Download e-book for iPad: Modeling of Curves and Surfaces with MATLAB (Springer by Vladimir Rovenski

By Vladimir Rovenski

ISBN-10: 038771278X

ISBN-13: 9780387712789

This article on geometry is dedicated to numerous important geometrical subject matters together with: graphs of services, ameliorations, (non-)Euclidean geometries, curves and surfaces in addition to their functions in numerous disciplines. This ebook provides user-friendly tools for analytical modeling and demonstrates the opportunity of symbolic computational instruments to help the improvement of analytical options.

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The graph of max (min) of several linear functions is a convex (concave) polygon. One may illustrate this using the program 32 1 Functions and Graphs Fig. 13 Graphs of piecewise differentiable functions. 13(d), hold on; plot(x, M, x, m, ’b’, ’LineWidth’, 2); for i = 1 : n; plot(x, eval(F(i)), ’g’); end % upper/lower green polygons. 3 Functions containing the operation abs Example. Given sequences of points {ai } and of weights {bi }, where 1 ≤ i ≤ n, find the minimum of the function fn (x) = ∑ni=1 bn |x − ai |.

If ρ = f (ϕ ) is the equation of the given curve in polar coordinates, then the equation of its conchoid is ρ = f (ϕ ) ± l. In other words, Nicomedes studied the conchoid of a line. 34(c). 40 1 Functions and Graphs Fig. 20 Strophoid, cissoid, and conchoid of line. 3. Fix a point A and a line l at distance a from this point; AC is the perpendicular from A onto the line l. On a ray rotating around the point A place the segments BM1 and BM2 from the point B of its intersection with the line l. Moreover, BM1 = BM2 = CB is assumed.

Another example: f = x2 + 1 and g = x3 + x have a common divisor x2 + 1. a = [1, 0, 1]; b = [1, 0, 1, 0]; res2 = det(Resfg(a, b)) % obtain 0. The polynomial f = x3 − 4x2 + 4x has a root x = 2 of multiplicity 2. We detect this fact using Resfg. a = [1, -4, 4 0]; b = polyder(a) r3 = det(Resfg(a, b)) % Answer: b = [3, −8, 4] % obtain 0. 2. 2 Lagrange polynomials We all know that two points determine a first-degree polynomial, P1 (x) = p1 x + p2 (a line). Given n + 1 points {(x1, y1 ), . . , (xn+1, yn+1 )} where xi are n + 1 distinct real numbers, we wish to find an nth-degree interpolation Lagrange polynomial Pn (x) = n+1− j whose coefficients p satisfy the following linear system of n + 1 equa∑n+1 i j=1 p j x tions: p1 xni + · · · + pn xi + pn+1 = yi , i = 1, .

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Modeling of Curves and Surfaces with MATLAB (Springer Undergraduate Texts in Mathematics and Technology) by Vladimir Rovenski

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