Showing papers on "Centroid published in 1969"

Centroid Method of Integration

Abstract: THE usefulness, for numerical integration in p dimensions, of the following “centroid method” seems to have been generally underestimated. The method is a generalization of the mid-point method for one dimension. The first approximation to the integral is f(x) multiplied by the “volume” of the region of integration, where x is the centroid of the region. This first approximation is itself often more accurate than might be expected. (It is exact if f is linear.) If f has a p-dimensional Taylor expansion about the centroid, valid in the region, then the integral is given exactly by the following equation where n = ν1 + ν2 + … + νp, where the terms for n = 1 vanish, and where m(ν) is the generalized moment of inertia Because any non-pathological domain can be “triangulated” into simplexes, it is convenient to have a formula for the generalized moments of inertia of a simplex the vertices of which are xk = (xk1, …, xkp), (k = 0,1, …, p). It can be shown that m(ν) is equal to the coefficient of … in where V is the “volume” of the simplex, given by a familiar determinant, and where If 1≤n≤3 (but not if n≥4), it can be readily deduced that and if p = 2 and 1≤n≤5 (but not if n≥6), where c2=1/12, c3 = c4=1/30, c5 = 2/105. An example of a fourth moment for arbitrary p is

The centroid and circumcentre of a plane convex set

Abstract: © Foundation Compositio Mathematica, 1969, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.