Authors: Arman Maesumi
Comments: 5 Pages.
Given a triangle ABC, the average area of an inscribed triangle RST whose vertices are uniformly distributed on BC, CA and AB, is proven to be one-fourth of the area of ABC. The average of the square of the area of RST is shown to be one-twelfth of the square of the area of ABC, and the average of the cube of the ratio of the areas is 5/144. A Monte Carlo simulation confirms the theoretical results, as well as a Maxima program which computes the exact averages.
The derivation of multilinear forms used to compute the moments of sets bounded by subdivision surfaces requires solving a number of systems of linear equations. As the support of the subdivision mask or the degree of the moment grows, the corresponding linear system becomes intractable to construct, let alone to solve by Gaussian elimination. In the paper, we argue that the power iteration and the geometric series are feasible methods to approximate the multilinear forms. The tensor iterations investigated in this work are shown to converge at favorable rates, achieve arbitrary numerical accuracy, and have a small memory footprint. In particular, our approach makes it possible to compute the volume, centroid, and inertia of spatial domains bounded by Catmull-Clark and Loop subdivision surfaces.
Authors: Gaurav S. Biraris
Comments: 20 Pages.
The paper proposes a generalization of geometric notion of vectors concerning dimensionality of the configuration space. In certain dimensional spaces, certain types of ordered directions exist along which elements of vector spaces can be interpreted. Scalars along the ordered directions form Banach spaces. Different types of geometrical vectors are algebraically identical, the difference arises in the configuration space geometrically. In the universe four types of vectors exists. Thus any physical quantity in the universe comes in four types of vectors. Though All the types of vectors belong to different Banach spaces (& their directions can’t be compared), their magnitudes can be compared. A gross comparison between the magnitudes of the different typed geometric vectors is obtained at end of the paper.