GAmazing

GAmazing is a playground in order to show that geometric algebra is amazing.

The aim of this project is to realize multiple forms and fractals with geometric algebra using GARAMON to obtain beautiful things.

Amazing geometric algebra

How to install Garamon ?

Download this GARAMON

cd garamon
mkdir build
cd build
cmake ..
sudo make install

What is the dependencies of the project ?

• CMake (>3.10)
• Eigen (>3.3.4)
• C3GA
• OpenGL
• glm
• glew
• glfw
• (optional) Doxygen

Generate documentation (optional)

cd doxygen
doxygen Doxyfile

Getting started

mkdir build
cd build
cmake ..
make
./bin/GAmazing

Explanation

Square

4 CGA points a, b, c, d

c3ga::Mvec<float> a = c3ga::point<float>(-0.5f, -0.5f, 0);
c3ga::Mvec<float> b = c3ga::point<float>(0.5f, -0.5f, 0);
c3ga::Mvec<float> c = c3ga::point<float>(0.5f, 0.5f, 0);
c3ga::Mvec<float> d = c3ga::point<float>(-0.5f, 0.5f, 0);

Constrution of the plane

c3ga::Mvec<float> plane = a ^ b ^ c ^ c3ga::ei<float>();

Normals construction

c3ga::Mvec<float> normal = plane * a;

Retriving vertices position in euclidean space

c3ga::Mvec<float> a0 = a / a[c3ga::E0];
c3ga::Mvec<float> b0 = b / b[c3ga::E0];
c3ga::Mvec<float> c0 = c / c[c3ga::E0];
c3ga::Mvec<float> d0 = d / d[c3ga::E0];
glm::vec3 aPos = { a0[c3ga::E1], a0[c3ga::E2], a0[c3ga::E3] };
glm::vec3 bPos = { b0[c3ga::E1], b0[c3ga::E2], b0[c3ga::E3] };
glm::vec3 cPos = { c0[c3ga::E1], c0[c3ga::E2], c0[c3ga::E3] };
glm::vec3 dPos = { d0[c3ga::E1], d0[c3ga::E2], d0[c3ga::E3] };

Bouncing circle

Simple circle

c3ga::Mvec<double> C = c3ga::point<double>(1,2,2.0) 
                      ^ c3ga::point<double>(2,0,2.0) 
                      ^ c3ga::point<double>(1,0,2.0);  

Circle is displayed with a point x, y and a radius.

extractDualCircle use the center as x and y coord. The radius is extracted for the dual circle too and multiply by 100, otherwise the value was too small.

Flower

Simple sphere

c3ga::Mvec<double> sphere = c3ga::point<double>(0,0,2.0) 
                          ^ c3ga::point<double>(2,0,2.0) 
                          ^ c3ga::point<double>(1,1,2.0) 
                          ^ c3ga::point<double>(1,0,3.0);

We dualize the sphere and extract some parameters.

sphere /= sphere[c3ga::E0]; // back to correct scale
double squareRadius = sphere | sphere;
double radius = sqrt(squareRadius);
c3ga::Mvec<double> center = sphere;
center /= center[c3ga::E0];
c3ga::Mvec<double> direction;
direction[c3ga::E0] =  center[c3ga::E0];

The direction is used to compute the angle of the flower and the radius is obviously for the flower's radius.

Fractal 1

I played with some multivectors.

c3ga::Mvec<double> p;
p[c3ga::E0] = c3ga::e2<double>();
p[c3ga::E1] = imageXf;
p[c3ga::E2] = imageYf;

c3ga::Mvec<double> x = p;
for (int i = 0; i < maxIter; i++) {
    x = x * x + c; // n = 2
    if (x.norm()/10.0f > 20) {
        break;
    }
}

Each value is then stored into a buffer

float valF = (x.norm() / 1e10f) * 100;
float val = (valF > 255) ? 255 : (valF);
rgbBuffer[idx + 0] = 0.0f;
rgbBuffer[idx + 1] = int(val/5);
rgbBuffer[idx + 2] = val;

Fractal 2

Same as above except that we play with the condition on x and the buffer storage.

Ressources

Nom du site our de l'article	Auteur(s)	Lien
GARAMON : Geometric Algebra Recursive and Adaptative Monster	Stephane Breuils. Vincent Nozick. Laurent Fuchs	http://www-igm.univ-mlv.fr/~vnozick/data/garamon.pdf
Computer graphics using conformal geometric algebra	Richard James Wareham	https://rjw57.github.io/phd-thesis/rjw-thesis.pdf
Euclidean Geometry and Geometric Algebra	Chris Doran	http://geometry.mrao.cam.ac.uk/2020/06/euclidean-geometry-and-geometric-algebra/
Geometric algebra for computer science	Leo Dorst. Daniel Fontijne. Stephen Mann	https://geometricalgebra.org/sandbox.html
Ganja	Steven De Keninck	https://github.com/enkimute/ganja.js
Polyèdres	Mathcurve	https://mathcurve.com/polyedres/polyedres.shtml
Clifford Algebra: A visual introduction	Slehar	https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/