Tessellating Algebraic Curves and Surfaces using A-patches

An implicit surface is defined as the zero set of a scalar function over 3-space. Sign of implicit function determines whether a point is inside or outside the implicit surface: f (P )  = 0 P is on the surface. < 0 P is inside the surface. > 0 P is outside the surface. An algebraic function is an implicit function where the function is polynomial. By using an A-patch representation of algebraic curves and surfaces, we are able to identify regions in which the curve/surface does not lie, and efficiently tessellate the curve/surface in the regions in which it does lie. A-PATCHES Bivariate Bernstein polynomials: B ~i (P ) = ( n ~i ) p0 0 p i1 1 p i2 2 , where~i = (i0, i1, i2) with i0, i1, i2,≥ n and i0 + i1 + i2 = n, and where (p0, p1, p2) are the Barycentric coordinates. An A-patch weights scalar values with Bernstein basis: