Questions tagged [mathematics]

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24
votes
1answer
819 views

Is my perspective math correct?

I've got a homework in which I have to calculate and plot some points using a pespective transformation, but I'm not sure my results are correct, since the 3d plot using Camera coordinates looks very ...
13
votes
3answers
3k views

Correct Specular Term of the Cook-Torrance / Torrance-Sparrow Model

For a while I have been doing some research on the topic of Physically Based Rendering. One reflection model that is mentioned over and over is the Cook-Torrance / Torrance-Sparrow model. It seems ...
13
votes
1answer
114 views

Thorough, accessible material about Fourier basis functions and Spherical Harmonics?

Spherical harmonics appear in several computer graphics techniques. I feel that in order to be a better computer graphics developer, I need to have a deep understanding of what they are and how they ...
11
votes
0answers
623 views

How correct is rendering in XYZ / linear RGB space?

I have a spectral power distribution (SPD, 5 nm steps) for all light sources in the scene and SPDs for the reflectance of all surfaces under any light / viewer angle in question. I'm going to ...
9
votes
2answers
2k views

What makes a good permutation table?

I'm implementing improved Perlin noise. Its key feature for randomisation is the hardcoded permutation table, which gives essentially random but reproducible gradients at the cells of the grid. The ...
8
votes
3answers
1k views

Does premultiplied alpha give order independent transparency?

I've heard that pre-multiplied alpha gives you order independent transparency but when I sit down and do the math, it doesn't seem to be working. Is that untrue, or am I doing something incorrectly? ...
8
votes
1answer
272 views

Moving each point of a surface in direction of corresponding normal

So let assume that I have some convex smooth and unclosed surface. I'm moving each point of it in a normal direction by some constant factor (This factor is same for all points on surface). Can I ...
8
votes
1answer
149 views

2D sampling with multidimensional transformations

I'm currently learning mathemetical concepts of distribution and the way to use them in a ray tracer with the book "Physically Based Rendering". Let's start by uniformly sampling an hemisphere: As ...
8
votes
1answer
210 views

Find the longest straight line between two points on surface of polygon

My shape is a slightly concave polygon, and I'd like to know the maximal diameter. I imagine a straight line between two points on the surface of the polygon, such that the line does not pass outside ...
7
votes
2answers
7k views

How to achieve gimbal lock with Euler angles?

I was studying about Euler angles and I came across the problem gimbal lock which can occur by using Euler angles. In gimbal lock we lose one axis of rotation. But how? And when does it happen? On ...
7
votes
2answers
1k views

How am I able to perform perspective projection without a near plane?

I'm learning about 3D computer graphics, but I'm having a hard time understanding why the near plane of a viewing frustum can not be placed at z position $0$ (right at the camera). I can understand ...
7
votes
1answer
467 views

How to invert an affine matrix with small values?

I'm using an affine transformation matrix to transform 2D coordinates from screen (magnitude 10e3) to small parts of fractal sets (magnitude as little as 10e-15). I also need to map the other way ...
7
votes
2answers
1k views

Estimating real world object distance from photo

I'm not really sure if this is possible but it should be. I currently have a photo of a sphere, whose radius I've measures physically. Knowing the apparent size of the sphere from the photo and all ...
7
votes
1answer
2k views

Understanding Jump Flooding Algorithm (JFA) for Voronoi Diagrams

I'm having trouble understanding the JFA. As far as I understood the algorithm, it walks log(n) times through every pixel (no matter if it is a seed or not) and looks at that pixel's neighbors in $(x+...
6
votes
1answer
296 views

Logarithmic spiral with equal vertex spacing, what equations?

I need to draw a logarithmic spiral (or close approximation) whose vertices are equally spaced, such that the lines between any two consecutive vertices are of equal length. (Actually, that spiral is ...
6
votes
1answer
1k views

Microfacet shading for diffuse materials

I recently replaced the Lambertian BRDF in my path-tracer with Oren-Nayar, under the assumption that I could adjust it to use the GGX distribution model with appropriate masking/shadowing. PBR ...
6
votes
1answer
756 views

What does it *mean* to scale in an arbitrary direction?

My graphics text explains rotating along the cardinal axes, then generalizes the math to show how to make a rotation matrix for rotating around an arbitrary vector. That all makes sense and I can both ...
6
votes
2answers
667 views

Rotate line around center

I have to use a propriertary graphics-engine for drawing a line. I can rotate the whole drawing by its origin point (P1). What I want, is to rotate it around its center point(M). So basically that it ...
6
votes
1answer
375 views

Where does the cosine factor comes from in the GGX PDF?

The GGX NDF, as it appears on the paper where it is presented is: $$D(m)=\frac{\alpha_g^2\space\chi^+(m\cdot n)}{\pi\cos^4(\theta_m)(\alpha_g^2+\tan^2(\theta_m))^2}$$ It is equivalent, in the range $...
6
votes
2answers
1k views

OpenGL with SFML, create an n-pointed star?

Completely stuck on how to do this. Yes, it is a homework problem, but its the first we've had in the class so not too familiar with graphics. Also, seeing as its homework, please don't just give the ...
6
votes
1answer
122 views

Monte Carlo Integration

Let $I$ = the value of integration and $p$ = probability distribution. The estimator is denoted as $\left\langle I \right >$ and is $$\left\langle I \right >=\frac{1}{N}\sum_{i=1}^{N}\frac{f(...
5
votes
2answers
203 views

Why is $pdf(\Psi) = \frac{1}{2\pi}$ when picking from hemisphere uniformly?

I'm trying to figure out why the probability density function for picking a direction uniformly on the hemisphere is $\frac{1}{2\pi}$. Something tells me this is related to the number of steradians ...
5
votes
1answer
219 views

Sine-based Tiled Procedural Bump

I am trying to implement this formula to generate bump but I am facing some issue. The result doesn't look the same it's much darker. Here is my result (without same parameters) but it is much darker ...
5
votes
1answer
698 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:...
5
votes
1answer
92 views

The variance for the Monte Carlo estimator

I was wondering about the variance for Monte Carlo. The book (Advanced Global Illumination 2ed, p.60) writes the equation as follows: $\sigma^2 = \frac{1}{N} \int(\frac{f(x)}{p(x)} - I)^2p(x)dx$ $\...
5
votes
1answer
89 views

Best way to group 3d points into planes, if any

If I have some 3d points, how can I discover parallel planes within them (not just arbitrary planes). I want to know how many "levels" or "floors" something can travel within, after it has done so, ...
5
votes
1answer
135 views

Correct post-displacement normal calculation (Y-component and epsilon)

I often use a 2D Perlin noise texture to displace a grid in the vertex shader (e.g. terrain, ocean). However, every time I google the method to calculate post-displacement normals there is a slight ...
5
votes
1answer
2k views

Draw angles lines in raster graphics using bresenham line algorithm

I at the moment trying to draw some angled lines using bresenham line algorithm which can circulate a 2d array of size 21x21, as a line angled from 0 - 2pi. So the idea is that the program has to ...
5
votes
0answers
178 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 ...
4
votes
1answer
132 views

How does a rigged model work?

What black magic is behind this? Starting from a pile of polygons, a skeleton, and an initial conformation, how is a new conformation turned into a new pile of polygons?
4
votes
1answer
57 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, \...
4
votes
1answer
80 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, ...
4
votes
2answers
69 views

Modulo vs Scaling vs Capping when performing filters

I'm currently taking a graphics course and we've recently covered cross-correlation with regards to using a filter matrix that is applied to a region of pixels in a sort of continual raster-type scan ...
4
votes
2answers
131 views

Correctness of logical steps for create the matrix of viewing transformation

In general, I understand the creation of viewing transformation matrix, but I have my doubts. For example, let's say that a world coordinate (called $A$) has a center at $(1,1,1)$, and the view ...
4
votes
1answer
45 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 ...
4
votes
1answer
209 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 ...
4
votes
2answers
490 views

What is a Gaussian Lobe concerning BRDF and NDF?

I was reading a paper about voxel cone tracing and came across this term. So, what is a Gaussian Lobe or a Lobe in general? I am asking in the context of Computer Graphics, BRDF and NDF.
4
votes
1answer
152 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] ...
4
votes
1answer
260 views

Meaning of the parameter in a line-plane intersection in the space

The scenery is the following: I have a polygon in the space, defined (the polygon) from your vertices. By 3 vertices of the polygon (non-aligned) revenue equation of the plane containing the polygon. ...
4
votes
2answers
143 views

intersection between line segments - narrowed precondition

I believe that the issue may already have been discussed here. I want to find if two line segments does intersect and if they do then find and store and intesection points. Now it is already confirmed ...
4
votes
1answer
174 views

Calculate Transformation between two point sets - but with constrained Degrees of Freedom

How can I calculate the rigid transformation [R|t] between two 3d triangles, but restricted to a given N degrees of freedom (for N = 1..6) ? I know for N=6 I can get a least-squares solution via SVD ...
4
votes
3answers
102 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, ...
4
votes
0answers
165 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): ...
4
votes
0answers
54 views

Making surface with common parametrization from bunch of other surfaces

I have bunch of surfaces, like in these two images: Currently to make single surface with common parametrization I sample points uniformly from all source surfaces, and fit single NURBS surface from ...
3
votes
1answer
309 views

Generate this kind of 2d burst or pulse algorithm?

I'm currently trying to develop a game using Cocos2d-x using C++. I really need to generate this burst or (donut-shaped) pulse. I'm having a difficult time how to generate such shape. At first, I was ...
3
votes
1answer
251 views

Calculate a rotation around an arbitrary axis

Suppose I have two 3D triangles before & after a single rotation around an unknown axis. How would I go about finding this axis?
3
votes
3answers
84 views

Does the blending matrix change between calculating various curve segments in a uniform cubic B-splines approximation?

I would like to ask about uniform (periodic) cubic B-splines (approximation, no interpolation). $$B=1/6\begin{bmatrix}-1&3&-3&1\\3&-6&3&0\\-3&0&3&0\\1&4&1&...
3
votes
1answer
606 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 ...
3
votes
2answers
102 views

Why is the approximation valid, in the formula provided by Brian Karis?

I'm sorry I'm bringing this topic up again, but I need to expound some of the topics. With reference to this question, I was wondering if someone can help me out in expanding some of the details. In ...
3
votes
1answer
617 views

Object rotating around origin instead of object center?

I am implementing simple rotation but the object is not rotating around the local center instead it is rotating around world origin. This is the code: ...