lightxbulb
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Fundamental drawbacks of rasterization over ray tracing
8 votes

Rasterization is based on the idea of projecting various primitives (e.g. triangles, line segments, points, quads, maybe even some curved surfaces like bezier patches) on the screen and then ...

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Rendering equation in terms of paths rather than directions
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8 votes

This is a very good question. There is a common misconception that Monte Carlo, or integration is applied "recursively" on the rendering equation. That is not what's happening. Numerical integration ...

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Difference betwen Rendering Equation, Lighting model, Ray Tracing, Global Illumination and Shadows?
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7 votes

The rendering equation aims to describe what the light distribution for a specific scene is, under several assumptions. The most important assumption is that we are working in a geometrical optics ...

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Why wasn't CryTek SSAO multiplied by 2?
5 votes

Ambient occlusion is the direct illumination at a Lambertian surface point due to a homogeneous light source at infinity. That means that you assume a light source with $L_e(\pm\infty, \omega_i) = 1$, ...

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How to compute the following integral over a polygon?
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5 votes

Triangulate the Voronoi cell then write the integral as a sum over the triangles: $$\int_{\Omega}\|P - Pi\|\,dP = \sum_{k=1}^{N}\int_{\Delta_k}\|P-P_i\|\,dP.$$ Write the integration over the ...

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$(x, y, 1)$ is 2D homogenous coordinates or 3D homogenous coordinates?
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4 votes

If you have $(x,y,z) \in \mathbb{R}^3$ and you relate it to $(x/z, y/z) \in \mathbb{R}^2$ then you have interpreted $(x,y,z)$ as one possible representation of the 2D vector $(x/z, y/z)$ in ...

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How to Sample 3D Points Outside and Inside the Mesh Surface
4 votes

I have a few suggestions: Partition the volume by using tetrahedra within the mesh, and outside of it (in the cube). Set the probability to sample each tetrahedron to its volume divided by the inside/...

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Probability density for explicit light sampling
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4 votes

It comes from the relation between the area formulation and the solid angle formulation of the rendering equation: $d\omega = \frac{\cos\theta_L}{r^2}dA$: https://arxiv.org/abs/1205.4447

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How can I transform an ellipse into a circle?
4 votes

You can use a shear, or equivalently a rotation by 45, scale along one of the axes, rotate back (which is equivalent to the shear which you are looking for). EDIT: On second thought, you don't need ...

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Do straight lines always remain straight when projected with a perspective camera?
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4 votes

"A not so simple approach". I may have messed a little bit too much with grouping the terms, do forgive my elementary math skills, it's a side effect of using tools like wolfram and mathematica too ...

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Strange stripes on the gradient
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4 votes

The 'jumps' to 'a bit darker' are an optical illusion due to how human perception works. Check out Mach bands in wikipedia. Now as for why you get a step function even though you have a smooth ...

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Why is the symbol for solid angle a small omega in the definition of the BRDF?
3 votes

The $\omega$ is a direction. Whether you parametrise this direction in spherical coordinates $(\phi, \theta)$, in Cartesian coordinates $(x,y,z)$, or some other coordinate system is irrelevant. Thus ...

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Dynamic Ray-Triangle Intersection
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3 votes

There are 2 ways to go about intersecting the triangle. Let the vertices of the triangle have positions $v_1, v_2, v_3$. Let the ray have origin $o$ and direction $d$. Let the model (4x4) matrix be $M$...

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Intrepret path/light tracing by rewriting light transport equation
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3 votes

Having the rendering equation one can rewrite it in operator form: $$L = L_e + TL$$ $$(I-T)L = L_e$$ $$L = (I-T)^{-1}L_e$$ $$L = \sum_{k=0}^{\infty}T^kL_e$$ The last equality holds if $T$ is a ...

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Why isn't rasterization combined with raytracing more often?
3 votes

Rasterization is sometimes used for primary rays. However, it limits greatly what you can do - depth of field, motion blur, participating media, refraction, only basic camera models, no adaptive or ...

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Multiple Importance Sampling in Path tracer produces Dark Images
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3 votes

Throughout my answer I'll sometimes refer to some results in https://sites.fas.harvard.edu/~cs278/papers/veach.pdf by using [MIS,section_number]. You can skip the following derivation if you don't ...

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When sampling direct light, what to do if testing ray been blocked by transparent object?
2 votes

If a ray intersect a transparent object (regardless of the order) multiply by the colour to get proper attenuation, then continue. Note that the order does not matter. The other options is to ignore ...

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What is exactly the third component in homogeneous coordinate system?
2 votes

The relationship between standard coordinates $(x,y)$ and homogeneous coordinates $(X,Y,Z)$ is $x = X / Z, y = Y/Z$. Homogeneous coordinates are a type of projective coordinates. All points on the ...

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Screen Space anti aliasing?
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2 votes

The easiest way to deal with this would be to provide thickness for the edges in the continuous setting. That is, make your edges out of solid capsules/cylinders, then you would not have this issue. ...

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

It is cancelled out by the probability density function in the estimator. The pdf in their case is exactly: $\frac{\cos\theta}{\pi}$, which is in the denominator: att = albedo * cos_theta / pdf = ...

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Inverted Normals in Raytracer
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2 votes

The issue is, that you considered that the rays coming from the camera to be the light carrying rays. Instead, the rays bounce off the surface, and return to the light source (which you have decided ...

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Douglas-Peuker and equal distances
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2 votes

I would consider the curvature in that case too. If the curvature is small - then it is a flat region - so you can safely remove it - your 4th point for example. If the curvature is large (your 5th ...

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Why cubic curves provide the minimum curvature interpolants?
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2 votes

For a function $y = f(x)$ the (signed) curvature at $x$ is given by: $$ \kappa(x) = \frac{f''(x)}{(1+f'^2(x))^{\frac{3}{2}}} $$ If you assume that the slope is very small compared to $1$: $ f'^2<\!...

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What is the purpose of the projected solid angle $dA^\perp$ term in the definition of Radiance?
2 votes

Since you seem to want an explanation in terms of irradiance, consider both the radiance and irradiance definitions: $$E = \frac{d\Phi}{dA}, \quad L = \frac{d^2\Phi}{d\omega dA^{\perp}}$$ We can ...

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Why is it easier to inverse transform every object in a scene than to transform a camera?
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2 votes

That's actually incorrect. You can transform every ray of your camera if you wish (and numerous implementations do so). There are some advantages and disadvantages to each method (e.g. if your rays ...

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Identify plane of symmetry in 3D mesh
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2 votes

You can look into this: https://cescg.org/wp-content/uploads/2017/03/Dvo%C5%99%C3%A1k-Estimating-Approximate-Plane-of-Symmetry-of-3D-Triangle-Meshes.pdf There are definitely other papers on the ...

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smallpt: ray scattering and Importance Sampling
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2 votes

Let me first address some misconceptions you have: the theory states that we need to shoot x number of rays for each intersection No, the "theory" doesn't state such a thing. Note also that the ...

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Simple Two Point Perspective of a Cube
2 votes

Your first image employs an orthographics projection, while the second uses a perspective projection. You can look up the perspective matrix derivation in: http://www.songho.ca/opengl/...

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Finding the normals of a tileable 2D surface extracted from 4D space
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2 votes

Since the question was somewhat clarified I will formalize both the question and the answer for future readers. Having a differentiable scalar field $f : \mathbb{R}^4 \rightarrow \mathbb{R}$ we want ...

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Half Edge criterion to check if an edge flip is illegal?
1 votes

Let the edge to be flipped be made up of $v_l, \, v_r$. This edge needs to be part of 2 triangles (cannot be a boundary edge). Let the remaining vertices of the two triangles be $v_t, \, v_b$. Now ...

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