Huahao Shou

Comparison of interval methods for plotting algebraic curves

This paper compares the performance and efficiency of different function range interval methods for plotting f(x, y)=0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD. The methods considered are interval arithmetic methods (using the power basis, Bernstein basis, Homer form and centred form), an affine arithmetic method, a Bernstein coefficient method, Taubins method, Rivlins method, Gopalsamys method, and related methods which also take into account derivative information. Our experimental results show that the affine arithmetic method, interval arithmetic using the centred form, the Bernstein coefficient method, Taubins method, Rivlins method, and their related derivative methods have similar performance, and generally they are more accurate and efficient than Gopalsamys method and interval arithmetic using the power basis, the Bernstein basis, and Horner form methods.

Modified Affine Arithmetic Is More Accurate than Centered Interval Arithmetic or Affine Arithmetic

In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a box-shaped region: (i) using interval arithmetic in centered form is always more accurate than standard affine arithmetic, and (ii) modified affine arithmetic is always more accurate than interval arithmetic in centered form. Test results show that modified affine arithmetic is not only more accurate but also much faster than standard affine arithmetic. We thus suggest that modified affine arithmetic is the method of choice for evaluating algebraic functions, such as implicit surfaces, over a box.

AFFINE INTERVALS IN A CSG GEOMETRIC MODELLER

Our CSG modeller, svLis, uses interval arithmetic to categorize implicit functions representing primitive shapes against boxes; this allows an efficient implementation of recursive spatial division to localize the primitives for a variety of purposes, such as rendering or the computation of integral properties.

A Recursive Taylor Method for Algebraic Curves and Surfaces

This paper examines recursive Taylor methods for multivariate polynomial evaluation over an interval, in the context of algebraic curve and surface plotting as a particular application representative of similar problems in CAGD. The modified affine arithmetic method (MAA), previously shown to be one of the best methods for polynomial evaluation over an interval, is used as a benchmark; experimental results show that a second order recursive Taylor method (i) achieves the same or better graphical quality compared to MAA when used for plotting, and (ii) needs fewer arithmetic operations in many cases. Furthermore, this method is simple and very easy to implement. We also consider which order of Taylor method is best to use, and propose that second order Taylor expansion is generally best. Finally, we briefly examine theoretically the relation between the Taylor method and the MAA method.

Robust Plotting of Polar Algebraic Curves, Space Algebraic Curves, and Offsets of Planar Algebraic Curves

In this paper, based on modifications to the well-known subdivision scheme in combination with modified affine arithmetic method to guide the subdivision, we propose four robust and reliable algorithms for plotting polar algebraic curves, space algebraic curves and offsets of planar algebraic curves.

Affine Arithmetic and Bernstein Hull Methods for Algebraic Curve Drawing

We compare approaches to the location of the algebraic curve f(x,y) = 0 in a rectangular region of the plane, based on recursive use of conservative estimates of the range of the function over a rectangle. Previous work showed that performing interval arithmetic in the Bernstein basis is more accurate than using the power basis, and that affine arithmetic in the power basis is better than using interval arithmetic in the Bernstein basis. This paper shows that using affine arithmetic with the Bernstein basis gives no advantage over affine arithmetic with the power basis. It also considers the Bernstein coefficient method based on the convex hull property, which has similar performance to affine arithmetic.

What order Taylor model is sufficient for subdivision based implicit curve plotting

In this paper, we conducted some well chosen experimental tests to determine what order is sufficient for Taylor model applied in subdivision based implicit curve plotting algorithm. Test results show that order 1 is sufficient for most test examples with simple expressions, and some more complicated test examples need order 2 or 3. Order 3 is sufficient for all test examples we tested.