A description of how a subset of planar rational Bezier curves can be used to define single-valued curves in polar coordinates is given. For this subset, an angular parameter is derived from the recursive algorithm for evaluating a rational curve. The resulting parametrization admits an expression in terms of sinusoidal basis functions. Curves of degree-two are conics with focus at the origin of coordinates.
geometry, curves, polar coordinates, rational B~zier curves
Certain planar curves can be described as the graph of a single-valued function y=l(x) in Cartesian coordinates (x, y). This nonparametrical representation is the easiest one with which to work, but only curves that project one to one onto the x-axis can be described. For the approximation of these univariate functions, nonparametric B&zier curves ~ are used. Control points in such Bezier curves lie over abscissae regularly spaced along the x-axis.
Similarly, single-valued curves r = r(O) can be defined in a system of polar coordinates (r, 0). This coordinate system is the base of spherical and cylindrical coordinate systems, and it is very suitable when representing closed curves. If a closed curve is expressed in the form r = r(0), the point classification can be performed with little effort: a given point P0 = (r0, 0o) lies within the area defined by the curve if r 0 < r(Oo). Furthermore, in some engineering applications, like the designing of cam profiles, it is very convenient to define the profile in terms of r = r(O), to analyse the mechanism as the cam rotates.
This article develops a method for the approximation of single-valued curves in polar coordinates by using a subset of planar rational B~zier curves. In complete analogy to the Cartesian case, to define a segment of a single-valued curve that spans a given angle between two radial directions, the corresponding control points will be regularly spaced with respect to the angular coordinate 0 between both radial directions. These single-valued curves in polar coordinates are constructed by identifying an angular parameter in the recursion algorithm for evaluating a rational B~zier curve. An expression for r(O) in terms of sinusoidal basis functions is developed. These basis functions are analogous to