Download e-book for kindle: Curves and surfaces in geometric modeling : theory and by Jean Gallier

By Jean Gallier

ISBN-10: 1558605991

ISBN-13: 9781558605992

Curves and Surfaces for Geometric Design

deals either a theoretically unifying realizing of polynomial curves and surfaces and an efficient method of implementation so you might carry to undergo by yourself work-whether you are a graduate pupil, scientist, or practitioner.

Inside, the focal point is on "blossoming"-the technique of changing a polynomial to its polar form-as a common, simply geometric rationalization of the habit of curves and surfaces. This perception is critical for a lot greater than its theoretical attractiveness, for the writer proceeds to illustrate the worth of blossoming as a pragmatic algorithmic software for producing and manipulating curves and surfaces that meet many various standards. you are going to discover ways to use this and similar suggestions drawn from affine geometry for computing and adjusting keep watch over issues, deriving the continuity stipulations for splines, growing subdivision surfaces, and more.

The made of groundbreaking learn through a noteworthy computing device scientist and mathematician, this ebook is destined to turn out to be a vintage paintings in this complicated topic. it is going to be an important acquisition for readers in lots of diversified components, together with special effects and animation, robotics, digital truth, geometric modeling and layout, clinical imaging, desktop imaginative and prescient, and movement planning.

* Achieves a intensity of assurance now not present in the other publication during this field.

* deals a mathematically rigorous, unifying method of the algorithmic iteration 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 entire implementations, explaining how they produce hugely non-stop curves and surfaces.

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

* comprises appendices on linear algebra, uncomplicated topology, and differential calculus.

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Sample text

Conversely, the uniqueness of m (λ0 , . . , λm ) such that x = i=0 λi ai implies the uniqueness of (λ1 , . . 10 again, (− a− 0 a1 , . . , a0 am ) is linearly independent. 3 suggests the notion of affine frame. Affine frames are the affine analogs − → of bases in vector spaces. Let E, E , + be a nonempty affine space, and let (a0 , . . , am ) be a family of m + 1 points in E. The family (a0 , . . , am ) determines the family of m − → → −−→ vectors (− a− 0 a1 , . . , a0 am ) in E . Conversely, given a point a0 in E and a family of m vectors − → (u1 , .

A0 am ) is a basis of E , then, for every x ∈ E, since x = a0 + a0 x, there is a unique family (x1 , . . , xm ) of scalars, such that −−→ → a− x = a0 + x1 − 0 a1 + · · · + xm a0 am . → −−→ The scalars (x1 , . . , xm ) are coordinates with respect to (a0 , (− a− 0 a1 , . . , a0 am )). 6. AFFINE INDEPENDENCE AND AFFINE FRAMES 35 x ∈ E can also be expressed uniquely as m x= λi ai i=0 m with m i=1 xi , and λi = xi for 1 ≤ i ≤ m. The scalars i=0 λi = 1, and where λ0 = 1 − (λ0 , . . , λm ) are also certain kinds of coordinates with respect to (a0 , .

Am ) in E, where ai = a0 + ui , 1 ≤ i ≤ m. Thus, for any m ≥ 1, it is equivalent to consider a family of m + 1 points (a0 , . . , am ) in − → E, and a pair (a0 , (u1, . . , um )), where the ui are vectors in E . − → → −−→ −→ When (− a− 0 a1 , . . , a0 am ) is a basis of E , then, for every x ∈ E, since x = a0 + a0 x, there is a unique family (x1 , . . , xm ) of scalars, such that −−→ → a− x = a0 + x1 − 0 a1 + · · · + xm a0 am . → −−→ The scalars (x1 , . . , xm ) are coordinates with respect to (a0 , (− a− 0 a1 , .

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

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