Ebook: Bézier and B-Spline Techniques

Computer-aided modeling techniques have been developed since the advent of NC milling machines in the late 40's. Since the early 60's Bezier and B­ spline representations evolved as the major tool to handle curves and surfaces. These representations are geometrically intuitive and meaningful and they lead to constructive numerically robust algorithms. It is the purpose of this book to provide a solid and unified derivation of the various properties of Bezier and B-spline representations and to show the beauty of the underlying rich mathematical structure. The book focuses on the core concepts of Computer-aided Geometric Design (CAGD) with the intent to provide a clear and illustrative presentation of the basic principles as well as a treatment of advanced material, including multivariate splines, some subdivision techniques and constructions of arbitrarily smooth free-form surfaces. In order to keep the book focused, many further CAGD methods are ex­ cluded. In particular, rational Bezier and B-spline techniques are not ad­ dressed since a rigorous treatment within the appropriate context of projec­ tive geometry would have been beyond the scope of this book.

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This book provides a solid and uniform derivation of the various properties B?zier and B-spline representations have, and shows the beauty of the underlying rich mathematical structure. The book focuses on the core concepts of Computer Aided Geometric Design with the intension to give a clear and illustrative presentation of the basic principles, as well as a treatment of advanced material including multivariate splines, some subdivision techniques and constructions of free form surfaces with arbitrary smoothness.
The text is beautifully illustrated with many excellent figures to emphasize the geometric constructive approach of this book.

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This book provides a solid and uniform derivation of the various properties B?zier and B-spline representations have, and shows the beauty of the underlying rich mathematical structure. The book focuses on the core concepts of Computer Aided Geometric Design with the intension to give a clear and illustrative presentation of the basic principles, as well as a treatment of advanced material including multivariate splines, some subdivision techniques and constructions of free form surfaces with arbitrary smoothness.
The text is beautifully illustrated with many excellent figures to emphasize the geometric constructive approach of this book.
Content:
Front Matter....Pages I-XIV
Front Matter....Pages 1-1
Geometric fundamentals....Pages 3-8
B?zier representation....Pages 9-23
B?zier techniques....Pages 25-41
Interpolation and approximation....Pages 43-57
B-spline representation....Pages 59-75
B-spline techniques....Pages 77-89
Smooth curves....Pages 91-107
Uniform subdivision....Pages 109-121
Front Matter....Pages 123-123
Tensor product surfaces....Pages 125-140
B?zier representation of triangular patches....Pages 141-153
B?zier techniques for triangular patches....Pages 155-169
Interpolation....Pages 171-178
Constructing smooth surfaces....Pages 179-188
Gk-constructions....Pages 189-203
Stationary subdivision for regular nets....Pages 205-223
Stationary subdivision for arbitrary nets....Pages 225-236
Front Matter....Pages 237-237
Box splines....Pages 239-258
Simplex splines....Pages 259-269
Multivariate splines....Pages 271-285
Back Matter....Pages 287-304

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This book provides a solid and uniform derivation of the various properties B?zier and B-spline representations have, and shows the beauty of the underlying rich mathematical structure. The book focuses on the core concepts of Computer Aided Geometric Design with the intension to give a clear and illustrative presentation of the basic principles, as well as a treatment of advanced material including multivariate splines, some subdivision techniques and constructions of free form surfaces with arbitrary smoothness.
The text is beautifully illustrated with many excellent figures to emphasize the geometric constructive approach of this book.
Content:
Front Matter....Pages I-XIV
Front Matter....Pages 1-1
Geometric fundamentals....Pages 3-8
B?zier representation....Pages 9-23
B?zier techniques....Pages 25-41
Interpolation and approximation....Pages 43-57
B-spline representation....Pages 59-75
B-spline techniques....Pages 77-89
Smooth curves....Pages 91-107
Uniform subdivision....Pages 109-121
Front Matter....Pages 123-123
Tensor product surfaces....Pages 125-140
B?zier representation of triangular patches....Pages 141-153
B?zier techniques for triangular patches....Pages 155-169
Interpolation....Pages 171-178
Constructing smooth surfaces....Pages 179-188
Gk-constructions....Pages 189-203
Stationary subdivision for regular nets....Pages 205-223
Stationary subdivision for arbitrary nets....Pages 225-236
Front Matter....Pages 237-237
Box splines....Pages 239-258
Simplex splines....Pages 259-269
Multivariate splines....Pages 271-285
Back Matter....Pages 287-304
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