Computer Graphics

1. Analytic geometry overview and review. Construction of geometries, the main characteristics of various important geometries. Analytical geometry of Euclidean space: point, vector, coordinate systems. Meaning and implementation of vector operations in C++. Combination of points: parametric equations of a segment, line and circle. Distance: plane, sphere implicit equation. Algebras: vector, matrix, complex number, Clifford algebra.

2. Geometric modeling. Classic curves: implicit, parametric and explicit forms. Freeform curves with a combination of points. Lagrangian interpolation. Hermite interpolation to two points. Bezier approximation. Catmull-Rom spline. Parametric surfaces. Surface with extraction and rotation. Catmull-Clark split curve and surface.

3. Geometric transformations: Elementary transformations and their matrix formalism. Homogeneous coordinates. Projective geometry (ideal point, relation between Cartesian and homogeneous coordinates). Homogeneous linear transformations and their properties. Shift, scale, rotate (Rodriguez formula).

4.  2D image synthesis: Vectorization of curves. Dividing polygons into triangles. Modeling transformation. View transformation. Cutting sections and areas. Section drawing. Area filling.

5.  GLUT/OpenGL 3 and 4/GLSL: Syntax, connection with the windowing system. Open window, register event handler functions. Vertex array object and vertex buffer object. The GPU pipeline for 2D graphics. GLSL shaders. Implementation of "Hello triangle" in OpenGL/GLSL environment.

6.  Basic optical model of 3D image synthesis: Beam density. BRDF. Optically smooth materials, law of reflection and refraction of geometrical optics, Fresnel equations. Wrinkled surfaces, diffuse and shimmering surface. Direction and point light source. Concept of colors, color matching.

7.  Ray tracing: Solving the visibility problem, normal vector of surfaces, shadow calculation. Recursive ray tracing: reflection and refraction.

8.  Incremental 3D image synthesis. Tessellation of surfaces. Modeling transformation. View transformation in the case of perspective projection. Cutting in homogeneous coordinates. Masking problem in screen coordinate system, z-buffer algorithm. Gouraud and Phong shading. Texture mapping. 3D graphics application of OpenGL and graphics hardware. Texture rendering, filtering. GPU programming.

 9. Computer games: Construction of virtual reality systems and games. The avatar. Game engine. Posters, particle systems. The physics of games. Field modeling. Movement of characters.