APPLIED GEOMETRY FOR COMPUTER GRAPHICS AND CAD

By A. ELORA

CONTENTS 1 TRANSFORMATIONS OF THE PLANE INTRODUCTION TRANSLATIONS SCALING ABOUT THE ORIGIN RE?ECTIONS ROTATION ABOUT THE ORIGIN SHEARS CONCATENATION OF TRANSFORMATIONS APPLICATIONS INSTANCING ROBOTICS 2 HOMOGENEOUS COORDINATES AND TRANSFORMATIONS OF THE PLANE HOMOGENEOUS COORDINATES POINTS AT IN?NITY VISUALIZATION OF THE PROJECTIVE PLANE LINE MODEL OF THE PROJECTIVE PLANE SPHERICAL MODEL OF THE PROJECTIVE PLANE TRANSFORMATIONS IN HOMOGENEOUS COORDINATES INVERSE TRANSFORMATIONS ROTATION ABOUT AN ARBITRARY POINT RE?ECTION IN AN ARBITRARY LINE DEVICE COORDINATE TRANSFORMATION POINT AND LINE GEOMETRY IN HOMOGENEOUS COORDINATES 3 HOMOGENEOUS COORDINATES AND TRANSFORMATIONS OF SPACE TRANSFORMATIONS OF SPACE SCALINGS AND RE?ECTIONS ROTATIONS ABOUT THE COORDINATE AXES ROTATION ABOUT AN ARBITRARY LINE RE?ECTION IN AN ARBITRARY PLANE COMPUTER-AIDED DESIGN ORIENTATION OF A RIGID BODY GEOMETRIC METHODS FOR LINES AND PLANES IN SPACE QUATERNIONS 4 PROJECTIONS AND THE VIEWING PIPELINE PROJECTIONS OF THE PLANE PROJECTIONS OF THREE-DIMENSIONAL SPACE THE VIEWPLANE COORDINATE MAPPING THE VIEWING PIPELINE CLASSI?CATION OF PROJECTIONS CLASSI?CATION OF PARALLEL PROJECTIONS CLASSI?CATION OF PERSPECTIVE PROJECTIONS 5 CURVES CURVE RENDERING PARAMETRIC CURVES ARCLENGTH AND REPARAMETRIZATION APPLICATION: NUMERICAL CONTROLLED MACHINING AND O?SETS CONICS CLASSI?CATION OF CONICS CONICS IN STANDARD FORM INTERSECTIONS OF A CONIC WITH A LINE PARAMETRIZATION OF AN IRREDUCIBLE CONIC CONVERTING FROM PARAMETRIC FORM TO IMPLICIT FORM CONICS IN SPACE APPLICATIONS OF CONICS 6 B´EZIER CURVES I B´EZIER CURVES OF LOW DEGREE LINEAR B´EZIER CURVES QUADRATIC B´EZIER CURVES CUBIC B´EZIER CURVES THE E?ECT OF ADJUSTING A CONTROL POINT THE GENERAL B´EZIER CURVE PROPERTIES OF THE BERNSTEIN POLYNOMIALS CONVEX HULLS PROPERTIES OF B´EZIER CURVES THE DE CASTELJAU ALGORITHM SUBDIVISION OF A B´EZIER CURVE 6 10 APPLICATIONS 6 10 1 RENDERING 6 10 2 INTERSECTION OF A PLANAR B´EZIER CURVE AND A LINE 6 10 3 INTERSECTION OF TWO B´EZIER CURVES 7 B´EZIER CURVES II SPATIAL B´EZIER CURVES DERIVATIVES OF B´EZIER CURVES CONVERSIONS BETWEEN REPRESENTATIONS PIECEWISE B´EZIER CURVES RATIONAL B´EZIER CURVES PROPERTIES OF RATIONAL B´EZIER CURVES DE CASTELJAU ALGORITHM FOR RATIONAL CURVES PROJECTIONS OF RATIONAL B´EZIER CURVES DERIVATIVES OF RATIONAL B´EZIER CURVES 8 B-SPLINES INTEGRAL B-SPLINE CURVES PROPERTIES OF THE B-SPLINE CURVE B-SPLINE TYPES APPLICATION: FONT DESIGN APPLICATION: MORPHING OR SOFT OBJECT ANIMATION THE DE BOOR ALGORITHM DERIVATIVES OF A B-SPLINE NON-UNIFORM RATIONAL B-SPLINES (NURBS) PROJECTIONS OF NURBS CURVES DERIVATIVES OF NURBS RATIONAL DE BOOR ALGORITHM KNOT INSERTION 9 SURFACES QUADRIC SURFACES O?SET SURFACES B´EZIER AND B-SPLINE SURFACES PROPERTIES OF B´EZIER AND B-SPLINE SURFACES DERIVATIVES OF B´EZIER AND B-SPLINE SURFACES SURFACE CONSTRUCTIONS EXTRUDED SURFACES RULED SURFACES TRANSLATIONALLY SWEPT SURFACES SURFACES OF REVOLUTION SURFACE SUBDIVISION SKIN AND LOFT SURFACES GEOMETRIC MODELLING AND CAD WIREFRAME MODELLER SURFACE MODELLER CONSTRUCTIVE SOLID GEOMETRY (CSG) MODELLERS BOUNDARY REPRESENTATIONS (B-REP) 10 CURVE AND SURFACE CURVATURES 10 1 CURVATURE OF A PLANE CURVE 10 2 CURVATURE AND TORSION OF A SPACE CURVE 10 3 CURVATURE OF B´EZIER CURVES 10 4 SURFACE CURVATURES 11 RENDERING 11 1 INTRODUCTION 11 2 COLOUR 11 3 AN ILLUMINATION MODEL FOR RE?ECTED LIGHT 11 3 1 DI?USE RE?ECTION 11 3 2 SPECULAR RE?ECTION 11 3 3 AMBIENT RE?ECTION 11 3 4 ATTENUATION 11 3 5 TOTAL INTENSITY 11 4 SHADING ALGORITHMS 11 4 1 FLAT SHADING 11 4 2 GOURAUD SHADING 11 4 3 PHONG SHADING 11 5 SILHOUETTES 11 6 SHADOWS SOLUTIONS REFERENCES