Download A Sampler of Useful Computational Tools for Applied by Daniel Cohen-Or, Chen Greif, Tao Ju, Niloy J. Mitra, Ariel PDF

By Daniel Cohen-Or, Chen Greif, Tao Ju, Niloy J. Mitra, Ariel Shamir, Olga Sorkine-Hornung, Hao (Richard) Zhang

A Sampler of precious Computational instruments for utilized Geometry, special effects, and photo Processing indicates the best way to use a set of mathematical ideas to resolve very important difficulties in utilized arithmetic and machine technology components. The e-book discusses primary instruments in analytical geometry and linear algebra. It covers a variety of themes, from matrix decomposition to curvature research and relevant part research to dimensionality reduction.

Written by way of a staff of hugely revered professors, the publication can be utilized in a one-semester, intermediate-level path in computing device technological know-how. It takes a realistic problem-solving process, averting specific proofs and research. compatible for readers with out a deep educational history in arithmetic, the textual content explains find out how to remedy non-trivial geometric difficulties. It speedy will get readers in control on a number of instruments hired in visible computing and utilized geometry.

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Extra info for A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

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An   ..  = b1 a1  + b2 a2  + · · · + bn an  . 11), and an important Linear Algebra? 11: Transposing a matrix. equality is (AB) = B A . The transposition operation is useful for expressing an inner product between two vectors in a matrix form: v, w = v w = w v . As we shall see in later chapters, we often solve problems by finding a vector x that satisfies a set of equations represented in a matrix form, Ax = b. The system of equations Ax = b has a unique solution if the matrix A is non-singular (we assume A is a square matrix).

In Chapter 4 we will review the singular value decomposition (SVD), which will tell us how general linear transformations behave, among other things. Finding the coordinates of a vector Suppose we are given a basis v1 , v2 , v3 in R3 . An arbitrary vector a ∈ R3 has three coordinates denoted by, say, α1 , α2 , α3 and associated with the basis vectors: n a= α i vi . i=1 Linear Algebra? 27 To extract the αi , we can use the linearity of the space and form inner products that give us n a, vj = αi vi , vj , i=1 and thus we have a, v1 = α1 v1 , v1 + α1 v2 , v1 + α3 v3 , v1 ; a, v2 = α2 v1 , v2 + α1 v2 , v2 + α3 v3 , v2 ; a, v3 = α3 v1 , v3 + α1 v2 , v3 + α3 v3 , v3 .

That is Ax = λx ⇔ Ax − λx = 0 ⇔ Ax − λIx = 0 ⇔ (A − λI)x = 0 . So, a nontrivial solution exists if and only if det(A − λI) = 0. The expression det(A − λI) can be developed into a polynomial of degree n called the characteristic polynomial of A. The roots of this characteristic polynomial are the eigenvalues of A. Therefore, there are always n eigenvalues (some or all of which may be complex). If n is odd, there is at least one real eigenvalue. Let us look at an example:   1 0 2 A =  3 0 −3 .

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