cppd/math

The more dimensions, the better the geometry.

Math Viewer

Just for fun with mathematics and stuff

Contents

• File types
• Rendering
  • GPU
  • CPU
• Algorithms
  • 2-space
  • Spaces with arbitrary number of dimensions
• Technical details
• Images
  • 3-manifold reconstruction and path tracing in 4-space
  • 4D Cornell box
  • Real-time rendering
  • 3D models
  • 4D polytope

File types

• OBJ and STL files with objects in 3-space
• Extended OBJ and STL files with objects in n-space
• Text files with the coordinates of points in n-space

Rendering

GPU

• Triangle meshes (rasterization)
• Volumes (ray marching)
• Isosurfaces (ray marching)
• Shadows (ray tracing, ray marching, shadow mapping)
• Order-independent transparency (lists of fragments)

CPU

• Bidirectional path tracing with multiple importance sampling
• (n-1)-simplex meshes in n-space on (n-1)-dimensional screen
• n-dimensional microfacet BRDF
• Spectral rendering

Algorithms

2-space

C++ and GLSL

Subject	Algorithms
Discrete Fourier transform for arbitrary sizes	Bluestein's algorithm and radix-2 fast Fourier transform
Optical flow	Pyramidal Lucas-Kanade
Convex hull	Divide et impera

Spaces with arbitrary number of dimensions

C++

Subject	Algorithms
Convex hull	Randomized incremental
Delaunay triangulation	Convex hull of paraboloid
Voronoi diagram	The Delaunay triangulation
Manifold reconstruction	Cocone
Manifold reconstruction with boundary detection	BoundCocone
Euclidean minimum spanning tree	Kruskal’s algorithm and the Delaunay triangulation
Intersection of hyperplanes	Gaussian elimination
Intersection of convex polytopes	The simplex algorithm
Ray intersection acceleration using spatial subdivision	2d-trees
Ray intersection acceleration using object subdivision	Bounding volume hierarchies and the surface area heuristic
Eigenvalues and eigenvectors of symmetric matrices	Jacobi method
Least squares fitting a plane to points	Principal component analysis
State estimation	The Kalman filter The extended Kalman filter (EKF) The unscented Kalman filter (UKF) The H∞ filter

Technical details

Property	Value
Programming languages	C++23, GLSL 4.60
C++ compilers	GCC 13, Clang 18
Platforms	Linux
Graphics and compute APIs	Vulkan 1.3 with ray tracing extensions
Frameworks	Qt 6
Libraries	FreeType, GMP
Optional libraries	cuFFT (for DFT tests), FFTW (for DFT tests)
Build systems	CMake

Images

3-manifold reconstruction and path tracing in 4-space

2D images of a 3D image of the 3-manifold reconstructed from the points on the 3-torus T³ (S¹×S¹×S¹)

• 5-space
  • The convex hull
  • 4-simplex facets
• 4-space
  • 3-simplex mesh without boundary
  • 4-parallelotope hierarchy
  • 4-dimensional microfacet BRDF
  • 3-ball light source
  • 4-dimensional soft shadows
  • Parallel projection onto 3D screen

Volumetric rendering

4D Cornell box

2D images of a 3D image of the 4-dimensional Cornell box

• Seven 3-parallelotopes (the sides of the box)
• One 3-parallelotope light source
• One 3-simplex mesh without boundary (3-sphere)
• One 4-parallelotope
• 4-dimensional microfacet BRDF
• 4-dimensional soft shadows
• Perspective projection onto 3D screen

Real-time rendering

Semitransparent objects

• Triangle mesh (green) and its ray tracing shadows
• Isosurface (blue) and its ray marching shadows

3D models

Spectral rendering

1. Blackbody 2500 K
2. Daylight 4000 K
3. Daylight D65
4. Blackbody 12000 K

4D polytope

4-dimensional cross-polytope

v -11   0   0   0
v   0 -12   0   0
v   0   0 -13   0
v   0   0   0 -14
v   0   0   0  15
v   0   0  16   0
v   0  17   0   0
v  18   0   0   0
f 1 2 3 4
f 1 2 3 5
f 1 2 6 4
f 1 2 6 5
f 1 7 3 4
f 1 7 3 5
f 1 7 6 4
f 1 7 6 5
f 8 2 3 4
f 8 2 3 5
f 8 2 6 4
f 8 2 6 5
f 8 7 3 4
f 8 7 3 5
f 8 7 6 4
f 8 7 6 5