By Jean Gallier
Curves and Surfaces for Geometric Design
deals either a theoretically unifying realizing of polynomial curves and surfaces and an efficient method of implementation for you to convey to endure by yourself work-whether you are a graduate scholar, scientist, or practitioner.
Inside, the focal point is on "blossoming"-the means of changing a polynomial to its polar form-as a ordinary, only geometric clarification of the habit of curves and surfaces. This perception is necessary for much greater than its theoretical splendor, 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 alternative standards. you will discover ways to use this and comparable strategies drawn from affine geometry for computing and adjusting keep an eye on issues, deriving the continuity stipulations for splines, growing subdivision surfaces, and more.
The manufactured from groundbreaking study via a noteworthy desktop scientist and mathematician, this e-book is destined to grow to be a vintage paintings in this complicated topic. will probably be a vital acquisition for readers in lots of diversified parts, together with special effects and animation, robotics, digital fact, geometric modeling and layout, clinical imaging, machine imaginative and prescient, and movement planning.
* Achieves a intensity of insurance now not present in the other ebook during this field.
* deals a mathematically rigorous, unifying method of the algorithmic iteration and manipulation of curves and surfaces.
* Covers easy techniques of affine geometry, the correct framework for facing curves and surfaces by way of keep watch over points.
* information (in Mathematica) many whole implementations, explaining how they produce hugely non-stop curves and surfaces.
* offers the first thoughts for developing and studying the convergence of subdivision surfaces (Doo-Sabin, Catmull-Clark, Loop).
* comprises appendices on linear algebra, easy topology, and differential calculus.
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Extra resources for Curves and surfaces in geometric modeling : theory and algorithms
10). Thus, if (− a− 0 a1 , . . 10, (λ1 , . . , λm ) is unique, and since λ0 = 1 − m i=1 λi , λ0 is also unique. 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 , .
7. 13: The effect of an affine map is an affine map. Since we can write 1 1 1 3 = √ √ 2 2 2 √ 2 2 − √ √ 2 2 2 2 1 2 , 0 1 this affine map is the composition of a shear, followed by a rotation of angle π/4, followed by √ a magnification of ratio 2, followed by a translation. 13. The image of the square (a, b, c, d) is the parallelogram (a′ , b′ , c′ , d′ ). The following lemma shows the converse of what we just showed. Every affine map is determined by the image of any point and a linear map. 2.
The terms affine dilatation and central dilatation are used by Pedoe . Snapper and Troyer use the term dilation for an affine dilatation and magnification for a central dilatation . Samuel uses homothety for a central dilatation, a direct translation of the French “homoth´etie” . Since dilation is shorter than dilatation and somewhat easier to pronounce, perhaps we use use that! Observe that Ha,λ (a) = a, and when λ = 0 and x = a, Ha,λ (x) is on the line defined by → by λ. 16 shows the effect of a central dilatation of center d.
Curves and surfaces in geometric modeling : theory and algorithms by Jean Gallier