Computational Geometry

Contents

1. Two Dimensional Transformations
1. Introduction
2. Representation of Points
3. Transformation and Matrices
4. Transformation of Points (Scaling, Reflection & Shearing)
5. Rotation (about the Origin)
5.1 Inverse Rotation
5.2 Concatenation of Transformations (Or Combined Transformations)
6. Transformation of Straight Line
7. Midpoint Transformation
8. Transformation of Parallel Lines
9. Transformation of Intersecting Lines
10. Transformation and Area of Plane Figure
11. Solid Body Transformations
12. Homogeneous Co-ordinates and Translation
13. Rotation about an Arbitrary Point
14. Reflection through an Arbitrary Line
14.1 Algorithm for reflection Through the Line y = mx + c
15. Overall Scaling
16. Points at Infinity

2. Three Dimensional Transformations
1. Introduction
1.1 Geometric Effects of Elements of [T]
2. Three Dimensional Rotation
3. Multiple Transformations
3.1 Rotation about an axis Parallel to Co-ordinate axes
3.2 Rotation about an Arbitrary Axis in the Space
4. Reflections through Co-ordinate Planes
5. Reflection through an Arbitrary Plane

3. Projection
1. Introduction
2. Affine and Perspective Geometry
2.1 Two Basic Types of Projection
3. Orthographic Projection
3.1 Six Different Views of an Object
4. Axonometric Projections
4.1 Principal Foreshortening Factors
4.2 Trimetric, Dimetric, Isometric Projections
5. Oblique Projections
5.1 Standard Oblique Projection
5.2 Transformation Matrix for Oblique Projection
6. Perspective Projection
6.1 Transformation Matrix for Perspective Projection
6.2 Properties of Perspective Projection
7. Vanishing Points

4. Plane Curves
1. Introduction
2. Curve Representation
3. Non-Parametric Curves
4. Parametric Curves
5. Parametric Representation of Circle
5.1 Generation of n-points on the Circle x2 + y2 = r2
5.2 Generation of an arc of Circle
5.3 Generation of n-points on the Circle (x–h)2+(y–k)2= r2
6. Parametric Representation of a Parabola and Generation of Parabola
6.1 Iterative Technique to Generate n-points on Parabola
6.2 Algorithm to Generate n-points on Parabola y2=4ax