# Failed Shadertoy “Dust” Game – Browsers Need To Compile Shaders Off the Main Thread!

I was working on a Dust Game on shadertoy for a few weeks and got a decent amount of the way through it. Unfortunately I hit a snag.

There’s a sad fact that browsers compile WebGL shaders on the main thread. If ever WebGL blocks for too long, browsers consider WebGL hung and reports to the user that WebGL crashed.

The shaders for my game got complex enough that they were hitting the timeout while compiling, even though I wasn’t even close to being finished. So, I simplified a bit and published it unfinished, thinking that perhaps when WebGL 2.0 comes out this summer, I can revisit it.

Unfortunately, but totally understandably, different people have different machines in the wild, and some people have computers that take longer to compile shaders than my computer does, which means that plenty of people were still “crashing”. Because of this, the shadertoy people moved my shader back to draft/private until I simplified it more since crashes are no good, even if they are not realy real crashes in the normal sense. You can still see it by direct link here if you are interested: Shadertoy: Dust Sim Sandbox (Incomplete)

I’ve spent enough time on this thing so I’m going to leave it private for now, but I really think this shows a fundamental problem with WebGL.

Since the shaders compile on the main thread (apparently in many browsers if not all!), and detect a crash/hang by timing out, it means that when you write WebGL code, you really have to either keep things super simple in your shaders, or you have to be ok with having some amount of users experiencing “crashes” when the shader compiles time out on their machine.

In other words, you have to choose between innovation and stability.

That doesn’t really bode well for a platform like WebGL that basically is all about innovation! I’m sure this same problem plagues WebVR, which means that if you want to make a WebVR experience that works well on all machines, you better not do anything very complex 😛

Despite this failure, I did succeed in making a minesweeper game though, check it out here: Shadertoy: Minesweeper

Anyways, onto the next next shadertoy! I think I’m going to try to make one of those side view physics based car/motorcycle/bicycle hill climbing games.

## Update!

It turns out my understanding of the shader compiling process was a bit simplified. Read the link below for the full scoop, but in the end, the problem remains, because there doesn’t seem to be any way to query the status of the async work, so you seem to have to guess and hope you have enough work to do. In the case of shadertoy, there is no geometry to load, or things of that nature, so that makes it especially difficult to deal with.

Asynchronous Shader Compilation

Thanks to @tuan_kuranes for the better information (:

# Making a Ray Traced Snake Game in Shadertoy

Shadertoy.com recently added multipass rendering, and one of the founders – iq – introduced the feature to us by (ab)using an off screen texture to store state from previous frames. This was interesting because up until that point, you had to render each frame using only the current time, and current input from the user – you had no knowledge whatsoever of the past. He ended up making a brick breaking game, which is pretty awesome. You can see it here: Shadertoy: Bricks Game.

I decided to follow that path too and make a snake game in shadertoy. This is a high level overview of how the game works and various considerations that came up while creating the game.

Play Shadertoy: Snake Game

## Simulation vs. Presentation

When you start making a shadertoy shader, you have a single pixel shader program to work in. In the past this is where you did all your work to display the image to the user.

Now with multipass rendering you have the option of creating an off screen buffer (up to four of them actually, but we only need one for this discussion), which you write a pixel shader program for as well. This program is able to write to the off screen buffer by emitting pixels (like usual), but interestingly, it can also read from the same off screen buffer. Since it can both read and write the image, this allows it to preserve state across frames.

Since this pixel shader has state (unlike the main on screen pixel shader), this is where all your “simulation code” has to go. This is where you do all your game logic. Strangely, even though this buffer is a full screen buffer, you only need to process as many pixels as you need for storage of your game state. You can use the “discard” keyword for all other pixels, to get a quick “early out” in the pixel processing.

The on screen pixel shader is able to read this off screen buffer which stores game state, but it isn’t able to write to it. Because of this, the on screen pixel shader is essentially the visualization of the simulation.

## Simulation Considerations

The simulation had quite a few considerations. Some of these were a little mind bending at first, but made sense after thinking about it for a while.

Perhaps the single strangest thing about the simulation pixel shader program is this: The pixel shader program can only write the value of the pixel that it represents. Let’s talk some details about what that means.

Reading and Writing Variables

In the snake game, there is a grid of 16×16 pixels, where each pixel stores the state of a game cell. Besides this, there are four other pixels that store various pieces of game state – like what direction the snake is moving in, whether the snake is dead or not, and the location of the apple.

One complication of this is that to spawn an apple, we have to set the apple’s location variable, but we also have to mark that an apple is in the specific grid cell where it should be. What we essentially have to do is pick a random location for the apple to go to, and then we essentially have to say “if the pixel that this shader program is running for, is the specific grid cell where the apple should spawn at, then emit a pixel color that represents the necessary data to represent an apple”.

That is kind of weird. Even weirder perhaps, you have to remember that this pixel shader program is being run for several pixels per frame (once per variable you have in your state!), and they all have to be in agreement about the random position of the apple so that only one apple gets written to the board, so your random numbers have to be deterministic, yet still seemingly random to the player.

It’s odd to think that the entire game simulation is running per variable in your simulation, but that is what is happening.

There’s a trap that’s easy to fall in here too, where you might be tempted to write code like this: “If the pixel that we are running the shader program for is the new location that the snake wants to move to, and that grid cell contains a snake body part, then set the game over game state variable flag to true”

The problem there is that our if statement guarantees that we are running the pixel shader program for a grid cell, not for the game over game state variable, so when we change the game over state variable, it doesn’t get written out, since this pixel shader program instance isn’t running for that variable.

I actually wrote that exact code and it took me a little while to figure out why the game wouldn’t end when the snake crashed into itself, but it would end if it went out of bounds. The code looked very similar, and was very simple, and one worked, but the other didn’t.

The answer here is that instead of only running the logic for the grid cell that the snake head wants to enter, you have to make the logic run for all instances. So, unfortunately that means a texture read to read the grid cell that the snake head wants to enter into. That way, all instances have the information they need to run the same logic.

Technically, I guess it’s only the “game over” variable that needs to do the texture read, but it’s a lot easier and safer if you make all pixels run the same code deterministically. Also, having each pixel run the same code is likely a performance gain.

You might be able to squeeze some more performance out of being clever, but it’s real easy to bite you in the end too! In this case, the snake game runs well enough (60fps on my machine, even with the camera edge on which is the worst case) so I’m calling it good.

Numerical Problems

One gotcha to be wary of is that since you are storing game state variables in pixels, is that every variable is a vec4 of floats, and that they can only store floating point values between 0 and 1. So, if you are storing the position of a cell in a grid, you need to “normalize it” to be between 0 and 1.

Sadly, the floating point isn’t really guaranteed to be a full 32 bit floating point number either, and might be an 8 bit fixed point number or similar on some platforms. This makes it real easy to have numerical problems when you normalize your values into the 0 to 1 range.

Random Apple Spawning

When an apple is eaten (or the game is starting) an apple needs to spawn in a random location that isn’t yet taken by the snake’s body. This would mean generating random numbers in a loop until we found an empty spot. This could possibly be a perf problem for the pixel shader, possibly doing a large number of iterations, and possibly not finding a valid location by the time it ran out of it’s fixed size for loop indices.

I went with a different solution. When there is no apple spawned, the simulation generates a random position to put the apple at, and if it’s empty, it puts the apple there. If the position already was taken by a snake body part, it leaves the apple unspawned and tries again next frame. With a suitably high frame rate (it runs at 60fps for me!) it ought to be able to find an empty spot pretty quickly, even if you have a really large snake.

Frame Rate Independent Gameplay

When making games on any platform, it’s very easy to have problems with objects moving faster on machines that have higher FPS. The typical way to solve this is to keep track of how much time has elapsed between frames, keeping a total as the frames progress, and when the total gets above a certain point, you do a game update and reset the total back to zero (or the remainder after subtracting your “tick time” out of the total).

This was luckily very easy to do in shadertoy as well. There is a value you are provided called “iTimeDelta” which is how long the last frame took to render, in seconds. I divide this number by how many ticks i want per second so that the total will always be between 0 and 1, and then add it into a state variable (which is stored in a pixel). When doing the add, if the result is ever greater than 1, i do a tick, and reset it to 0.

Doing the tick just means moving the snake and handling whether the snake ate an apple, died, etc.

Low Latency Input

At first I had my input handling be handled inside of the tick. This was a problem because the tick happens 12 times a second, which means if you want the snake to change directions, you better have the key pressed down during one of those ticks. If you press it and then release it between ticks, the key press is lost and the snake doesn’t change direction and you die. This was extremely noticeable and made for really bad controls.

To solve this, I moved the key press handling to be OUTSIDE the tick, and had it be processed every frame. Instead of changing the snake’s direction right away though, I queued it up to be handled on the next tick. This way, you can quickly tap keys and they are registered as they should be on the next tick. The controls feel fine now and all is well.

## Visualization & Rendering

The rendering of the snake game had fewer considerations than the simulation, but there are a couple things worth mentioning.

There are just a few parts to the rendering:

1. If the ray hits the game board, make it look like the game board
2. If the ray enters the small box of “play area” above the game board, it raytraces the contents of the game board
3. If the ray misses everything, it does a lookup in a cube map texture to get the background
4. If the ray hits anything, it uses some variables (normal, diffuse color, shinyness) from the collision to do the shading. It also does a lookup in the cube map texture to do some environment mapping to give the impression that reflection is happening, even though it isn’t

The second bullet item is the most interesting. What the code does is find the grid cells that the ray enters and leaves the play area, and then it walks from the start to the end, checking each cell to see if it has anything in it, and if so, doing a raytrace test against it.

At first I used the Bresenham line algorithm to walk the grid cells, but when there were holes in some of my spheres in strange places, I realized my problem. Bresenham isn’t meant to draw to (visit) every grid cell that a line passes through – it is only meant to draw to the most important ones!

So, I replaced Bresenham with a grid traversal algorithm and all was well.

The game is rendered using real time raytracing, but it only does PRIMARY rays. It doesn’t do reflection, refraction or shadows. This is because doing any of those things means raytracing against the game board (grid cells) more than once. Since we have to do a texture lookup per grid cell we want to raytrace, this could equal a lot of texture lookups as you can imagine. There is a hard limit to the number you can do (I’m pretty sure, although I don’t know what it is, and it probably varies from machine to machine), so I wanted to try and not push things too hard, and just stayed with primary rays only.

## That’s All!

That’s basically it. Multipass rendering is an awesome feature in shadertoy. On one hand it makes me a bit sad getting it, because people don’t have to work within such tight constraints when doing the amazing things they do in shadertoy. On the other hand though, I think multipass rendering really ought to empower people to do some much more amazing things than in previous days – including but not limited to games.

I really look forward to learning from what other people are doing to learn more about how to leverage multipass rendering in interesting ways.

I also look forward to contributing my own game and non game multipass shaders. I have some ideas for both, keep an eye out (:

# Gaussian Blur

In this post we are going to take the concepts we went over in the last post (Box Blur) and apply them to Gaussian blurring.

At a high level, Gaussian blurring works just like box blurring in that there is a weight per pixel and that for each pixel, you apply the weights to that pixel and it’s neighbors to come up with the final value for the blurred pixel.

With true Gaussian blurring however, the function that defines the weights for each pixel technically never reaches zero, but gets smaller and smaller over distance. In theory this makes a Gaussian kernel infinitely large. In practice though, you can choose a cut off point and call it good enough.

The parameters to a Gaussian blur are:

• Sigma ($\sigma$) – This defines how much blur there is. A larger number is a higher amount of blur.
• Radius – The size of the kernel in pixels. The appropriate pixel size can be calculated for a specific sigma, but more information on that lower down.

Just like a box blur, a Gaussian blur is separable which means that you can either apply a 2d convolution kernel, or you can apply a 1d convolution kernel on each axis. Doing a single 2d convolution means more calculations, but you only need one buffer to put the results into. Doing two 1d convolutions (one on each axis), ends up being fewer calculations, but requires two buffers to put the results into (one intermediate buffer to hold the first axis results).

Here is a 3 pixel 1d Gaussian Kernel for a sigma of 1.0:

Below is a 3×3 pixel 2d Gaussian Kernel also with a sigma of 1.0. Note that this can be calculated as an outer product (tensor product) of 1d kernels!

An interesting property of Gaussian blurs is that you can apply multiple smaller blurs and it will come up with the result as if you did a larger Blur. Unfortunately it’s more calculations doing multiple smaller blurs so is not usually worth while.

If you apply multiple blurs, the equivalent blur is the square root of the sum of the squares of the blur. Taking wikipedia’s example, if you applied a blur with radius 6 and a blur with a radius of 8, you’d end up with the equivelant of a radius 10 blur. This is because $\sqrt{6^2 + 8^2} = 10$.

## Calculating The Kernel

There are a couple ways to calculate a Gaussian kernel.

Believe it or not, Pascal’s triangle approaches the Gaussian bell curve as the row number reaches infinity. If you remember, Pascal’s triangle also represents the numbers that each term is calculated by after expanding binomials $(x+y)^N$. So technically, you could use a row from Pascal’s triangle as a 1d kernel and normalize the result, but it isn’t the most accurate.

A better way is to use the Gaussian function which is this:

$e^{-x^2/(2*\sigma^2)}$

Where the sigma is your blur amount and x ranges across your values from the negative to the positive. For instance if your kernel was 5 values, it would range from -2 to +2.

An even better way would be to integrate the Gaussian function instead of just taking point samples. You can read about it in the link at the bottom “Gaussian Kernel Calculator”, but it’s also what we do in the example code.

Whatever way you do it, make sure and normalize the result so that the weights add up to 1. This makes sure that your blurring doesn’t make the image get brighter (greater than 1) or dimmer (less than 1).

## Calculating The Kernel Size

Given a sigma value, you can calculate the size of the kernel you need by using this formula:

$1+2 \sqrt{-2 \sigma^2 \ln 0.005}$

That formula makes a Kernel large enough such that it cuts off when the value in the kernel is less than 0.5%. You can adjust the number in there to higher or lower depending on your desires for speed versus quality.

## Examples

Once again, here is the unaltered image we are working with:

Here is the image blurred with a sigma of 3,3 (3 on the x axis and 3 on the y axis):

Here is the image blurred with a sigma of 20,3:

Here is the image blurred with a sigma of 50,50:

## Shadertoy

Here’s a shadertoy implementing Gaussian Blur: Shadertoy:DF Gaussian Blur

## Code

Here’s the source code I used to blur the examples above:

#define _CRT_SECURE_NO_WARNINGS

#include <stdio.h>
#include <stdint.h>
#include <array>
#include <vector>
#include <functional>
#include <windows.h>  // for bitmap headers.  Sorry non windows people!

typedef uint8_t uint8;

const float c_pi = 3.14159265359f;

struct SImageData
{
SImageData()
: m_width(0)
, m_height(0)
{ }

long m_width;
long m_height;
long m_pitch;
std::vector<uint8> m_pixels;
};

void WaitForEnter ()
{
printf("Press Enter to quit");
fflush(stdin);
getchar();
}

bool LoadImage (const char *fileName, SImageData& imageData)
{
// open the file if we can
FILE *file;
file = fopen(fileName, "rb");
if (!file)
return false;

// read the headers if we can
BITMAPFILEHEADER header;
BITMAPINFOHEADER infoHeader;
if (fread(&header, sizeof(header), 1, file) != 1 ||
fread(&infoHeader, sizeof(infoHeader), 1, file) != 1 ||
header.bfType != 0x4D42 || infoHeader.biBitCount != 24)
{
fclose(file);
return false;
}

// read in our pixel data if we can. Note that it's in BGR order, and width is padded to the next power of 4
imageData.m_pixels.resize(infoHeader.biSizeImage);
fseek(file, header.bfOffBits, SEEK_SET);
if (fread(&imageData.m_pixels[0], imageData.m_pixels.size(), 1, file) != 1)
{
fclose(file);
return false;
}

imageData.m_width = infoHeader.biWidth;
imageData.m_height = infoHeader.biHeight;

imageData.m_pitch = imageData.m_width*3;
if (imageData.m_pitch & 3)
{
imageData.m_pitch &= ~3;
imageData.m_pitch += 4;
}

fclose(file);
return true;
}

bool SaveImage (const char *fileName, const SImageData &image)
{
// open the file if we can
FILE *file;
file = fopen(fileName, "wb");
if (!file)
return false;

// make the header info
BITMAPFILEHEADER header;
BITMAPINFOHEADER infoHeader;

header.bfType = 0x4D42;
header.bfReserved1 = 0;
header.bfReserved2 = 0;
header.bfOffBits = 54;

infoHeader.biSize = 40;
infoHeader.biWidth = image.m_width;
infoHeader.biHeight = image.m_height;
infoHeader.biPlanes = 1;
infoHeader.biBitCount = 24;
infoHeader.biCompression = 0;
infoHeader.biSizeImage = image.m_pixels.size();
infoHeader.biXPelsPerMeter = 0;
infoHeader.biYPelsPerMeter = 0;
infoHeader.biClrUsed = 0;
infoHeader.biClrImportant = 0;

header.bfSize = infoHeader.biSizeImage + header.bfOffBits;

// write the data and close the file
fwrite(&header, sizeof(header), 1, file);
fwrite(&infoHeader, sizeof(infoHeader), 1, file);
fwrite(&image.m_pixels[0], infoHeader.biSizeImage, 1, file);
fclose(file);
return true;
}

int PixelsNeededForSigma (float sigma)
{
// returns the number of pixels needed to represent a gaussian kernal that has values
// down to the threshold amount.  A gaussian function technically has values everywhere
// on the image, but the threshold lets us cut it off where the pixels contribute to
// only small amounts that aren't as noticeable.
const float c_threshold = 0.005f; // 0.5%
return int(floor(1.0f + 2.0f * sqrtf(-2.0f * sigma * sigma * log(c_threshold)))) + 1;
}

float Gaussian (float sigma, float x)
{
return expf(-(x*x) / (2.0f * sigma*sigma));
}

float GaussianSimpsonIntegration (float sigma, float a, float b)
{
return
((b - a) / 6.0f) *
(Gaussian(sigma, a) + 4.0f * Gaussian(sigma, (a + b) / 2.0f) + Gaussian(sigma, b));
}

std::vector<float> GaussianKernelIntegrals (float sigma, int taps)
{
std::vector<float> ret;
float total = 0.0f;
for (int i = 0; i < taps; ++i)
{
float x = float(i) - float(taps / 2);
float value = GaussianSimpsonIntegration(sigma, x - 0.5f, x + 0.5f);
ret.push_back(value);
total += value;
}
// normalize it
for (unsigned int i = 0; i < ret.size(); ++i)
{
ret[i] /= total;
}
return ret;
}

const uint8* GetPixelOrBlack (const SImageData& image, int x, int y)
{
static const uint8 black[3] = { 0, 0, 0 };
if (x < 0 || x >= image.m_width ||
y < 0 || y >= image.m_height)
{
return black;
}

return &image.m_pixels[(y * image.m_pitch) + x * 3];
}

void BlurImage (const SImageData& srcImage, SImageData &destImage, float xblursigma, float yblursigma, unsigned int xblursize, unsigned int yblursize)
{
// allocate space for copying the image for destImage and tmpImage
destImage.m_width = srcImage.m_width;
destImage.m_height = srcImage.m_height;
destImage.m_pitch = srcImage.m_pitch;
destImage.m_pixels.resize(destImage.m_height * destImage.m_pitch);

SImageData tmpImage;
tmpImage.m_width = srcImage.m_width;
tmpImage.m_height = srcImage.m_height;
tmpImage.m_pitch = srcImage.m_pitch;
tmpImage.m_pixels.resize(tmpImage.m_height * tmpImage.m_pitch);

// horizontal blur from srcImage into tmpImage
{
auto row = GaussianKernelIntegrals(xblursigma, xblursize);

int startOffset = -1 * int(row.size() / 2);

for (int y = 0; y < tmpImage.m_height; ++y)
{
for (int x = 0; x < tmpImage.m_width; ++x)
{
std::array<float, 3> blurredPixel = { 0.0f, 0.0f, 0.0f };
for (unsigned int i = 0; i < row.size(); ++i)
{
const uint8 *pixel = GetPixelOrBlack(srcImage, x + startOffset + i, y);
blurredPixel[0] += float(pixel[0]) * row[i];
blurredPixel[1] += float(pixel[1]) * row[i];
blurredPixel[2] += float(pixel[2]) * row[i];
}

uint8 *destPixel = &tmpImage.m_pixels[y * tmpImage.m_pitch + x * 3];

destPixel[0] = uint8(blurredPixel[0]);
destPixel[1] = uint8(blurredPixel[1]);
destPixel[2] = uint8(blurredPixel[2]);
}
}
}

// vertical blur from tmpImage into destImage
{
auto row = GaussianKernelIntegrals(yblursigma, yblursize);

int startOffset = -1 * int(row.size() / 2);

for (int y = 0; y < destImage.m_height; ++y)
{
for (int x = 0; x < destImage.m_width; ++x)
{
std::array<float, 3> blurredPixel = { 0.0f, 0.0f, 0.0f };
for (unsigned int i = 0; i < row.size(); ++i)
{
const uint8 *pixel = GetPixelOrBlack(tmpImage, x, y + startOffset + i);
blurredPixel[0] += float(pixel[0]) * row[i];
blurredPixel[1] += float(pixel[1]) * row[i];
blurredPixel[2] += float(pixel[2]) * row[i];
}

uint8 *destPixel = &destImage.m_pixels[y * destImage.m_pitch + x * 3];

destPixel[0] = uint8(blurredPixel[0]);
destPixel[1] = uint8(blurredPixel[1]);
destPixel[2] = uint8(blurredPixel[2]);
}
}
}
}

int main (int argc, char **argv)
{
float xblursigma, yblursigma;

bool showUsage = argc < 5 ||
(sscanf(argv[3], "%f", &xblursigma) != 1) ||
(sscanf(argv[4], "%f", &yblursigma) != 1);

char *srcFileName = argv[1];
char *destFileName = argv[2];

if (showUsage)
{
printf("Usage: <source> <dest> <xblur> <yblur>nBlur values are sigmann");
WaitForEnter();
return 1;
}

// calculate pixel sizes, and make sure they are odd
int xblursize = PixelsNeededForSigma(xblursigma) | 1;
int yblursize = PixelsNeededForSigma(yblursigma) | 1;

printf("Attempting to blur a 24 bit image.n");
printf("  Source=%sn  Dest=%sn  blur=[%0.1f, %0.1f] px=[%d,%d]nn", srcFileName, destFileName, xblursigma, yblursigma, xblursize, yblursize);

SImageData srcImage;
if (LoadImage(srcFileName, srcImage))
{
printf("%s loadedn", srcFileName);
SImageData destImage;
BlurImage(srcImage, destImage, xblursigma, yblursigma, xblursize, yblursize);
if (SaveImage(destFileName, destImage))
printf("Blurred image saved as %sn", destFileName);
else
{
printf("Could not save blurred image as %sn", destFileName);
WaitForEnter();
return 1;
}
}
else
{
printf("could not read 24 bit bmp file %snn", srcFileName);
WaitForEnter();
return 1;
}
return 0;
}


## Links

Here is a really great explanation of the Gaussian blur.
Gaussian Blur – Image processing for scientists and engineers, Part 4
I highly recommend reading the 6 part series about image processing (DSP) from the beginning because it’s really informative and very easy to read!
Images are data – Image processing for scientists and engineers, Part 1

If you want to take this from theory / hobby level up to pro level, give this link a read from intel:
Intel: An investigation of fast real-time GPU-based image blur algorithms

# Box Blur

If you ever have heard the terms “Box Blur”, “Boxcar Function”, “Box Filter”, “Boxcar Integrator” or other various combinations of those words, you may have thought it was some advanced concept that is hard to understand and hard to implement. If that’s what you thought, prepare to be surprised!

A box filter is nothing more than taking N samples of data (or NxN samples of data, or NxNxN etc) and averaging them! Yes, that is all there is to it 😛

In this post, we are going to implement a box blur by averaging pixels.

## 1D Case

For the case of a 1d box filter, let’s say we wanted every data point to be the result of averaging it with it’s two neighbors. It’d be easy enough to program that by just doing it, but let’s look at it a different way. What weight would we need to multiply each of the three values by (the value and it’s two neighbors) to make it come up with the average?

Yep, you guessed it! For every data value, you multiply it and it’s neighbors by 1/3 to come up with the average value. We could easily increase the size of the filter to 5 pixels, and multiply each pixel by 1/5 instead. We could continue the pattern as high as we wanted.

One thing you might notice is that if we want a buffer with all the results, we can’t just alter the source data as we go, because we want the unaltered source values of the data to use those weights with, to get the correct results. Because of that, we need to make a second buffer to put the results of the filtering into.

Believe it or not, that diagram above is a convolution kernel, and how we talked about applying it is how you do convolution in 1d! It just so happens that this convolution kernel averages three pixels into one, which also happens to provide a low pass filter type effect.

Low pass filtering is what is done before down sampling audio data to prevent aliasing (frequencies higher than the sample rate can handle, which makes audio sound bad).

Surprise… blurring can also be seen as low pass filtering, which is something you can do before scaling an image down in size, to prevent aliasing.

## 2D Case

The 2d case isn’t much more difficult to understand than the 1d case. Instead of only averaging on one axis, we average on two instead:

Something interesting to note is that you can either use this 3×3 2d convolution kernel, or, you could apply the 1d convolution kernel described above on the X axis and then the Y axis. The methods are mathematically equivalent.

Using the 2d convolution kernel would result in 9 multiplications per pixel, but if going with the separated axis X and then Y 1d kernel, you’d only end up doing 6 multiplications per pixel (3 multiplications per axis). In general, if you have a seperable 2d convolution kernel (meaning that you can break it into a per axis 1d convolution), you will end up doing N^2 multiplications when using the 2d kernel, versus N*2 multiplications when using the 1d kernels. You can see that this would add up quickly in favor of using 1d kernels, but unfortunately not all kernels are separable.

Doing two passes does come at a cost though. Since you have to use a temporary buffer for each pass, you end up having to create two temporary buffers instead of one.

You can build 2d kernels from 1d kernels by multiplying them as a row vector, by a column vector. For instance, you can see how multiplying the (1/3,1/3,1/3) kernel by itself as a column vector would create the 2nd kernel, that is 3×3 and has 1/9 in every spot.

The resulting 3×3 matrix is called an outer product, or a tensor product. Something interesting to note is that you don’t have to do the same operation on each axis!

## Examples

Here are some examples of box blurring with different values, using the sample code provided below.

The source image:

Now blurred by a 10×10 box car convolution kernel:

Now blurred by a 100×10 box car convolution kernel:

## Shadertoy

You can find a shadertoy implementation of box blurring here: Shadertoy:DF Box Blur

## Code

Here’s the code I used to blur the example images above:

#define _CRT_SECURE_NO_WARNINGS

#include <stdio.h>
#include <stdint.h>
#include <array>
#include <vector>
#include <functional>
#include <windows.h>  // for bitmap headers.  Sorry non windows people!

typedef uint8_t uint8;

const float c_pi = 3.14159265359f;

struct SImageData
{
SImageData()
: m_width(0)
, m_height(0)
{ }

long m_width;
long m_height;
long m_pitch;
std::vector<uint8> m_pixels;
};

void WaitForEnter ()
{
printf("Press Enter to quit");
fflush(stdin);
getchar();
}

bool LoadImage (const char *fileName, SImageData& imageData)
{
// open the file if we can
FILE *file;
file = fopen(fileName, "rb");
if (!file)
return false;

// read the headers if we can
BITMAPFILEHEADER header;
BITMAPINFOHEADER infoHeader;
if (fread(&header, sizeof(header), 1, file) != 1 ||
fread(&infoHeader, sizeof(infoHeader), 1, file) != 1 ||
header.bfType != 0x4D42 || infoHeader.biBitCount != 24)
{
fclose(file);
return false;
}

// read in our pixel data if we can. Note that it's in BGR order, and width is padded to the next power of 4
imageData.m_pixels.resize(infoHeader.biSizeImage);
fseek(file, header.bfOffBits, SEEK_SET);
if (fread(&imageData.m_pixels[0], imageData.m_pixels.size(), 1, file) != 1)
{
fclose(file);
return false;
}

imageData.m_width = infoHeader.biWidth;
imageData.m_height = infoHeader.biHeight;

imageData.m_pitch = imageData.m_width*3;
if (imageData.m_pitch & 3)
{
imageData.m_pitch &= ~3;
imageData.m_pitch += 4;
}

fclose(file);
return true;
}

bool SaveImage (const char *fileName, const SImageData &image)
{
// open the file if we can
FILE *file;
file = fopen(fileName, "wb");
if (!file)
return false;

// make the header info
BITMAPFILEHEADER header;
BITMAPINFOHEADER infoHeader;

header.bfType = 0x4D42;
header.bfReserved1 = 0;
header.bfReserved2 = 0;
header.bfOffBits = 54;

infoHeader.biSize = 40;
infoHeader.biWidth = image.m_width;
infoHeader.biHeight = image.m_height;
infoHeader.biPlanes = 1;
infoHeader.biBitCount = 24;
infoHeader.biCompression = 0;
infoHeader.biSizeImage = image.m_pixels.size();
infoHeader.biXPelsPerMeter = 0;
infoHeader.biYPelsPerMeter = 0;
infoHeader.biClrUsed = 0;
infoHeader.biClrImportant = 0;

header.bfSize = infoHeader.biSizeImage + header.bfOffBits;

// write the data and close the file
fwrite(&header, sizeof(header), 1, file);
fwrite(&infoHeader, sizeof(infoHeader), 1, file);
fwrite(&image.m_pixels[0], infoHeader.biSizeImage, 1, file);
fclose(file);
return true;
}

const uint8* GetPixelOrBlack (const SImageData& image, int x, int y)
{
static const uint8 black[3] = { 0, 0, 0 };
if (x < 0 || x >= image.m_width ||
y < 0 || y >= image.m_height)
{
return black;
}

return &image.m_pixels[(y * image.m_pitch) + x * 3];
}

void BlurImage (const SImageData& srcImage, SImageData &destImage, unsigned int xblur, unsigned int yblur)
{
// allocate space for copying the image for destImage and tmpImage
destImage.m_width = srcImage.m_width;
destImage.m_height = srcImage.m_height;
destImage.m_pitch = srcImage.m_pitch;
destImage.m_pixels.resize(destImage.m_height * destImage.m_pitch);

SImageData tmpImage;
tmpImage.m_width = srcImage.m_width;
tmpImage.m_height = srcImage.m_height;
tmpImage.m_pitch = srcImage.m_pitch;
tmpImage.m_pixels.resize(tmpImage.m_height * tmpImage.m_pitch);

// horizontal blur from srcImage into tmpImage
{
float weight = 1.0f / float(xblur);
int half = xblur / 2;
for (int y = 0; y < tmpImage.m_height; ++y)
{
for (int x = 0; x < tmpImage.m_width; ++x)
{
std::array<float, 3> blurredPixel = { 0.0f, 0.0f, 0.0f };
for (int i = -half; i <= half; ++i)
{
const uint8 *pixel = GetPixelOrBlack(srcImage, x + i, y);
blurredPixel[0] += float(pixel[0]) * weight;
blurredPixel[1] += float(pixel[1]) * weight;
blurredPixel[2] += float(pixel[2]) * weight;
}

uint8 *destPixel = &tmpImage.m_pixels[y * tmpImage.m_pitch + x * 3];

destPixel[0] = uint8(blurredPixel[0]);
destPixel[1] = uint8(blurredPixel[1]);
destPixel[2] = uint8(blurredPixel[2]);
}
}
}

// vertical blur from tmpImage into destImage
{
float weight = 1.0f / float(yblur);
int half = yblur / 2;

for (int y = 0; y < destImage.m_height; ++y)
{
for (int x = 0; x < destImage.m_width; ++x)
{
std::array<float, 3> blurredPixel = { 0.0f, 0.0f, 0.0f };
for (int i = -half; i <= half; ++i)
{
const uint8 *pixel = GetPixelOrBlack(tmpImage, x, y + i);
blurredPixel[0] += float(pixel[0]) * weight;
blurredPixel[1] += float(pixel[1]) * weight;
blurredPixel[2] += float(pixel[2]) * weight;
}

uint8 *destPixel = &destImage.m_pixels[y * destImage.m_pitch + x * 3];

destPixel[0] = uint8(blurredPixel[0]);
destPixel[1] = uint8(blurredPixel[1]);
destPixel[2] = uint8(blurredPixel[2]);
}
}
}
}

int main (int argc, char **argv)
{
int xblur, yblur;

bool showUsage = argc < 5 ||
(sscanf(argv[3], "%i", &xblur) != 1) ||
(sscanf(argv[4], "%i", &yblur) != 1);

char *srcFileName = argv[1];
char *destFileName = argv[2];

if (showUsage)
{
printf("Usage: <source> <dest> <xblur> <yblur>nn");
WaitForEnter();
return 1;
}

// make sure blur size is odd
xblur = xblur | 1;
yblur = yblur | 1;

printf("Attempting to blur a 24 bit image.n");
printf("  Source=%sn  Dest=%sn  blur=[%d,%d]nn", srcFileName, destFileName, xblur, yblur);

SImageData srcImage;
if (LoadImage(srcFileName, srcImage))
{
printf("%s loadedn", srcFileName);
SImageData destImage;
BlurImage(srcImage, destImage, xblur, yblur);
if (SaveImage(destFileName, destImage))
printf("Blurred image saved as %sn", destFileName);
else
{
printf("Could not save blurred image as %sn", destFileName);
WaitForEnter();
return 1;
}
}
else
{
printf("could not read 24 bit bmp file %snn", srcFileName);
WaitForEnter();
return 1;
}
return 0;
}


## Next Up

Next up will be a Gaussian blur, and I’m nearly done w/ that post but wanted to make this one first as an introductory step!

Before we get there, I wanted to mention that if you do multiple box blurs in a row, it will start to approach Gaussian blurring. I’ve heard that three blurs in a row will make it basically indistinguishable from a Gaussian blur.

# Resizing Images With Bicubic Interpolation

In the last post we saw how to do cubic interpolation on a grid of data.

Strangely enough, when that grid is a grid of pixel data, bicubic interpolation is a common method for resizing images!

Bicubic interpolation can also used in realtime rendering to make textures look nicer when scaled than standard bilinear texture interpolation.

This technique works when making images larger as well as smaller, but when making images smaller, you can still have problems with aliasing. There are are better algorithms to use when making an image smaller. Check the links section at the bottom for more details!

## Example

Here’s the old man from The Legend of Zelda who gives you the sword.

Here he is scaled up 4x with nearest neighbor, bilinear interpolation and bicubic interpolation.

Here he is scaled up 16x with nearest neighbor, bilinear interpolation and bicubic interpolation.

## Shadertoy

I made a shadertoy to show you how to do this in a GLSL pixel shader as well. Shadertoy: Bicubic Texture Filtering

In the screenshot below, going from left to right it uses: Nearest Neighbor, Bilinear, Lagrange Bicubic interpolation (only interpolates values, not slopes), Hermite Bicubic interpolation.

## Sample Code

Here’s the code that I used to resize the images in the examples above.

#define _CRT_SECURE_NO_WARNINGS

#include <stdio.h>
#include <stdint.h>
#include <array>
#include <vector>
#include <windows.h>  // for bitmap headers.  Sorry non windows people!

#define CLAMP(v, min, max) if (v < min) { v = min; } else if (v > max) { v = max; }

typedef uint8_t uint8;

struct SImageData
{
SImageData()
: m_width(0)
, m_height(0)
{ }

long m_width;
long m_height;
long m_pitch;
std::vector<uint8> m_pixels;
};

void WaitForEnter ()
{
printf("Press Enter to quit");
fflush(stdin);
getchar();
}

bool LoadImage (const char *fileName, SImageData& imageData)
{
// open the file if we can
FILE *file;
file = fopen(fileName, "rb");
if (!file)
return false;

// read the headers if we can
BITMAPFILEHEADER header;
BITMAPINFOHEADER infoHeader;
if (fread(&header, sizeof(header), 1, file) != 1 ||
fread(&infoHeader, sizeof(infoHeader), 1, file) != 1 ||
header.bfType != 0x4D42 || infoHeader.biBitCount != 24)
{
fclose(file);
return false;
}

// read in our pixel data if we can. Note that it's in BGR order, and width is padded to the next power of 4
imageData.m_pixels.resize(infoHeader.biSizeImage);
fseek(file, header.bfOffBits, SEEK_SET);
if (fread(&imageData.m_pixels[0], imageData.m_pixels.size(), 1, file) != 1)
{
fclose(file);
return false;
}

imageData.m_width = infoHeader.biWidth;
imageData.m_height = infoHeader.biHeight;

imageData.m_pitch = imageData.m_width*3;
if (imageData.m_pitch & 3)
{
imageData.m_pitch &= ~3;
imageData.m_pitch += 4;
}

fclose(file);
return true;
}

bool SaveImage (const char *fileName, const SImageData &image)
{
// open the file if we can
FILE *file;
file = fopen(fileName, "wb");
if (!file)
return false;

// make the header info
BITMAPFILEHEADER header;
BITMAPINFOHEADER infoHeader;

header.bfType = 0x4D42;
header.bfReserved1 = 0;
header.bfReserved2 = 0;
header.bfOffBits = 54;

infoHeader.biSize = 40;
infoHeader.biWidth = image.m_width;
infoHeader.biHeight = image.m_height;
infoHeader.biPlanes = 1;
infoHeader.biBitCount = 24;
infoHeader.biCompression = 0;
infoHeader.biSizeImage = image.m_pixels.size();
infoHeader.biXPelsPerMeter = 0;
infoHeader.biYPelsPerMeter = 0;
infoHeader.biClrUsed = 0;
infoHeader.biClrImportant = 0;

header.bfSize = infoHeader.biSizeImage + header.bfOffBits;

// write the data and close the file
fwrite(&header, sizeof(header), 1, file);
fwrite(&infoHeader, sizeof(infoHeader), 1, file);
fwrite(&image.m_pixels[0], infoHeader.biSizeImage, 1, file);
fclose(file);
return true;
}

// t is a value that goes from 0 to 1 to interpolate in a C1 continuous way across uniformly sampled data points.
// when t is 0, this will return B.  When t is 1, this will return C.  Inbetween values will return an interpolation
// between B and C.  A and B are used to calculate slopes at the edges.
float CubicHermite (float A, float B, float C, float D, float t)
{
float a = -A / 2.0f + (3.0f*B) / 2.0f - (3.0f*C) / 2.0f + D / 2.0f;
float b = A - (5.0f*B) / 2.0f + 2.0f*C - D / 2.0f;
float c = -A / 2.0f + C / 2.0f;
float d = B;

return a*t*t*t + b*t*t + c*t + d;
}

float Lerp (float A, float B, float t)
{
return A * (1.0f - t) + B * t;
}

const uint8* GetPixelClamped (const SImageData& image, int x, int y)
{
CLAMP(x, 0, image.m_width - 1);
CLAMP(y, 0, image.m_height - 1);
return &image.m_pixels[(y * image.m_pitch) + x * 3];
}

std::array<uint8, 3> SampleNearest (const SImageData& image, float u, float v)
{
// calculate coordinates
int xint = int(u * image.m_width);
int yint = int(v * image.m_height);

// return pixel
auto pixel = GetPixelClamped(image, xint, yint);
std::array<uint8, 3> ret;
ret[0] = pixel[0];
ret[1] = pixel[1];
ret[2] = pixel[2];
return ret;
}

std::array<uint8, 3> SampleLinear (const SImageData& image, float u, float v)
{
// calculate coordinates -> also need to offset by half a pixel to keep image from shifting down and left half a pixel
float x = (u * image.m_width) - 0.5f;
int xint = int(x);
float xfract = x - floor(x);

float y = (v * image.m_height) - 0.5f;
int yint = int(y);
float yfract = y - floor(y);

// get pixels
auto p00 = GetPixelClamped(image, xint + 0, yint + 0);
auto p10 = GetPixelClamped(image, xint + 1, yint + 0);
auto p01 = GetPixelClamped(image, xint + 0, yint + 1);
auto p11 = GetPixelClamped(image, xint + 1, yint + 1);

// interpolate bi-linearly!
std::array<uint8, 3> ret;
for (int i = 0; i < 3; ++i)
{
float col0 = Lerp(p00[i], p10[i], xfract);
float col1 = Lerp(p01[i], p11[i], xfract);
float value = Lerp(col0, col1, yfract);
CLAMP(value, 0.0f, 255.0f);
ret[i] = uint8(value);
}
return ret;
}

std::array<uint8, 3> SampleBicubic (const SImageData& image, float u, float v)
{
// calculate coordinates -> also need to offset by half a pixel to keep image from shifting down and left half a pixel
float x = (u * image.m_width) - 0.5;
int xint = int(x);
float xfract = x - floor(x);

float y = (v * image.m_height) - 0.5;
int yint = int(y);
float yfract = y - floor(y);

// 1st row
auto p00 = GetPixelClamped(image, xint - 1, yint - 1);
auto p10 = GetPixelClamped(image, xint + 0, yint - 1);
auto p20 = GetPixelClamped(image, xint + 1, yint - 1);
auto p30 = GetPixelClamped(image, xint + 2, yint - 1);

// 2nd row
auto p01 = GetPixelClamped(image, xint - 1, yint + 0);
auto p11 = GetPixelClamped(image, xint + 0, yint + 0);
auto p21 = GetPixelClamped(image, xint + 1, yint + 0);
auto p31 = GetPixelClamped(image, xint + 2, yint + 0);

// 3rd row
auto p02 = GetPixelClamped(image, xint - 1, yint + 1);
auto p12 = GetPixelClamped(image, xint + 0, yint + 1);
auto p22 = GetPixelClamped(image, xint + 1, yint + 1);
auto p32 = GetPixelClamped(image, xint + 2, yint + 1);

// 4th row
auto p03 = GetPixelClamped(image, xint - 1, yint + 2);
auto p13 = GetPixelClamped(image, xint + 0, yint + 2);
auto p23 = GetPixelClamped(image, xint + 1, yint + 2);
auto p33 = GetPixelClamped(image, xint + 2, yint + 2);

// interpolate bi-cubically!
// Clamp the values since the curve can put the value below 0 or above 255
std::array<uint8, 3> ret;
for (int i = 0; i < 3; ++i)
{
float col0 = CubicHermite(p00[i], p10[i], p20[i], p30[i], xfract);
float col1 = CubicHermite(p01[i], p11[i], p21[i], p31[i], xfract);
float col2 = CubicHermite(p02[i], p12[i], p22[i], p32[i], xfract);
float col3 = CubicHermite(p03[i], p13[i], p23[i], p33[i], xfract);
float value = CubicHermite(col0, col1, col2, col3, yfract);
CLAMP(value, 0.0f, 255.0f);
ret[i] = uint8(value);
}
return ret;
}

void ResizeImage (const SImageData &srcImage, SImageData &destImage, float scale, int degree)
{
destImage.m_width = long(float(srcImage.m_width)*scale);
destImage.m_height = long(float(srcImage.m_height)*scale);
destImage.m_pitch = destImage.m_width * 3;
if (destImage.m_pitch & 3)
{
destImage.m_pitch &= ~3;
destImage.m_pitch += 4;
}
destImage.m_pixels.resize(destImage.m_pitch*destImage.m_height);

uint8 *row = &destImage.m_pixels[0];
for (int y = 0; y < destImage.m_height; ++y)
{
uint8 *destPixel = row;
float v = float(y) / float(destImage.m_height - 1);
for (int x = 0; x < destImage.m_width; ++x)
{
float u = float(x) / float(destImage.m_width - 1);
std::array<uint8, 3> sample;

if (degree == 0)
sample = SampleNearest(srcImage, u, v);
else if (degree == 1)
sample = SampleLinear(srcImage, u, v);
else if (degree == 2)
sample = SampleBicubic(srcImage, u, v);

destPixel[0] = sample[0];
destPixel[1] = sample[1];
destPixel[2] = sample[2];
destPixel += 3;
}
row += destImage.m_pitch;
}
}

int main (int argc, char **argv)
{
float scale = 1.0f;
int degree = 0;

bool showUsage = argc < 5 ||
(sscanf(argv[3], "%f", &scale) != 1) ||
(sscanf(argv[4], "%i", &degree) != 1);

char *srcFileName = argv[1];
char *destFileName = argv[2];

if (showUsage)
{
printf("Usage: <source> <dest> <scale> <degree>ndegree 0 = nearest, 1 = bilinear, 2 = bicubic.nn");
WaitForEnter();
return 1;
}

printf("Attempting to resize a 24 bit image.n");
printf("  Source = %sn  Dest = %sn  Scale = %0.2fnn", srcFileName, destFileName, scale);

SImageData srcImage;
if (LoadImage(srcFileName, srcImage))
{
printf("%s loadedn", srcFileName);
SImageData destImage;
ResizeImage(srcImage, destImage, scale, degree);
if (SaveImage(destFileName, destImage))
printf("Resized image saved as %sn", destFileName);
else
printf("Could not save resized image as %sn", destFileName);
}
else
printf("could not read 24 bit bmp file %snn", srcFileName);
return 0;
}


## Links

The link below talks about how to do cubic texture sampling on the GPU without having to do 16 texture reads!
GPU Gems 2 Chapter 20. Fast Third-Order Texture Filtering

This link is from Inigo Quilez, where he transforms a texture coordinate before passing it to the bilinear filtering, to get higher quality texture sampling without having to do extra texture reads. That is pretty cool.
IQ: improved texture interpolation

# Cubic Hermite Rectangles

Time for another Frankenstein post. This time we are going to combine the following:

The end result is going to be a Cubic Hermite Rectangle Surface like the below. Note that the curve only passes through the inner four control points, and the outer ring of 12 control points are used to determine the slope.

Just like the cubic hermite curve counterpart, a cubic hermite rectangle surface is C1 continuous everywhere, which is great for use as a way of modeling geometry, as well as just for interpolation of multidimensional data. In the image below, each checkerboard square is an individual hermite rectangle.

The links section at the bottom has links to the shadertoys I made that I got the screenshots from.

## Code

Here’s some C++ code that does bicubic hermite interpolation

#include <stdio.h>
#include <array>

typedef std::array<float, 4> TFloat4;
typedef std::array<TFloat4, 4> TFloat4x4;

const TFloat4x4 c_ControlPointsX =
{
{
{ 0.7f, 0.8f, 0.9f, 0.3f },
{ 0.2f, 0.5f, 0.4f, 0.1f },
{ 0.6f, 0.3f, 0.1f, 0.4f },
{ 0.8f, 0.4f, 0.2f, 0.7f },
}
};

const TFloat4x4 c_ControlPointsY =
{
{
{ 0.2f, 0.8f, 0.5f, 0.6f },
{ 0.6f, 0.9f, 0.3f, 0.8f },
{ 0.7f, 0.1f, 0.4f, 0.9f },
{ 0.6f, 0.5f, 0.3f, 0.2f },
}
};

const TFloat4x4 c_ControlPointsZ =
{
{
{ 0.6f, 0.5f, 0.3f, 0.2f },
{ 0.7f, 0.1f, 0.9f, 0.5f },
{ 0.8f, 0.4f, 0.2f, 0.7f },
{ 0.6f, 0.3f, 0.1f, 0.4f },
}
};

void WaitForEnter ()
{
printf("Press Enter to quit");
fflush(stdin);
getchar();
}

// t is a value that goes from 0 to 1 to interpolate in a C1 continuous way across uniformly sampled data points.
// when t is 0, this will return p[1].  When t is 1, this will return p[2].
// p[0] and p[3] are used to calculate slopes at the edges.
float CubicHermite(const TFloat4& p, float t)
{
float a = -p[0] / 2.0f + (3.0f*p[1]) / 2.0f - (3.0f*p[2]) / 2.0f + p[3] / 2.0f;
float b = p[0] - (5.0f*p[1]) / 2.0f + 2.0f*p[2] - p[3] / 2.0f;
float c = -p[0] / 2.0f + p[2] / 2.0f;
float d = p[1];

return a*t*t*t + b*t*t + c*t + d;
}

float BicubicHermitePatch(const TFloat4x4& p, float u, float v)
{
TFloat4 uValues;
uValues[0] = CubicHermite(p[0], u);
uValues[1] = CubicHermite(p[1], u);
uValues[2] = CubicHermite(p[2], u);
uValues[3] = CubicHermite(p[2], u);
return CubicHermite(uValues, v);
}

int main(int argc, char **argv)
{
// how many values to display on each axis. Limited by console resolution!
const int c_numValues = 4;

printf("Cubic Hermite rectangle:n");
for (int i = 0; i < c_numValues; ++i)
{
float iPercent = ((float)i) / ((float)(c_numValues - 1));
for (int j = 0; j < c_numValues; ++j)
{
if (j == 0)
printf("  ");
float jPercent = ((float)j) / ((float)(c_numValues - 1));
float valueX = BicubicHermitePatch(c_ControlPointsX, jPercent, iPercent);
float valueY = BicubicHermitePatch(c_ControlPointsY, jPercent, iPercent);
float valueZ = BicubicHermitePatch(c_ControlPointsZ, jPercent, iPercent);
printf("(%0.2f, %0.2f, %0.2f) ", valueX, valueY, valueZ);
}
printf("n");
}
printf("n");

WaitForEnter();
return 0;
}


And here’s the output. Note that the four corners of the output correspond to the four inner most points defined in the data!

## On The GPU / Links

While cubic Hermite rectangles pass through all of their control points like Lagrange surfaces do (and like Bezier rectangle’s don’t), they don’t suffer from Runge’s Phenomenon like Lagrange surfaces do.

However, just like Lagrange surfaces, Hermite surfaces don’t have the nice property that Bezier surfaces have, where the surface is guaranteed to stay inside of the convex hull defined by the control points.

Since Hermite surfaces are just cubic functions though, you could calculate the minimum and maximum value that they can reach using some calculus and come up with a bounding box by going that direction. The same thing is technically true of Lagrange surfaces as well for what it’s worth.

Check out the links below to see cubic Hermite rectangles rendered in real time in WebGL using raytracing and raymarching:
Shadertoy: Cubic Hermite Rectangle
Shadertoy: Infinite Hermite Rectangles

# Cubic Hermite Interpolation

It’s a big wide world of curves out there and I have to say that most of the time, I consider myself a Bezier man.

Well let me tell you… cubic Hermite splines are technically representable in Bezier form, but they have some really awesome properties that I never fully appreciated until recently.

Usefulness For Interpolation

If you have a set of data points on some fixed interval (like for audio data, but could be anything), you can use a cubic Hermite spline to interpolate between any two data points. It interpolates the value between those points (as in, it passes through both end points), but it also interpolates a derivative that is consistent if you approach the point from the left or the right.

In short, this means you can use cubic Hermite splines to interpolate data such that the result has $C1$ continuity everywhere!

Usefulness As Curves

If you have any number $N$ control points on a fixed interval, you can treat it as a bunch of piece wise cubic Hermite splines and evaluate it that way.

The end result is that you have a curve that is $C1$ continuous everywhere, it has local control (moving any control point only affects the two curve sections to the left and the two curve sections to the right), and best of all, the computational complexity doesn’t rise as you increase the number of control points!

The image below was taken as a screenshot from one of the HTML5 demos I made for you to play with. You can find links to them at the end of this post.

## Cubic Hermite Splines

Cubic Hermite splines have four control points but how it uses the control points is a bit different than you’d expect.

The curve itself passes only through the middle two control points, and the end control points are there to help calculate the tangent at the middle control points.

Let’s say you have control points $P_{-1}, P_0, P_1, P_2$. The curve at time 0 will be at point $P_0$ and the slope will be the same slope as a line would have if going from $P_{-1}$ to $P_1$. The curve at time 1 will be at point $P_1$ and the slope will be the same slope as a line would have if going from $P_0$ to $P_2$.

Check out the picture below to see what I mean visually.

That sounds like a strange set of properties, but they are actually super useful.

What this means is that you can treat any group of 4 control points / data points as a separate cubic hermite spline, but when you put it all together, it is a single smooth curve.

Note that you can either interpolate 1d data, or you can interpolate 2d data points by doing this interpolation on each axis. You could also use this to make a surface, which will likely be the next blog post!

## The Math

I won’t go into how the formula is derived, but if you are interested you should check out Signal Processing: Bicubic Interpolation.

The formula is:

$a*t^3+b*t^2+c*t+d$

Where…

$a = \frac{-P_{-1} + 3*P_0 - 3*P_1 + P_2}{2}$
$b = P_{-1} - \frac{5*P_0}{2} + 2*P_1 - \frac{P_2}{2}$
$c = \frac{-P_{-1} + P_1}{2}$
$d = P_0$

Note that t is a value that goes from 0 to 1. When t is 0, your curve will be at $P_1$ and when t is 1, your curve will be at $P_2$. $P_{-1}$ and $P_{2}$ are used to be able to make this interpolation $C1$ continuous.

Here it is in some simple C++:

// t is a value that goes from 0 to 1 to interpolate in a C1 continuous way across uniformly sampled data points.
// when t is 0, this will return B.  When t is 1, this will return C.
static float CubicHermite (float A, float B, float C, float D, float t)
{
float a = -A/2.0f + (3.0f*B)/2.0f - (3.0f*C)/2.0f + D/2.0f;
float b = A - (5.0f*B)/2.0f + 2.0f*C - D / 2.0f;
float c = -A/2.0f + C/2.0f;
float d = B;

return a*t*t*t + b*t*t + c*t + d;
}


## Code

Here is an example C++ program that interpolates both 1D and 2D data.

#include <stdio.h>
#include <vector>
#include <array>

typedef std::vector<float> TPointList1D;
typedef std::vector<std::array<float,2>> TPointList2D;

void WaitForEnter ()
{
printf("Press Enter to quit");
fflush(stdin);
getchar();
}

// t is a value that goes from 0 to 1 to interpolate in a C1 continuous way across uniformly sampled data points.
// when t is 0, this will return B.  When t is 1, this will return C.
float CubicHermite (float A, float B, float C, float D, float t)
{
float a = -A/2.0f + (3.0f*B)/2.0f - (3.0f*C)/2.0f + D/2.0f;
float b = A - (5.0f*B)/2.0f + 2.0f*C - D / 2.0f;
float c = -A/2.0f + C/2.0f;
float d = B;

return a*t*t*t + b*t*t + c*t + d;
}

template <typename T>
inline T GetIndexClamped(const std::vector<T>& points, int index)
{
if (index < 0)
return points[0];
else if (index >= int(points.size()))
return points.back();
else
return points[index];
}

int main (int argc, char **argv)
{
const float c_numSamples = 13;

// show some 1d interpolated values
{
const TPointList1D points =
{
0.0f,
1.6f,
2.3f,
3.5f,
4.3f,
5.9f,
6.8f
};

printf("1d interpolated values.  y = f(t)n");
for (int i = 0; i < c_numSamples; ++i)
{
float percent = ((float)i) / (float(c_numSamples - 1));
float x = (points.size()-1) * percent;

int index = int(x);
float t = x - floor(x);
float A = GetIndexClamped(points, index - 1);
float B = GetIndexClamped(points, index + 0);
float C = GetIndexClamped(points, index + 1);
float D = GetIndexClamped(points, index + 2);

float y = CubicHermite(A, B, C, D, t);
printf("  Value at %0.2f = %0.2fn", x, y);
}
printf("n");
}

// show some 2d interpolated values
{
const TPointList2D points =
{
{ 0.0f, 1.1f },
{ 1.6f, 8.3f },
{ 2.3f, 6.5f },
{ 3.5f, 4.7f },
{ 4.3f, 3.1f },
{ 5.9f, 7.5f },
{ 6.8f, 0.0f }
};

printf("2d interpolated values.  x = f(t), y = f(t)n");
for (int i = 0; i < c_numSamples; ++i)
{
float percent = ((float)i) / (float(c_numSamples - 1));
float x = 0.0f;
float y = 0.0f;

float tx = (points.size() -1) * percent;
int index = int(tx);
float t = tx - floor(tx);

std::array<float, 2> A = GetIndexClamped(points, index - 1);
std::array<float, 2> B = GetIndexClamped(points, index + 0);
std::array<float, 2> C = GetIndexClamped(points, index + 1);
std::array<float, 2> D = GetIndexClamped(points, index + 2);
x = CubicHermite(A[0], B[0], C[0], D[0], t);
y = CubicHermite(A[1], B[1], C[1], D[1], t);

printf("  Value at %0.2f = (%0.2f, %0.2f)n", tx, x, y);
}
printf("n");
}

WaitForEnter();
return 0;
}


The output of the program is below:

## Links

Here are some interactive HTML5 demos i made:
1D cubic hermite interpolation
2D cubic hermite interpolation

More info here:
Wikipedia: Cubic Hermite Spline

Closely related to cubic hermite splines, catmull-rom splines allow you to specify a “tension” parameter to make the result more or less curvy:
Catmull-Rom spline

# Lagrange Rectangles

In this post we are going to Frankenstein ideas from two other recent posts. If you haven’t seen these yet you should probably give them a read!

Ingredient 1: Lagrange interpolation
Ingredient 2: Rectangular Bezier Patches

## Lagrange Surface

Lets say you have a grid of size MxN and you want to make a 3d surface for that grid.

You could use a Bezier rectangle but lets say that you really need the surface to pass through the control points. Bezier curves and surfaces only generally pass through the end / edge control points.

So what do you do? How about using Lagrange interpolation instead?

Just like how Bezier rectangles work, you interpolate on one axis, and then take those values and interpolate on the other axis.

Doing that, you get something like the below:

This comes at a price though. Whereas a Bezier curve or surface will be completely contained by it’s control points, a Lagrange rectangle isn’t always. Also, they are subject to something called Runge’s Phenomenon which basically means that the more control points you add, the more likely a surface is to get a bit “squirly”. You can see this effect when you add a lot of control points to my 1d lagrange interpolation demo as well: HTML5 1d Lagrange Interpolation.

Below is a picture of a bicubic Lagrange rectangle using the same control points the cubic Bezier rectangles used. Notice how much more extreme the peaks and valleys are! In the screenshot above, i scaled down the control points to 1/3 of what they were in the Bezier demo to make it look more reasonably well behaved.

## Code

#include <stdio.h>
#include <array>

typedef std::array<float, 3> TFloat3;
typedef std::array<TFloat3, 3> TFloat3x3;

const TFloat3x3 c_ControlPointsX =
{
{
{ 0.7f, 0.8f, 0.9f },
{ 0.2f, 0.5f, 0.4f },
{ 0.6f, 0.3f, 0.1f },
}
};

const TFloat3x3 c_ControlPointsY =
{
{
{ 0.2f, 0.8f, 0.5f },
{ 0.6f, 0.9f, 0.3f },
{ 0.7f, 0.1f, 0.4f },
}
};

const TFloat3x3 c_ControlPointsZ =
{
{
{ 0.6f, 0.5f, 0.3f },
{ 0.7f, 0.1f, 0.9f },
{ 0.8f, 0.4f, 0.2f },
}
};

void WaitForEnter ()
{
printf("Press Enter to quit");
fflush(stdin);
getchar();
}

//=======================================================================================
float QuadraticLagrange (const TFloat3& p, float t)
{
float c_x0 = 0.0 / 2.0;
float c_x1 = 1.0 / 2.0;
float c_x2 = 2.0 / 2.0;

return
p[0] *
(
(t - c_x1) / (c_x0 - c_x1) *
(t - c_x2) / (c_x0 - c_x2)
) +
p[1] *
(
(t - c_x0) / (c_x1 - c_x0) *
(t - c_x2) / (c_x1 - c_x2)
) +
p[2] *
(
(t - c_x0) / (c_x2 - c_x0) *
(t - c_x1) / (c_x2 - c_x1)
);
}

float BiquadraticLagrangePatch(const TFloat3x3& p, float u, float v)
{
TFloat3 uValues;
uValues[0] = QuadraticLagrange(p[0], u);
uValues[1] = QuadraticLagrange(p[1], u);
uValues[2] = QuadraticLagrange(p[2], u);
return QuadraticLagrange(uValues, v);
}

int main(int argc, char **argv)
{
// how many values to display on each axis. Limited by console resolution!
const int c_numValues = 4;

printf("Lagrange rectangle:n");
for (int i = 0; i < c_numValues; ++i)
{
float iPercent = ((float)i) / ((float)(c_numValues - 1));
for (int j = 0; j < c_numValues; ++j)
{
if (j == 0)
printf("  ");
float jPercent = ((float)j) / ((float)(c_numValues - 1));
float valueX = BiquadraticLagrangePatch(c_ControlPointsX, jPercent, iPercent);
float valueY = BiquadraticLagrangePatch(c_ControlPointsY, jPercent, iPercent);
float valueZ = BiquadraticLagrangePatch(c_ControlPointsZ, jPercent, iPercent);
printf("(%0.2f, %0.2f, %0.2f) ", valueX, valueY, valueZ);
}
printf("n");
}
printf("n");

WaitForEnter();
return 0;
}


And here is the output:

Compare that to the output of the Bezier rectangles code which used the same control points:

## Links

Shadertoy: Cubic Lagrange Rectangle

Note that the above uses Lagrange interpolation on a grid. The paper below talks about a way to make a Lagrange surface without using a grid:
A Simple Expression for Multivariate Lagrange Interpolation

# Finite Differences

Finite differences are numerical methods for approximating function derivatives – otherwise known as the slope of a function at a specific point on the graph. This can be helpful if it’s undesirable or impossible to calculate the actual derivative of a specific function.

This post talks about three methods: central difference, backwards difference and forward difference. They are all based on evaluating the function at two points and using the slope between those points as the derivative estimate.

The distance between those sample points is called an epsilon, and the smaller it is, the more accurate the approximation is in theory. In practice, extremely small values (like FLT_MIN) may hit numerical problems due to floating points number usage, and also you could hit performance problems due to using floating point denormals. Check the links section at the bottom for more info on denormals.

## Central Difference

The central difference is the most accurate technique of the three. You can find information about comparitive accuracy of these three techniques in the links section at the end. In practical terms, this may also be the slowest method too – or the most computationally expensive – which i’ll explain further down.

If you want to know the derivative of some function $y=f(x)$ at a specific value of x, you pick an epsilon e and then you calculate both $f(x-e)$ and $f(x+e)$. You subtract the first one from the second and divide by 2*e to get an approximated slope of the function at the specific value of x.

Remembering that the slope is just rise over run, and that the derivative at a point on a function is just the slope of the function at that point, this should hopefully make sense and be pretty intuitive why it works.

The resulting equation looks like:

$m = \frac{f(x+e)-f(x-e)}{2e}$

This process is visualized below. The black line is the actual slope at 0.4 and the orange line is the estimated slope. The orange dots are the sample points taken. The epsilon in this case is 0.2.

Interestingly, when dealing with quadratic (or linear) functions, the central difference method will give you the correct result. The picture above uses a quadratic function, so you can see no matter what value of e we use, it will always be parallel to the actual slope at that point. For cubic and higher functions, that won’t always be true.

## Backward Difference

The backward difference works just like the central difference except uses different sample points. It evaluates $f(x-e)$ and $f(x)$, subtracts the 1st one from the second one and divides the result by e.

The resulting equation looks like this:

$m = \frac{f(x)-f(x-e)}{e}$

A neat property shared by both this and the forward difference is that many times you are already going to be evaluating f(x) for other uses, so in practice this will just mean that you only have to evaluate f(x-e), and will already have the f(x) value. That can make it more efficient than the central difference method, but it can be less precise.

Also, if you are walking down a function (say, rendering a Bezier curve, and wanting the slope at each point to do something with), you may very well be able to use the f(x) of the previous point as your f(x-e) function, which means that you could possibly calculate the backwards difference by using the previous point, instead of evaluating the function extra times in your loop!

Check out the image below to see how different values of e result in different quality approximations. The smaller the epsilon value, the more accurate the result. An infinitely small epsilon would give the exact right answer.

### Forward Difference

The forward difference is just like the backwards difference but it evaluates forward instead of backwards.

The equation looks like this:

$m = \frac{f(x+e)-f(x)}{e}$

Below you can see it visually. Note again that smaller values of e make the estimation closer to correct.

## On the GPU

If you’ve ever encountered the glsl functions dFdx and dFdy and wondered how they work, they actually use these same techniques.

Shaders run in groups, and using dFdx, the shader just looks to it’s neighbor for the value that was passed to it’s dFdx, then using “local differencing” (per the docs), gives each shader the derivative it was able to calculate.

## Links

GLSL: dFdx, dFdy
Wikipediate: Finite Difference – The wikipedia page talks about more details, including how to calculate 2nd derivatives and higher!
Floating Point Denormals, Insignificant But Controversial
Comparing Methods of First Derivative Approximation Forward, Backward and Central Divided Difference

# Rectangular Bezier Patches

Rectangular Bezier Patches are one way to bring Bezier curves into the 3rd dimension as a Bezier surface. Below is a rendered image of a quadratic Bezier rectangle (degree of (2,2)) and a cubic Bezier rectangle (degree of (3,3)) taken as screenshots from a shadertoy demo I created that renders these in real time. Links at bottom of post!

## Intuition

Imagine that you had a Bezier curve with some number of control points. Now, imagine that you wanted to animate those control points over time instead of having a static curve.

One way to do this would be to just have multiple sets of control points as key frames, and just linearly interpolate between the key frames over time. You’d get something that might look like the image below (lighter red = farther back in time).

That is a simple and intuitive way to animate a Bezier curve, and is probably what you thought of immediately. Interestingly though, since linear interpolation is really a degree 1 Bezier curve, this method is actually using a degree 1 Bezier curve to control each control point!

What if we tried a higher order curve to animate each control point? Well… we could have three sets of control points, so that each control point was controlled over time by a quadratic curve. We could also try having four sets of control points, so that each control point was controlled over time by a cubic curve.

We could have any number of sets of control points, to be able to animate the control points over time using any degree curve.

Now, instead of animating the curve over TIME, what if we controlled it over DISTANCE (like, say, the z-axis, or “depth”). Look at the image above and think of it like you are looking at a surface from the side. If you took a bunch of the time interpolations as slices and set them next to each other so that there were no gaps between them, you’d end up with a smooth surface. TA-DA! This is how a Rectangular Bezier Patch is made.

Note that the degree of the curve on one axis doesn’t have to match the degree of the curve on the other axis. You could have a cubic curve where each control point is controlled by a linear interpolation, or you could have a degree 5 curve where each control point is controlled by degree 7 curves. Since there are two degrees involved in a Bezier rectangle, you describe it’s order with two numbers. The first example is degree (3,1) and the second example is degree (5,7).

Higher Dimensions

While you thinking about this, I wanted to mention that you could animate a bezier rectangle over time, using bezier curves to control those control points. If you then laid that out over distance instead of time, you’d end up with a rectangular box Bezier solid. If you are having trouble visualizing that, don’t feel dumb, it’s actually four dimensional!

You can think of it like a box that has a value stored at every (x,y,z) location, and those values are controlled by Bezier formulas so are smooth and are based on control points. It’s kind of a strange concept but is useful in some situations.

Say you made a 3d hot air baloon game and wanted to model temperature of the air at differently locations to simulate thermals. One way you could do this would be to store a bunch of temperatures in a 3d grid. Another way might involve using a grid of rectangular box Bezier solids perhaps. One benefit to the Bezier solid representation is that the data points are much smoother than a grid would be, and another is that you could make the grid much less dense.

Now, let’s say that you wanted to animate the thermals over time. You could use a fifth dimensional bezier hypercube solid. Let’s move on, my brain hurts 😛

## Math

The equation for a Bezier Rectangle is:

$\mathbf{p}(u, v) = \sum_{i=0}^n \sum_{j=0}^m B_i^n(u) \; B_j^m(v) \; \mathbf{k}_{i,j}$

$\mathbf{p}(u, v)$ is the point on the surface that you get after you plug in the parameters. $u$ and $v$ are the parameters to the surface and should be within the range 0 to 1. These are the same thing as the $t$ you see in Bezier curves, but there are two of them since there are two axes.

There are two Sigmas (summations) which mean that it’s a double for loop.

One of the for loops make $i$ go from 0 to $n$ and the other makes $j$ go from 0 to $m$. $m$ and $n$ are the degree of each axis.

$B_i^n(u)$ and $B_i^n(u)$ are Bernstein polynomials (aka binomial expansion terms) just as you see in Bezier Curves – there is just one per axis.

Lastly comes the control points $\mathbf{k}_{i,j}$. The number of control on one axis are multiplied by the number of control points on the other axis.

A biquadratic Bezier patch has a degree of (2,2) and has 3 control points on one axis, and 3 control points on the other. That means that it has 9 control points total.

A bicubic Bezier patch has a degree of (3,3) with 4 control points on each axis, for a total of 16 control points.

If you had a patch of degree (7,1), it would have 8 control points on one axis and 2 control points on the other axis, and so would also have 16 control points total, but they would be laid out differently than a bicubic Bezier patch.

As far as actually calculating points on a curve, the above only calculates the value for a single axis for the final point on the curve. If you have three dimensional control points (X,Y,Z), you have to do the above math for each one to get the final result. This is the same as how it works for evaluating Bezier curves.

## Code

#include
#include

typedef std::array TFloat3;
typedef std::array TFloat3x3;

const TFloat3x3 c_ControlPointsX =
{
{
{ 0.7f, 0.8f, 0.9f },
{ 0.2f, 0.5f, 0.4f },
{ 0.6f, 0.3f, 0.1f },
}
};

const TFloat3x3 c_ControlPointsY =
{
{
{ 0.2f, 0.8f, 0.5f },
{ 0.6f, 0.9f, 0.3f },
{ 0.7f, 0.1f, 0.4f },
}
};

const TFloat3x3 c_ControlPointsZ =
{
{
{ 0.6f, 0.5f, 0.3f },
{ 0.7f, 0.1f, 0.9f },
{ 0.8f, 0.4f, 0.2f },
}
};

void WaitForEnter ()
{
printf("Press Enter to quit");
fflush(stdin);
getchar();
}

float QuadraticBezier (const TFloat3& p, float t)
{
float s = 1.0f - t;
float s2 = s * s;
float t2 = t * t;

return
p[0] * s2 +
p[1] * 2.0f * s * t +
p[2] * t2;
}

float BiquadraticBezierPatch(const TFloat3x3& p, float u, float v)
{
TFloat3 uValues;
uValues[0] = QuadraticBezier(p[0], u);
uValues[1] = QuadraticBezier(p[1], u);
uValues[2] = QuadraticBezier(p[2], u);
return QuadraticBezier(uValues, v);
}

int main(int argc, char **argv)
{
// how many values to display on each axis. Limited by console resolution!
const int c_numValues = 4;

printf("Bezier rectangle:n");
for (int i = 0; i < c_numValues; ++i)
{
float iPercent = ((float)i) / ((float)(c_numValues - 1));
for (int j = 0; j < c_numValues; ++j)
{
if (j == 0)
printf("  ");
float jPercent = ((float)j) / ((float)(c_numValues - 1));
float valueX = BiquadraticBezierPatch(c_ControlPointsX, jPercent, iPercent);
float valueY = BiquadraticBezierPatch(c_ControlPointsY, jPercent, iPercent);
float valueZ = BiquadraticBezierPatch(c_ControlPointsZ, jPercent, iPercent);
printf("(%0.2f, %0.2f, %0.2f) ", valueX, valueY, valueZ);
}
printf("n");
}
printf("n");

WaitForEnter();
return 0;
}


And here is the output it gives:

Note that in the program above, I evaluate the surface points by evaluating one axis and then the other. This is basically the same as how I explained it at the top, where I’m effectively animating the control points over distance, then evaluating the curve slice of the surface at that specific distance.

You could also write it another way though, where you literally expand the mathematical formula to get just one expression to evaluate that takes all control points at once. I like the simplicity (of understanding) of the method I used, but the other method works just as well.

## The Rendering

It’s easy enough to calculate values on a Bezier Rectangle, but what if you want to draw one?

One way is to tessellate it, or break it up into triangles and then render the triangles. You can think of it like trying to render a grid, where each point of the grid is moved to be where ever the Bezier rectangle function says it should be.

Raytracing against these objects in the general case is very difficult however, because it basically comes down to solving equations of very high degree.

Raymarching against these objects is also difficult unfortunately because while raymarching only needs to know “am i above the shape, or underneath it?”, knowing what u,v to plug into the equation to get the height most relevant to a random point in space is also very difficult. Not as difficult as the raytracing equations, but probably just as much out of reach.

But never fear, as always, you can cheat!

If you read my post about one dimensional (explicit) Bezier curves (One Dimensional Bezier Curves), you may remember that math gets easier if you use one dimensional control points. The same is actually true with Bezier rectangles!

For the ray marching case, you can march a point through space, and plug the x,z coordinate of the point into the Bezier rectangle function as u,v values and the number that comes out you can treat as a y coordinate.

Now, ray marching a Bezier rectangle is the same as ray marching any old height map (check links section for more info on that).

What I did in my demos, is since i knew that the curve was constrained to 0-1 on the x and z axis, and the y axis min and max was the control point min and maxes, I did a raytrace of that bounding box to get a minimum and maximum distance that the ray was inside that box. From there, I did raymarching from that min time to the max time along the ray, considering the ray as hitting the surface whenever the distance from the ray to the surface on the y axis (rayPos.y – bezierRectangle.y) changed sign.

After I had a hit, I got the height of the curve slightly offset on the x axis, then slightly offset on the z axis to get a triangle that I could calculate a surface normal from, to do lighting and shading with.

There is room for improvement in the ray marching though. I evenly divide the space through the box by a specific amount to control the size of the steps. A better way to do this I think would be to get the gradient of the function and use that to get a distance estimate (check links section below for more information). I could use that value to control the distance the ray marches at each step, and should be able to march through the box much quicker.

Also, as the link on terrain marching explains, you can usually take farther steps when the ray is farther from the camera, because the eye notices less detail. I removed that since the Bezier rectangles are pretty close to the camera, but it probably still would be helpful. Also, it would DEFINITELY be helpful in the case of the “Infinite Bezier Rectangles” scene.

I am pretty sure you could directly raytrace an explicit Bezier rectangle (one who has one dimensional control points) – at least for low degrees. I personally don’t know how you would do that, but I think it might boil down to solving a 4th degree function or something else “reasonable” based on a similar question I had about Bezier triangles on the mathematics stack exchange site (link below).

Another Way To Render

There is another way to render Bezier surfaces using ray based methods that I didn’t use but want to mention.

A property of Bezier curves and surfaces is that they are guaranteed to be completely contained by the convex hull created by their control points.

Another property of Bezier curves and surfaces is that you can use the De Casteljeau algorithm to cut them up. For instance you could cut a Bezier curve into two different Bezier curves, and the same holds for Bezier surfaces.

Using these two properties, there is an interesting way to be able to tell if a ray intersects a bezier curve or not, which is:

1. If the line misses the convex hull, return a miss
2. If the convex hull is smaller than a pixel, return a hit
3. Otherwise, cut the Bezier object into a couple smaller Bezier objects
4. Recurse for each smaller Bezier object

Yes, believe it or not, that is a real technique! It’s called Bezier Clipping and there is a research paper in the links section below that talks about some of the details of using that rendering technique.

## Links

Lastly, I wanted to mention that the above is completely about Bezier rectangles, but there is no reason you couldn’t extend these rectangles to use rational Bezier functions, or be based on B-splines or NURBS, or even go a different direction and make hermite surfaces or catmull-rom surfaces, or even make surfaces that used exotic basis functions of your own crafting based on trigonometric functions or whatever else!

Here are the shadertoy demos I made:
Shadertoy: Cubic Bezier Rectangle
Shadertoy: Quadratic Bezier Rectangle
Shadertoy: Infinite Bezier Rectangles