Approximation and Visualisation of 3D Objects
Contents
Introduction
1. Position Transformations and Rendering of Geometrical Models
1.1 2D Geometrical Models and Position Transformations
1.1.1 Homogeneous Coordinates
1.1.2 Elementary Transformations in 2D
1.1.3 Multiple Transformations in 2D
1.1.4 Animation of Motion of a Body in 2D
1.2 3D Geometrical Models and Position Transformations
1.2.1 Elementary and Multiple Transformations in 3D
1.2.2 Problem of 3D Rotation: Euler’s System of Angles
1.2.3 3D Rotation about a Given Axis
1.2.4 Animation of Motion of a Body in 3D
1.3 Creation of Images of 3D Geometrical Objects
1.3.1 Parallel Projections
1.3.2 Prospective Projections
1.3.3 Non-Linear Projections
2. Curves and Surfaces
2.1 Analytical Curves and Surfaces
2.1.1 Calculation and Visualisation of Curves
2.1.2 Visualising Surfaces in MATLAB
2.1.3 Parametric Representation of Surfaces
2.1.4 Adaptation of the Surface Shape to Given Curves
2.1.5 Animation of Motion of Parametric Surfaces
2.2 Curves and Surfaces Obtained by Interpolation
2.2.1 Polynomial Interpolation of Curves
2.2.2 Polynomial Interpolation of Curved Surfaces
2.2.2.1 Obtaining Interpolation Coefficients by Hemming Method
2.2.2.2 Interpolation by Doubled Lagrange Functions
2.2.2.3 Interpolation over Curvilinear Quadrilateral Domains
2.2.2.4 Serendipity Interpolation Functions
2.2.3 Hermitian Interpolation of Curves and Surfaces
2.2.4 Interpolation of Curves and Surfaces by Global Splines
2.3 Curves and Surfaces Obtained by Approximation
2.3.1 Least Squares Polynomial Approximation
2.3.2 Bezier Curves
2.3.3 Polynomial Base for Bezier Curves
2.3.4 Bezier Surface Approximation by Bernstein Polynomials
2.3.5 Cubic Uniform B-Splines
2.3.6 Uniform B-Spline Surfaces
2.3.7 Cubic Non-Uniform Rational B-Splines (NURBS)
2.4 Lighting and Textures of Surfaces
2.4.1 Concepts of Lighting
2.4.2 Elementary Models of Lighting
2.4.3 Lighting of Meshed Surfaces
2.4.4 Textures
2.4.4.1 Textures on Quadrilateral Meshgrids
2.4.4.2. Textures on freely triangulated surfaces
3. Skeleton Models of Graphical Objects
3.1 Position Transformations of Skeletons
3.1.1 3D Model of a Bone and Its Position Transformations
3.1.2 3D Model of a Chain and Its Position Transformations
3.1.3 Envelopes of Bones
3.1.4 Motion Animation of Systems of Chains
3.2 Inverse Kinematic Problem Based on Optimisation
3.2.1 General Case of Chain Kinematics Equation with Constraints
3.2.2 Penalty Function Formulation
3.2.3 Calculation of the Gradient
3.2.4 Search for a Solution along the Gradient Direction
3.3 Other Issues of Skeleton Modelling
References