Geometry and Algorithms for COMPUTER AIDED DESIGN Erich Hartmann Department of Mathematics Darmstadt University of Technology October 2003 2 Contents 1 INTRODUCTION 7 1.1 Methods for DISPLAYING objects . 7 1.2 On the contents . 8 1.3 Software for displaying curves and surfaces . 9 1.4 On literature . 10 2 TOOLS 11 2.1 Structure of a CAD-program . 11 2.1.1 Global constants: The file ”geoconst.pas” . 11 2.1.2 Globale types: The file ”geotype.pas” . 11 2.1.3 Globale variables: The file ”geovar.pas” . 11 2.1.4 Capability of the graphics software . 12 2.2 Functions on IR, operations with vectors . 14 2.2.1 Functions on IR . 14 2.2.2 Operations with vectors . 14 2.3 Routines to Analytic Geometry . 17 2.3.1 Polarangle and quadratic equation . 17 2.3.2 Intersection line–line, circle–line, circle–circle . 17 2.3.3 Equation of a plane . 18 2.3.4 Intersection line–plane . 18 2.3.5 Intersection of three planes . 18 2.3.6 Intersection of two planes . 19 2.3.7 ξ-η–coordinates of a point in a plane . 19 2.3.8 Coordinates with respect of a new 3D-coordinate system . 19 2.4 Numeric: GAUSS-elimination, NEWTON-iteration . 19 3 PARALLEL/CENTRAL–PROJECTION 21 3.1 Orthographic Projection . 21 3.1.1 The projection formulae . 21 3.1.2 Procedures for orthographic projection . 22 3.2 Central projection . 23 3.2.1 The projection formulae . 23 3.2.2 Procedures for central projection . 24 4 CURVES 27 4.1 Planar Curves . 27 4.1.1 Definition and Representations of Planar Curves . 27 4.1.2 Arc length and curvature of a planar curve . 28 4.1.3 Offset Curves . 30 4.1.4 The normalform of a planar curve . 30 3 4.2 Displaying Parametric Curves in IR2 ........................... 32 4.3 Displaying implicit curves . 33 4.3.1 Marching algorithm . 33 4.3.2 Raster algorithm . 35 4.4 Intersection of two planar curves . 37 4.4.1 Intersection of a parametric curve and an implicit curve . 38 4.4.2 Intersection of two implicit curves . 38 4.4.3 Intersection of two parametric curves . 39 4.5 Foot points on planar curves . 39 4.5.1 Foot point on a parametric curve, curve inversion . 39 4.5.2 Foot point on an implicit curve . 40 4.5.3 Stable first order foot point algorithms . 41 4.6 B´ezier–curves . 42 4.7 Applications of the normalform . 43 4.7.1 Offset curves of implicit curves . 44 4.7.2 Bisector curves . 45 4.7.3 Numerical Parameterization of Curves . 46 4.8 Space Curves . 46 5 SURFACES 49 5.1 Definition and Representations of Surfaces . 49 5.2 The First and Second Fundamental Forms of a Surface . 50 5.2.1 The first fundamental form, arc length . 50 5.2.2 The second fundamental form, curvature . 50 5.3 Offset surfaces . 52 5.4 Normalform of a surface . 52 5.4.1 Definition of the normalform . 52 5.4.2 On the first and second derivatives of the normalform of a surface . 53 5.4.3 Cn–contact, Gn–contact . 55 5.4.4 G2–continuity theorems . 56 5.4.5 The curvature of an intersection curve . 57 5.5 Normalform of an implicit surface . 58 5.6 Normalform of a parametric surface . 59 5.7 Foot point on a parametric surface, surface inversion . 60 5.8 Stable first order foot point algorithms for surfaces . 61 5.8.1 Foot point algorithm for parametric surfaces . 61 5.8.2 Foot point algorithm for implicit surfaces . 61 5.9 The normalform of a space curve . 62 5.9.1 Definition of the normalform . 62 5.9.2 Foot point algorithm and evaluation of the normalform of a space curve . 63 5.9.3 Determining foot points on an intersection curve . 64 5.10 Applications of normalforms . 64 5.10.1 Approximation of a set of intersecting surfaces . 64 5.10.2 Approximation of intersecting pipe surfaces . 65 5.10.3 Numerical parametrization of implicit surfaces . 65 6 HIDDENLINE–ALGORITHM FOR NON CONVEX POLYHEDRONS 67 6.1 The Hiddenline-Algorithm . 67 6.2 Auxiliary procedures for the hiddenline algorithm . 71 6.2.1 The procedure aux polyhedron .......................... 71 6.2.2 The procedures aux quadrangle, aux cylinder, aux torus and displaying a parametrized surface . 71 4 6.3 Intersection of faces, intersection of two polyhedrons . 74 6.3.1 Intersection of faces bounded by planar polygons in IR3 . 74 6.3.2 Intersection of two polyhedrons . 76 6.4 IS line–polygonal patch, IS polygon–polyhedron . 78 6.4.1 Intersection of a line segment and a planar polygonal patch . 78 6.4.2 Intersection points of a polygon and a polyhedron . 79 7 TRIANGULATION OF IMPLICIT SURFACES 81 7.1 The triangulation algorithm (marching method) . 81 7.1.1 The idea of the algorithm . 81 7.1.2 The procedure surfacepoint ........................... 83 7.1.3 The data structure . 83 7.1.4 Step S0 . 84 7.1.5 Step S1 . 84 7.1.6 Step S2 . 84 7.1.7 Step S3 . 85 7.2 Examples . 87 7.2.1 Sphere . 87 7.2.2 Cylinder . 88 7.2.3 Torus . 88 7.2.4 Set of two intersecting surfaces . 88 7.2.5 G2-continuous blending of three cylinders . 89 7.3 Sample program: trisample.p .............................. 91 7.4 Ray tracing of triangulated surfaces with POVRAY . 92 8 INTERSECTIONS: CURVE – SURFACE, SURFACE – SURFACE 93 8.1 Intersection Curve – Surface . 93 8.1.1 IS parametric curve – implicit surface . 93 8.1.2 IS implicit curve – implicit surface . 93 8.1.3 IS implicit curve – parametric surface . 94 8.1.4 IS parametric curve – parametric surface . 94 8.2 Intersection surface – surface . 94 8.2.1 IS of two implicit surfaces . 94 8.2.2 IS of an implicit and a parametric surface . 96 8.2.3 IS of two parametric surfaces . ..