code works great 90% of the time but in some instances, I'm getting faces and edges that flip ...
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0answers
112 views

Help with understanding the look-at function (the view matrix)

I've been reading this article about the look-at function. I don't understand the part when they're trying to inverse the matrix N. They say that R is orthogonal because all its row and column ...
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25 views

How can I create multivariable function plotting software

I have finished Dennis Ritchie's C book, and I would like to create my own plotting software for enhancing my programming skill,I want sources,books or anything to reach this goal.Thanks for answers.
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36 views

Algorithm for randomly generating polyhedrons?

I want to randomly generate polyhedrons which need not be convex. What is the most efficient way for doing so? I can think of an algorithm for generating random convex polyhedrons, but am unable to ...
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2answers
133 views

How to calculate interpolation between multiple skeleton poses?

I found out that I should use spherical linear interpolation to blend two poses of a joint. I cannot find what to do if I have hundreds of poses to blend except applying the SLERP iteratively. What ...
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0answers
158 views

Pre-filtered environment map, deriving the equation

I'm reading through this article, and more specifically I'm trying to derive the equation that would explain the implementation the following shader (still in the same article): ...
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0answers
47 views

NURBS Weights De Boor's Algorithm

I am having trouble implementing an algorithm for 3rd degree NURBS. I have been able to program 2nd degree ones from the equations described here but I am not able to derive an equation for Rational B-...
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0answers
70 views

Constraining a Polyline to Look Smooth

What are the mathematical constraints required for a polyline, i.e., a piecewise linear curve, to appear as a smooth curve to human vision? I've discovered one document that specifies only that the ...
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3answers
1k views

How to calculate ray

In ray-tracing technique critical point is to calculate rays which came out from eye $E$ to target $T$ through pixel $P_{ij}$ on viewport. The "viewport" is represented as rectangle divided to square ...
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0answers
99 views

How can you avoid thin triangles in a parametric shape in a smart way?

I have created a closed 3D mesh using a parametric function. The shape is constructed by taking an arbitrary closed b-spline and scaling it based on one of the parameters (similar idea to a parametric ...
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1answer
331 views

Translate an arcball camera upwards or left-right

This problem is giving me a really hard time. I'm always looking at the origin and have a camera that moves around it in an arcball mode. That part works beautifully: I'm calculating the camera ...
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1answer
271 views

How to match Blenders polynomial lens distortion model with GLSL?

I'm trying to re-implement Blenders (libmv) polynomial lens distortion model using shadertoy, in order to apply the lens distortion in another application ideally by generating a STMap for that. I ...
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1answer
148 views

What is the equivalent of rotation matrix in Euler angle space for Quaternions?

I have to load a mesh in Blender (which uses Z and Y as up and forward axes), modify it and export it (with setting up and ...
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1answer
66 views

How to rotate a shape in another 3D software so that the rotation matches with the rotation in Blender?

I import a mesh in Blender, export it with setting axis_forward='-Z', axis_up='Y', import in NVIDIA's FleX, rotate it and store it on disk; I call this mesh ...
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1answer
121 views

Wrong tangential directions for cubic bezier curve

I am well aware that there are several ways to find the tangents to a Bezier curve. However, I choose to just substitute the result of differentiating the Bezier curve equation. So, given four ...
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1answer
6k views

How to convert Euler angles to Quaternions and get the same Euler angles back from Quaternions?

I am rotating n 3D shape using Euler angles in the order of XYZ meaning that the object is first rotated along the X axis, then Y...
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1answer
103 views

How to randomly draw Quaternions within a specific range of Euler angles for rotation?

I am pretty new to Quaternions so please bear with me. I want to draw random Quaternion samples so that their Euler angle equivalent would range within [-30, +30] ...
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1answer
74 views

How are the three plane equations derived in the NVIDIA's CG Tutorial chapter 8?

I'm reading the chapter 8 of the Cg tutorial and I could not understand 8.4.1 : Because all these coordinates lie in the plane of the same triangle, it is possible to derive plane equations for x, ...
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0answers
167 views

Ray tracing implicit surfaces using Sturm's theorem

I am trying to render implicit surfaces given by a polynomial equation $F(x,y,z) = 0$ using Sturm's theorem. Plugging the parametric ray equation $r(t) = o + td$, where $o\in\mathbb{R}^3$ is the ray ...
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1answer
123 views

How to simulate elastic material with restoring shape

I'd like to simulate a rubber-like material. This material if deformed would experience a restoring force to its original shape, a bit like a school rubber ereaser. I tried with a spring-mass system ...
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1answer
62 views

How different can the number of pixels in a straight line be to its real length?

I'm working on a game with a programmable robot. It uses a laser range finder to get the distance to walls. I'm using DDA to generate a line from the robot, going in the direction it is pointing, ...
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1answer
141 views

BRDF ray directions

The BRDF takes two directions, $\omega_{o}$ and $\omega_{i}$. I've heard that both directions should point away from the surface (one towards the camera, one towards the light), but iq's Oren-Nayar ...
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1answer
45 views

Linear independence of three points

What is the best way to prove if three points are linear dependent? This is my current way to do it: $$\mathbf{v_a} = \mathbf{b}-\mathbf{a} \\ \mathbf{v_b} = \mathbf{c}-\mathbf{a} \\ \mathbf{v_c} = \...
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1answer
176 views

How to compute volume of intersection of non-axis-aligned cuboids in 3D?

Given two cuboids in 3D space (8 vertex coordinates each) that are arbitrarily oriented, how can we find the volume of overlap between them in the fastest manner?
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2answers
584 views

Geometric interpretation of this bilinear interpolation equation?

Bilinear interpolation can be achieved by interpolating two values across the x axis, and then interpolating between the results across the y axis, like in the image below (from https://blog.demofox....
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1answer
545 views

GLSL shapes signed distance field implementation explanation?

I'm trying to understand all primitives In this article.After hard work I just learn Sphere and Cube , I wrote description here. I could draw 2D shapes but I have trouble understanding 3D below codes:...
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1answer
147 views

Can someone explain this formula for parse RGB to HSL?

Im trying to complain the algorithm to calculate the HSL value form a RGB color. I understand the algorithm for the Hue, but im lost with the Lightness and saturation. Someone can explain why? There ...
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1answer
547 views

Delta angle of two vectors about positive x axis

How do I calculate angle C? I can only know what angle between vector and x positive vector {1, 0} Angle between green and red is 60 degrees. Angle between blue ...