Curves and surfaces in geometric modeling : theory and by Jean Gallier

By Jean Gallier

Curves and Surfaces for Geometric Design

deals either a theoretically unifying knowing of polynomial curves and surfaces and an efficient method of implementation for you to carry to endure by yourself work-whether you are a graduate scholar, scientist, or practitioner.

Inside, the point of interest is on "blossoming"-the strategy of changing a polynomial to its polar form-as a common, merely geometric clarification of the habit of curves and surfaces. This perception is critical for much greater than its theoretical beauty, for the writer proceeds to illustrate the price of blossoming as a realistic algorithmic software for producing and manipulating curves and surfaces that meet many alternative standards. you are going to learn how to use this and comparable concepts drawn from affine geometry for computing and adjusting keep watch over issues, deriving the continuity stipulations for splines, developing subdivision surfaces, and more.

The made from groundbreaking examine through a noteworthy computing device scientist and mathematician, this e-book is destined to grow to be a vintage paintings in this advanced topic. it is going to be a vital acquisition for readers in lots of assorted components, together with special effects and animation, robotics, digital fact, geometric modeling and layout, scientific imaging, desktop imaginative and prescient, and movement planning.

* Achieves a intensity of insurance no longer present in the other publication during this field.

* deals a mathematically rigorous, unifying method of the algorithmic new release and manipulation of curves and surfaces.

* Covers uncomplicated techniques of affine geometry, the perfect framework for facing curves and surfaces when it comes to keep an eye on points.

* information (in Mathematica) many whole implementations, explaining how they produce hugely non-stop curves and surfaces.

* offers the first suggestions for developing and reading the convergence of subdivision surfaces (Doo-Sabin, Catmull-Clark, Loop).

* includes appendices on linear algebra, easy topology, and differential calculus.

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Am ), we can 46 CHAPTER 2. BASICS OF AFFINE GEOMETRY −−→ → compute the determinant detB (− a− 0 a1 , . . t. the basis B. For any bijective affine map f : E → E, since − → −→ − → −→ − → → −−→ detB ( f (− a0 a1 ), . . , f (− a0 am )) = det( f )detB (− a− 0 a1 , . . , only depends on f , and not on the particular basis B), we conclude that the ratio − → −→ − → −→ − → detB ( f (− a0 a1 ), . . , f (− a0 am )) = det( f ) − − → − − → detB (a0 a1 , . . , a0 am ) → −−→ is independent of the basis B.

We say that four points → − − − → a, b, c, d are coplanar , if the vectors ab, → ac, and ad, are linearly dependent. Hyperplanes will be characterized a little later. 3. Given an affine space E, E , + , for any family (ai )i∈I of points in E, the set V of barycenters i∈I λi ai (where i∈I λi = 1) is the smallest affine subspace containing 32 CHAPTER 2. BASICS OF AFFINE GEOMETRY (ai )i∈I . Proof. If (ai )i∈I is empty, then V = ∅, because of the condition i∈I λi = 1. If (ai )i∈I is nonempty, then the smallest affine subspace containing (ai )i∈I must contain the set V of barycenters i∈I λi ai , and thus, it is enough to show that V is closed under affine combinations, which is immediately verified.

Aa Thus, letting a = c in Chasles’ identity, we get − → → − ba = −ab. Given any four points a, b, c, d ∈ E, since by Chasles’ identity − − → → − → − → → ab + bc = ad + dc = − ac, → − − → → − − → we have ab = dc iff bc = ad (the parallelogram law ). 4 Affine Combinations, Barycenters A fundamental concept in linear algebra is that of a linear combination. The corresponding concept in affine geometry is that of an affine combination, also called a barycenter . 1. 4. 5: Two coordinates systems in R2 certain readers, we give another example showing what goes wrong if we are not careful in defining linear combinations of points.

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