3D Transformations I: 3D Translation, Scaling, and Coordinate Systems

Lecture 17 of CS602 extends geometric transformations into 3D space using 4x4 homogeneous coordinate matrices. Students formulate mathematical matrix operators for spatial translation along x, y, and z axes, uniform scaling, and differential 3D scaling relative to fixed coordinates. Scholars evaluate three-dimensional coordinate conventions, distinguishing right-handed and left-handed systems during 3D modeling and camera viewing. The curriculum contrasts orthographic parallel projections with perspective projections, calculating center of projection rays and vanishing points. Surface normal vectors are computed via cross products of triangle edges, providing directional references for back-face culling and lighting. The Phong illumination model decomposes surface reflectance into ambient environmental glow, Lambertian diffuse scattering, and shiny specular highlights. Shading techniques are evaluated across computational cost and visual fidelity, contrasting flat faceted shading with Gouraud and Phong interpolations. Students examine the OpenGL graphics pipeline, analyzing how vertex attributes flow through modelview and projection matrix stacks toward fragment shaders. OpenGL state machine concepts illustrate how current color, material reflectance, and lighting parameters persist across immediate mode rendering calls. Parametric polynomial curves such as cubic Hermite splines model smooth continuous trajectories using endpoint coordinates and tangent vectors. Bezier curve mathematics using Bernstein basis polynomials and de Casteljau recursive division provide intuitive affine-invariant control polygons. Surface modeling principles illustrate how bicubic patches and triangle meshes approximate complex freeform organic geometry in 3D space. Fractal geometry principles demonstrate how recursive affine transformations and self-similarity generate natural phenomena like coastlines and mountains. Viewing transformations construct camera orientation matrices by computing look-at, camera position, and normalized up direction vectors. Barycentric coordinates interpolate vertex attributes including colors, normals, and texture coordinates smoothly across rasterized triangle interiors. By engaging with solved past examination papers by Moaaz and Waqar Siddhu, learners gain actionable insights into recurrent university exam questions. Interactive quiz assessments on VU SIGMA facilitate rapid knowledge verification and retention of critical computer graphics terminology. Comprehensive coverage of rasterization and illumination standards ensures learners communicate with technical precision across visual computing industries.