Moments Defined by Subdivision Curves

Figure: We compute the exact area, centroid, and inertia of the 2-dimensional sets bounded by subdivision curves. The illustration shows the principle axes of the inertia tensor drawn at the centroid of the area; five different subdivision schemes are used to demonstrate the universality of our derivation. † We derive the d + 2-linear forms that compute the moment of degree d of the area enclosed by a subdivision curve in the plane. We circumvent the need to solve integrals involving the basis function by exploiting a recur-sive relation and calibration that establishes the coefficients of the form within the nullspace of a matrix. For demonstration, we apply the technique to the dual three-point scheme, the interpolatory C1 four-point scheme, and the dual C2 four-point scheme. The first author was partially supported by personal savings accumulated during his visit to the Nanyang Technological University as a visiting research scientist in 2012-2013. He’d like to thank everyone who worked to make this opportunity available to him