Linear precision for parametric patches
- PDF / 551,809 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 7 Downloads / 204 Views
Linear precision for parametric patches Luis David Garcia-Puente · Frank Sottile
Received: 22 July 2008 / Accepted: 23 February 2009 / Published online: 21 April 2009 © Springer Science + Business Media, LLC 2009
Abstract We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary (and quite restrictive) conditions for a patch to possess linear precision. A main focus is on linear precision for Krasauskas’ toric patches, which we show is equivalent to a certain rational map on CPd being a birational isomorphism. Lastly, we establish the connection between linear precision for toric surface patches and maximum likelihood degree for discrete exponential families in algebraic statistics, and show how iterative proportional fitting may be used to compute toric patches. Keywords Tensor product Bézier surfaces · Triangular Bézier surface patches · Barycentric coordinates · Iterative proportional fitting Mathematics Subject Classifications (2000) 65D17 · 14M25
Communicated by Helmut Pottmann. Work of Sottile supported by NSF CAREER grant DMS-0538734, by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation, and by Peter Gritzmann of the Technische Universität München. L. D. Garcia-Puente Department of Mathematics and Statistics, Sam Houston State University, Huntsville, TX 77341, USA e-mail: [email protected] URL: http://www.shsu.edu/˜ ldg005 F. Sottile (B) Department of Mathematics, Texas A&M University, College Station, TX 77843, USA e-mail: [email protected] URL: http://www.math.tamu.edu/˜ sottile
192
L.D. Garcia-Puente, F. Sottile
1 Introduction Bézier curves and surfaces are the fundamental units in geometric modeling. There are two basic shapes for surfaces—triangular Bézier patches and rectangular tensor product patches. Multi-sided patches are needed for some applications, and there are several control point schemes for C∞ multi-sided patches. These include the S-patches of Loop and DeRose [16], Warren’s hexagon [22], Karˇciauskas’s M-patches [11], and the multi-sided toric Bézier patches of Krasauskas [13]. (Relationships between these and other patches are discussed in [12].) Parametric patches are general control point schemes for C∞ patches whose shape is a polygon or polytope. They include the patch schemes just mentioned, as well as barycentric coordinates for polygons and polytopes [9, 21, 23]. The success and widespread adoption of Bézier and tensor-product patches is due in part to their possessing many useful mathematical properties. Some, such as affine invariance and the convex hull property, are built into their definitions and also hold for the more general parametric patches. Other properties, such as de Castlejau’s algorithm for computing Bézier patches, come from the specific form of their Bernstein polynomial blending functions. Linear precision is the ability
Data Loading...
