# COMPUTER AIDED GEOMETRIC DESIGN. Thomas W. Sederberg

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1 COMPUTER AIDED GEOMETRIC DESIGN Thomas W. Sederberg January 31, 2011

2 ii

3 T. W. Sederberg iii Preface This semester is the 24 th time I have taught a course at Brigham Young University titled, Computer Aided Geometric Design. When I first taught such a course in 1983, the field was young enough that no textbook covered everything that I wanted to teach, and so these notes evolved. The field now has matured to the point that several semesters worth of valuable material could be compiled. These notes, admittedly biased towards my own interests, reflect my personal preferences as to which of that material is most beneficial to students in an introductory course. I welcome anyone who has an interest in studying this fascinating topic to make free use of these notes. I invite feedback on typos and on material that could be written more clearly. Thomas W. Sederberg Department of Computer Science Brigham Young University January 2007

4 iv T. W. Sederberg

5 Contents 1 Introduction Points, Vectors and Coordinate Systems Vector Algebra Points vs. Vectors Rotation About an Arbitrary Axis Matrix Form Parametric, Implicit, and Explicit Equations Lines Parametric equations of lines Implicit equations of lines Distance from a point to a line Conic Sections Parametric equations of conics Bézier Curves The Equation of a Bézier Curve Bézier Curves over Arbitrary Parameter Intervals The de Casteljau Algorithm Degree Elevation The Convex Hull Property of Bézier Curves Distance between Two Bézier Curves Derivatives Three Dimensional Bézier Curves Rational Bézier Curves De Casteljau Algorithm and Degree Elevation on Rational Bézier Curves First Derivative at the Endpoint of a Rational Bézier Curve Curvature at an Endpoint of a Rational Bézier Curve Continuity Circular Arcs Reparametrization of Bézier Curves Advantages of Rational Bézier Curves Explicit Bézier Curves Integrating Bernstein polynomials v

6 vi CONTENTS 3 Polynomial Evaluation and Basis Conversion Horner s Algorithm in Power Basis Horner s Algorithm in Bernstein Basis Basis Conversion Example Closed Form Expression Forward Differencing Choosing δ Properties of Blending Functions Timmer s Parametric Cubic Ball s Rational Cubic Overhauser Curves B-Spline Curves Polar Form Subdivision of Bézier Curves Knot Vectors Extracting Bézier Curves from B-splines Multiple knots Periodic B-splines Bézier end conditions Knot insertion The de Boor algorithm Explicit B-splines B-spline hodographs Symmetric polynomials Knot Intervals Knot Insertion Interval Halving Degree-Two B-Splines using Knot Intervals Hodographs Degree elevation B-spline Basis Functions B-Spline Basis-Functions using Knot Intervals Refinement of B-Spline Basis Functions Recurrence Relation Planar Curve Intersection Bezout s Theorem Homogeneous coordinates Circular Points at Infinity Homogeneous parameters The Fundamental Theorem of Algebra The Intersection of Two Lines Homogeneous Points and Lines Intersection of a Parametric Curve and an Implicit Curve Order of Contact

7 CONTENTS vii 7.4 Computing the Intersection of Two Bézier Curves Timing Comparisons Bézier subdivision Interval subdivision Bézier Clipping method Fat Lines Bézier Clipping Iterating Clipping to other fat lines Multiple Intersections Rational Curves Example of Finding a Fat Line Example of Clipping to a Fat Line Offset Curves 95 9 Polynomial Root Finding in Bernstein Form Convex Hull Marching Bernstein Combined Subdivide & Derivative Algorithm Multiplication of Polynomials in Bernstein Form Intersection between a Line and a Rational Bézier Curve Polynomial Interpolation Undetermined Coefficients Lagrange Interpolation Newton Polynomials Neville s Scheme Comparison Error Bounds Chebyshev Polynomials Interpolating Points and Normals Approximation Introduction L 2 Error Approximating a Set of Discrete Points with a B-Spline Curve Parametrization Knot vector Fairing Interpolation Constrained fairing Images Interval Bézier Curves Interval arithmetic and interval polynomials Interval Bézier curves Affine maps Centered form Error monotonicity

8 viii CONTENTS Envelopes of interval Bézier curves Interval hodographs Approximation by interval polynomials Remainder formulae and interval approximants Hermite interpolation Estimating bounds on derivatives Approximation by interval Bézier curves Floating Point Error Free-Form Deformation (FFD) TENSOR-PRODUCT SURFACES Tensor-Product Bézier Surface Patches The de Casteljau Algorithm for Bézier Surface Patches Tangents and Normals Tessellation of Bézier Curves and Surfaces The curve case The surface case C n Surface Patches NURBS Surface T-Splines Equation of a T-Spline T-spline Local Refinement Blending Function Refinement T-spline Spaces Local Refinement Algorithm Converting a T-spline into a B-spline surface Efficient Computation of Points and Tangents on a Bézier surface patch Curvature at the Corner of a Bézier Surface Patch Curvatures of tensor-product rational Bézier surfaces Curvatures of triangular rational Bézier surfaces Curvature of an Implicit Surface Algebraic Geometry for CAGD Implicitization Brute Force Implicization Polynomial Resultants Definition of the Resultant of Two Polynomials Resultant of Two Degree One Polynomials Resultants of Degree-Two Polynomials Resultants of Degree-Three Polynomials Resultants of Higher Degree Polynomials Determining the Common Root Implicitization and Inversion Implicitization in Bézier Form Inversion of Bézier Curves Curve Inversion Using Linear Algebra Curve-Curve Intersections

9 CONTENTS ix 16.9 Surfaces Base Points Ideals and Varieties Ideals of Integers Ideals of Polynomials in One Variable Polynomials in Several Variables Polynomial Ideals and Varieties Gröbner Bases Implicitization using Moving Lines Definition Homogeneous Points and Lines Curves and Moving Lines Weights and Equivalency Pencils and Quadratic Curves Pencils of lines Intersection of Two Pencils Pencils on Quadratic Curves Moving Lines Bernstein Form Moving Line which Follows Two Moving Points Intersection of Two Moving Lines Base Points Axial Moving Lines Curve Representation with Two Moving Lines Axial Moving Line on a Curve Axial Moving Line on a Double Point Cubic Curves Quartic Curves General Case Implicitization Tangent Moving Lines Tangent Moving Lines and Envelope Curves Reciprocal Curves Tangent Directions Genus and Parametrization of Planar Algebraic Curves Genus and Parametrization Detecting Double Points Implicit Curve Intersections Discriminants Parametrizing Unicursal Curves Undetermined Coefficients

10 x CONTENTS

11 List of Figures 1.1 Equivalent Vectors Vectors Vector Addition and Subtraction Vector Projection Rotation about an Arbitrary Axis Rotation about an Arbitrary Axis Using Vector Algebra Line given by A 0 + A 1 t Affine parametric equation of a line Line defined by point and normal Normalized line equation Examples of cubic Bézier curves Font definition using Bézier curves Bézier Curves in Terms of Center of Mass Cubic Bézier blending functions Bézier curves of various degree Subdividing a cubic Bézier curve Recursively subdividing a quadratic Bézier curve Subdividing a quadratic Bézier curve Degree Elevation of a Bézier Curve Convex Hull Property Difference curve Hodograph Rational Bézier curve Rational curve as the projection of a 3-D curve Osculating Circle Endpoint curvature C 2 Bézier curves Circular arcs Circle as Degree 5 Rational Bézier Curve Circle with negative weight Explicit Bézier curve Variation Diminishing Property Timmer s PC Ball s Cubic xi

12 xii LIST OF FIGURES 5.4 Overhauser curves Spline and ducks Polar Labels Affine map property of polar values Subdividing a cubic Bézier curve Böhm algorithm Double knot Special B-Spline Curves Knot Insertion B-spline with knot vector [ ] De Boor algorithm Sample cubic B-spline Periodic B-splines labelled with knot intervals Periodic B-splines with double and triple knots Inferring polar labels from knot intervals Knot Insertion using Knot Intervals Interval Splitting using Knot Intervals Interval Splitting using Knot Intervals Introducing Zero Knot Intervals Interval halving for a non-uniform quadratic B-spline curve Interval halving for a non-uniform cubic B-spline curve Quadratic B-Spline Curves Interval Splitting of a Quadratic B-Spline Curve Interval Splitting of a Quadratic B-Spline Curve Hodograph of a Degree 3 Polynomial B-Spline Curve Finding the Control Points of a B-Spline Hodograph Degree elevating a degree one and degree two B-spline Degree elevating a degree three B-spline Cubic B-Spline Curve Basis function B3(t) Sample Cubic B-Spline Curve B-Spline Basis Function for Control Point P i in Figure Convex Hulls Three iterations of Bézier subdivision Interval preprocess and subdivision Fat line bounding a quartic curve Bézier curve/fat line intersection Explicit Bézier curve After first Bézier clip Two intersections Two intersections, after a split Example of how to find fat lines Clipping to a fat line Clipping to L max Clipping example Additional examples

13 LIST OF FIGURES xiii 8.1 Offset Curves Offset Curves in which the Offset Radius Exceeds the Radius of Curvature for a Portion of the Base Curve Bernstein root finding Root isolation heuristic (a-d) Root isolation heuristic (e-h) Interpolating Four Points Interpolating Four Points Error Bounds Piecewise linear approximation of a Bézier curve Two cases of (x x 0 )(x x 1 ) (x x 9 ) for 0 x Uniform vs. bad parametrization Arc length vs. bad parametrization Uniform vs. bad knots The fairing effect The shrank curve The fairing effect of bad parameter Fairing and interpolation with different constant Constrained fairing Constrained fairing A cubic interval Bézier curve The affine map of two scalar points The affine map of two scalar intervals The affine map of two vector intervals Interval de Casteljau algorithm The envelope of an interval Bézier curve Approximate arc length parametrization of circle Affine map in floating point Affine map in floating point FFD example FFD example Continuity control FFD local coordinates FFD undisplaced control points Bézier surface patch of degree Surface in Figure 15.1.a viewed as a family of t-iso-parameter curves Applying the de Casteljau algorithm to the surface in Figure 15.1.a Partial derivative vectors for P [0,1] [0,1] (s, t)(assuming weights are unity) Surface Control Grid Teapot modeled using 32 bicubic Bézier surface patches Two C n bicubic Bézier surface patches Knot insertions into a NURBS surface

14 xiv LIST OF FIGURES 15.9 Splitting a NURBS surface into Bézier patches Head modeled (a) as a NURBS with 4712 control points and (b) as a T-spline with 1109 control points. The red NURBS control points are superfluous Car door modeled as a NURBS and as a T-spline NURBS head model, converted to a T-spline A gap between two B-spline surfaces, fixed with a T-spline Pre-image of a T-mesh Pre-image of a T-mesh Knot lines for blending function B i (s, t) Example T-Mesh Sample Refinement of B 1 (s, t) Nested sequence of T-spline spaces Local refinement example Semi-standard T-splines Curve example Curvature of a Bézier curve Part of a rectangular mesh Part of a triangular mesh Two cubic curves intersecting nine times Intersection of Two Pencils of Lines Dual Point and Line Pencil of Lines Pencil of Lines Pencil of Lines Intersection of Two Pencils of Lines Rational Quadratic Curve Quadratic Bézier Curve Cubic Moving Line which Follow Linear and Quadratic Moving Points Intersection of Linear and Quadratic Moving Lines Cubic Bézier Curve Quartic Bézier Curve Quartic Bézier Curve with a Triple Point Envelope curve Dual and Reciprocal Cusp Inflection Point Crunode Double Tangent Irreducible Cubic Curve Crunode: x 3 + 9x 2 12y 2 = Cusp: x 3 3y 2 = Acnode: x 3 3x 2 3y 2 = Circle and Hyperbola Parametrizing a Circle Parametrizing a Cubic Curve

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