LEARNING OUTCOMES

Learning the theoretical and methodological fundamentals of Computer Graphics

PREREQUISITES

Linear algebra: vectors, matrices, linear transformations. 

Imperative programming

TEACHING METHODS

In presence classes for theory. Autonomous work by students for homework & final project.

SYLLABUS/CONTENT

Linear algebra: Vectors, matrices, and related operations; coordinate frame, change of frame; linear systems; geometric interpretations.

Images: vector and raster; output devices; image coordinates; color spaces; image formats.

Ray tracing: parallel and perspective projection; basic geometric intersections; shading: diffuse, specular, and ambient; shadows, reflections, and refractions.

Spatial data structures: queries and classification; Spatial indexes: regular grid, kd-tree, quadtree/octree, BSP; Primitive sorting techniques: Bounding Volume Hierarchies; Geometric proxies: sphere, capsule, half-space, AABB, OABB, convex and general polyhedron; collision detection strategies: static, dynamic.

Procedural synthesis: procedural noise; Perlin noise; color maps; implicit modeling: combination of distances and angles.

Implicit solid modeling: CSG; Rendering implicit models: ray marching; Explicit meshing of implicit models: marching squares/cubes; Implicit modeling in additive manufacturing.

Geometric transformations: linear transformations: scaling, rotation, and shearing in 2D; translation; affine transformations and algebra; lines in the affine space; affine sum; the affine space; homogeneous coordinates; scale, rotation, and translation in homogeneous coordinates in 2D and 3D; concatenation of transformations; generic rotations about the origin; Euler coordinates; Euler-axis angle; Transformations of normals; 

Viewing transformations: pipeline of transformations; viewport transformation; orthographic projection; camera transformation; change of frame; examples; composition of transformation and transformation matrices. 

Rasterization theory: canonical view volume and pixel grid; implicit shape representation and detection of pixels inside a shape; point-in-triangle test; edge function; barycentric interpolation; triangle rasterization; interpolation of attributes;  clipping; depth sorting; z-buffering; super-sampling anti-aliasing. 

Rasterization implementation: GPU, content creation, window manager, CPU and GPU memory and bus, OS-specific aspects. Software rasterization: rasterization pipeline, vertex input, presentation of a software rasterizer: rasterization of lines; shaders; rasterization pipeline; attributes; uniforms; vertex attributes and color interpolation; view transformation and preservation of aspect ratio; depth test.

Picking through ray casting.

Perspective transformations: perspective projection; extension of homogeneous coordinates and of the affine space; perspective division; perspective projection from frustum to parallelepiped; composition with the orthographic projection; aperture and aspect ratio; perspective division for interpolation of attributes.

Texture mapping: examples of color mapping, bump mapping, and displacement mapping;  UV mapping and texture lookup; problems with seams and distortion; resampling: magnification and minification; derivatives in screen space; nearest filtering and bilinear filtering; Moire patterns; mipmapping.

View transformations: order of transformations; modeling transformations: how to place different objects; scene graph; stack of matrices; placing the camera; orbiting camera; camera on a car (POV); object viewer; Euler angles; trackball; implementation with quaternions.

RECOMMENDED READING/BIBLIOGRAPHY

Material and references provided by the instructors