Shape Interrogation for Computer Aided Design and Manufacturing Book

1	Representation of Curves and Surfaces	1
1.1	Analytic representation of curves	1
1.2	Analytic representation of surfaces	4
1.3	Bezier curves and surfaces	6
1.4	B-spline curves and surfaces	20
1.5	Generalization of B-spline to NURBS	30
2	Differential Geometry of Curves	35
2.1	Arc length and tangent vector	35
2.2	Principal normal and curvature	39
2.3	Binormal vector and torsion	43
2.4	Frenet-Serret formulae	47
3	Differential Geometry of Surfaces	49
3.1	Tangent plane and surface normal	49
3.2	First fundamental form I (metric)	52
3.3	Second fundamental form II (curvature)	55
3.4	Principal curvatures	59
3.5	Gaussian and mean curvatures	64
3.6	Euler's theorem and Dupin's indicatrix	68
4	Nonlinear Polynomial Solvers and Robustness Issues	73
4.1	Introduction	73
4.2	Local solution methods	74
4.3	Classification of global solution methods	76
4.4	Projected Polyhedron algorithm	78
4.5	Auxiliary variable method for nonlinear systems with square roots of polynomials	88
4.6	Robustness issues	90
4.7	Interval arithmetic	92
4.8	Rounded interval arithmetic and its implementation	95
4.9	Interval Projected Polyhedron algorithm	105
5	Intersection Problems	109
5.1	Overview of intersection problems	109
5.2	Intersection problem classification	111
5.3	Point/point intersection	114
5.4	Point/curve intersection	114
5.5	Point/surface intersection	121
5.6	Curve/curve intersection	126
5.7	Curve/surface intersection	134
5.8	Surface/surface intersections	137
5.9	Overlapping of curves and surfaces	155
5.10	Self-intersection of curves and surfaces	157
6	Differential Geometry of Intersection Curves	161
6.1	Introduction	161
6.2	More differential geometry of curves	162
6.3	Transversal intersection curve	164
6.4	Intersection curve at tangential intersection points	170
6.5	Examples	177
7	Distance Functions	181
7.1	Introduction	181
7.2	Problem formulation	182
7.3	More about stationary points	185
7.4	Examples	192
8	Curve and Surface Interrogation	195
8.1	Classification of interrogation methods	195
8.2	Stationary points of curvature of free-form parametric surfaces	210
8.3	Stationary points of curvature of explicit surfaces	215
8.4	Stationary points of curvature of implicit surfaces	221
8.5	Contouring constant curvature	223
9	Umbilics and Lines of Curvature	231
9.1	Introduction	231
9.2	Lines of curvature near umbilics	232
9.3	Conversion to Monge form	237
9.4	Integration of lines of curvature	242
9.5	Local extrema of principal curvatures at umbilics	244
9.6	Perturbation of generic umbilics	250
9.7	Inflection lines of developable surfaces	256
10	Geodesics	265
10.1	Introduction	265
10.2	Geodesic equation	266
10.3	Two point boundary value problem	272
10.4	Initial approximation	275
10.5	Shortest path between a point and a curve	278
10.6	Numerical applications	281
10.7	Geodesic offsets	284
10.8	Geodesics on developable surfaces	287
11	Offset Curves and Surfaces	293
11.1	Introduction	293
11.2	Planar offset curves	307
11.3	Offset surfaces	316
11.4	Pythagorean hodograph	349
11.5	General offsets	352
11.6	Pipe surfaces	353
	Problems	367
	A: Color Plates	377
	References	381
	Index	405