## Comparison of Interval Methods for Plotting Algebraic Curves (2002)

Venue: | Comput. Aided Geom. Des |

Citations: | 24 - 2 self |

### BibTeX

@ARTICLE{Martin02comparisonof,

author = {Ralph Martin and Huahao Shou and Irina Voiculescu and Adrian Bowyer and Guojin Wang},

title = {Comparison of Interval Methods for Plotting Algebraic Curves},

journal = {Comput. Aided Geom. Des},

year = {2002},

volume = {19},

pages = {553--587}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper compares the performance and e#ciency of di#erent 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.

### Citations

875 |
Interval Analysis
- Moore
- 1966
(Show Context)
Citation Context ...should be. This is clearly unacceptable for certain applications, which may prefer to approximate the curve in some other way in pixel-sized cells. The classical technique of interval arithmetic (IA) =-=[21, 22]-=- provides a natural tool for range analysis [23]; an overview is given in Section 3.1. Subdivision methods based on IA have been proposed for rasterization of implicit curves and surfaces in computer ... |

529 |
Methods and Applications of Interval Analysis
- Moore
- 1979
(Show Context)
Citation Context ...should be. This is clearly unacceptable for certain applications, which may prefer to approximate the curve in some other way in pixel-sized cells. The classical technique of interval arithmetic (IA) =-=[21, 22]-=- provides a natural tool for range analysis [23]; an overview is given in Section 3.1. Subdivision methods based on IA have been proposed for rasterization of implicit curves and surfaces in computer ... |

132 | Interval analysis for computer graphics
- Snyder
- 1992
(Show Context)
Citation Context ...ametrically described shapes. They can represent, for example, the intersection of two parametric surfaces in R 3 , or the silhouette edges of a parametric surface in R 3 with respect to a given view =-=[28].-=- Tracing the implicit curve f(x, y) = 0 in a rectangular region [x, x] × [y, y], where f is a polynomial in two variables, is of great interest in CAD, CAGD and computer graphics. We consider here as... |

131 |
The Chebyshev Polynomials
- Rivlin
- 1974
(Show Context)
Citation Context ...a particularly efficient way to construct inclusion functions for polynomials. A further method for bounding the range of a polynomial over an interval for the univariate case by Cargo [6] and Rivlin =-=[25]-=- is based on a simple estimate of the second derivative in a Taylor expansion of the polynomial. Garloff [15] extended the idea to the bivariate case. The bounds are found from the values of the polyn... |

96 | Interval arithmetic recursive subdivision for implicit functions and constructive solid geometry - Duff - 1992 |

70 |
Computer Methods for the Range of Functions
- Ratschek, Rokne
- 1984
(Show Context)
Citation Context ...lls reach pixel size (or a desired accuracy). In this way large 1sportions of the plot area can be discarded quickly and reliably at an early stage, again leading to efficient methods. Range analysis =-=[23]-=- provides a general test procedure for reliably rejecting certain cells at each subdivision step. In range analysis, a conservative interval is computed for the range of function values within a cell.... |

66 | Affine arithmetic and its applications to computer graphics
- Comba, Stolfi
- 1993
(Show Context)
Citation Context ...per two other polynomial forms—Horner form and centred form are added for comparison, in addition to other approaches we also consider. The main weakness of IA is that it tends to be too conservativ=-=e [8, 13, 14]-=-, i.e. the range output for the function by IA is sometimes much wider than the actual range of values the function takes over a given interval. To solve this problem, Comba and Stolfi [8] proposed a ... |

35 | Approximation by interval bezier curves
- Sederberg, Farouki
- 1992
(Show Context)
Citation Context ...]. This comes from the fundamental property of interval arithmetic: x ∈ x ⇒ f(x) ∈ f(x). In the current application, intervals are used to represent (large) regions of interest in curves and sur=-=faces [26, 33]-=-. Other types of application use intervals to represent (small) errors or uncertainty, using intervals for the coefficients of the polynomials in a suitable basis, rather than the variables. A signifi... |

30 |
Algorithms for polynomials
- Farouki, Rajan
- 1988
(Show Context)
Citation Context ... surfaces [10]. The Bernstein basis Bi j (u) = � � i j i−j j u (1 − u) , j = 0, 1, · · · , i has been shown to be numerically more stable and better conditioned for finding roots than the p=-=ower basis [11, 12]-=-. Bowyer et al. [2, 3, 4, 5, 35] have extensively considered IA applied to multivariate Bernsteinform polynomials. Conversion between the power basis and Bernstein basis for multivariate polynomials i... |

30 | Adaptive enumeration of implicit surfaces with affine arithmetic, Comput. Graphics Forum 15 (5
- Figueiredo, Stolfi
- 1996
(Show Context)
Citation Context ...ly test whether the curve passes through each pixel. Such a test can be performed by evaluating the approximate Euclidean distance from the center of each pixel to the curve [31] or by point sampling =-=[14]-=-. Clearly such methods are not efficient. Continuation methods [7] are usually efficient because they use one or more seed pixels on a curve and then trace the curve continuously. However these method... |

27 | Distance approximations for rasterizing implicit curves
- Taubin
- 1994
(Show Context)
Citation Context ...g problem is to exhaustively test whether the curve passes through each pixel. Such a test can be performed by evaluating the approximate Euclidean distance from the center of each pixel to the curve =-=[31]-=- or by point sampling [14]. Clearly such methods are not efficient. Continuation methods [7] are usually efficient because they use one or more seed pixels on a curve and then trace the curve continuo... |

26 |
Rasterizing algebraic curves and surfaces
- Taubin
- 1994
(Show Context)
Citation Context ...or more seed pixels on a curve and then trace the curve continuously. However these methods have one fundamental difficulty, that of finding a complete set of initial seed pixels. Subdivision methods =-=[9, 28, 29, 30, 31, 32]-=- start with the plot area itself as an initial cell. If a cell is proved to be empty, it is ignored; otherwise, it is subdivided into smaller cells, which are then visited recursively, until the cells... |

26 |
Approximation of measured data with interval b-splines
- Tuohy, Maekawa, et al.
- 1997
(Show Context)
Citation Context ...]. This comes from the fundamental property of interval arithmetic: x ∈ x ⇒ f(x) ∈ f(x). In the current application, intervals are used to represent (large) regions of interest in curves and sur=-=faces [26, 33]-=-. Other types of application use intervals to represent (small) errors or uncertainty, using intervals for the coefficients of the polynomials in a suitable basis, rather than the variables. A signifi... |

21 |
Convergent bounds for the range of multivariate polynomials
- Garloff
- 1986
(Show Context)
Citation Context ...the range of a polynomial over an interval for the univariate case by Cargo [6] and Rivlin [25] is based on a simple estimate of the second derivative in a Taylor expansion of the polynomial. Garloff =-=[15]-=- extended the idea to the bivariate case. The bounds are found from the values of the polynomial at points of a regular grid subdividing the unit square. Gopalsamy et al. [16] proposed a method of eva... |

20 |
The Bernstein form of a polynomial
- Cargo, Shisha
- 1966
(Show Context)
Citation Context ...is approach is a particularly efficient way to construct inclusion functions for polynomials. A further method for bounding the range of a polynomial over an interval for the univariate case by Cargo =-=[6]-=- and Rivlin [25] is based on a simple estimate of the second derivative in a Taylor expansion of the polynomial. Garloff [15] extended the idea to the bivariate case. The bounds are found from the val... |

18 | Surface intersection using affine arithmetic
- Figueiredo
- 1996
(Show Context)
Citation Context ...per two other polynomial forms—Horner form and centred form are added for comparison, in addition to other approaches we also consider. The main weakness of IA is that it tends to be too conservativ=-=e [8, 13, 14]-=-, i.e. the range output for the function by IA is sometimes much wider than the actual range of values the function takes over a given interval. To solve this problem, Comba and Stolfi [8] proposed a ... |

16 |
Ray tracing general parametric surfaces using interval arithmetic
- Barth, Lieger, et al.
- 1994
(Show Context)
Citation Context ...asterization of implicit curves and surfaces in computer graphics applications [9, 28, 30]. IA also has been used in computer graphics applications such as fast ray tracing and robust solid modelling =-=[1, 18, 19]-=-. Because of the way arithmetic operators work in IA (for example, the distributive law no longer holds), the form used to express the polynomial f(x, y) affects the range for the function output by a... |

12 |
A Tracking Algorithm for Implicitly Defined Curves
- Chandler
(Show Context)
Citation Context ...n be performed by evaluating the approximate Euclidean distance from the center of each pixel to the curve [31] or by point sampling [14]. Clearly such methods are not efficient. Continuation methods =-=[7]-=- are usually efficient because they use one or more seed pixels on a curve and then trace the curve continuously. However these methods have one fundamental difficulty, that of finding a complete set ... |

11 |
Robust Arithmetic for Multivariate Bernstein-form Polynomials
- Berchtold, Bowyer
- 2000
(Show Context)
Citation Context .... For example, clearly f(u) = 1 + 2u − u 2 = 1 + u(2 − u), giving the power form of the polynomial, and the Horner form respectively. Supposing u = [0, 1], using the power form to evaluate f(u) gi=-=ves [0, 3]-=- as the answer, while the Horner form gives [1, 3]. Both answers are correct, in the sense that the interval obtained is guaranteed to contain the actual range of the function (but neither is exact: t... |

11 |
Quadtree Algorithms for Contouring Functions of Two Variables
- Suffern
- 1990
(Show Context)
Citation Context ...or more seed pixels on a curve and then trace the curve continuously. However these methods have one fundamental difficulty, that of finding a complete set of initial seed pixels. Subdivision methods =-=[9, 28, 29, 30, 31, 32]-=- start with the plot area itself as an initial cell. If a cell is proved to be empty, it is ignored; otherwise, it is subdivided into smaller cells, which are then visited recursively, until the cells... |

8 |
Interval methods in geometric modeling
- Bowyer, Berchtold, et al.
- 2000
(Show Context)
Citation Context ... coefficients of the polynomials in a suitable basis, rather than the variables. A significant property of IA already noted is that the form in which the polynomial is expressed can affect the result =-=[5]. Fo-=-r example, clearly f(u) = 1 + 2u − u 2 = 1 + u(2 − u), giving the power form of the polynomial, and the Horner form respectively. Supposing u = [0, 1], using the power form to evaluate f(u) gives ... |

7 |
Interval arithmetic applied to multivariate bernstein-form polynomials
- BERCHTOLD, VOICULESCU, et al.
- 2000
(Show Context)
Citation Context ... added. While the rules for +, −, ×, / are exact, more generally, the resulting interval is too large. To see this, consider x = [−1, 2]. Computing x × x gives [−2, 4], but the exact range for=-= x 2 is [0, 4]-=-. This happens here because the two quantities being multiplied are not independent. 4sNote that even the computation of powers can be done in several ways. In this paper we use the exact interval res... |

7 | On the Numerical Condition of - Farouki, Rajan - 1987 |

5 |
On the Algorithms and Implementation of a Geometric Algebra System
- Milne
- 1990
(Show Context)
Citation Context ...[a, b] × [c, d] = [min(ac, ad, bc, bd), max(ac, ad, bc, bd)] , [a, b] / [c, d] = [a, b] × [1/d, 1/c] provided 0 /∈ [c, d] . A treatment of interval division for intervals containing 0 can be found=-= in [20]-=-. The natural interval extension of a bivariate polynomial f(x, y), denoted by f(x, y), is obtained by replacing each occurrence of x and y in f(x, y) by intervals x and y, and evaluating the resultin... |

5 |
Interval and affine arithmetic for surface location of power
- Voiculescu, Berchtold, et al.
- 2000
(Show Context)
Citation Context ...erators work in IA (for example, the distributive law no longer holds), the form used to express the polynomial f(x, y) affects the range for the function output by an IA evaluation. A previous paper =-=[35] s-=-howed that IA using the Bernstein basis is generally more accurate than IA using the power basis. In the current paper two other polynomial forms—Horner form and centred form are added for compariso... |

4 |
Polynomial evaluation using affine arithmetic for curve drawing
- Zhang, Martin
- 2000
(Show Context)
Citation Context ...pling for procedural shaders [8, 13, 14, 17]. As AA usually computes tighter intervals than IA, it is possible to draw algebraic curves using AA more efficiently and with higher quality than using IA =-=[14, 36]-=-. However AA is still too conservative sometimes, and it does not obey the distributive law either. To solve these problems, in this paper we use a modified Matrix AA (MAA) method proposed in [27]. An... |

3 | The Bernstein Form in Set-Theoretic Geometric Modelling - Berchtold - 2000 |

3 |
A New Method of Evaluating Compact Geometric Bounds for Use in Subdivision Algorithms, Computer Aided Geometric Design
- Gopalsamy, Khandekar, et al.
- 1991
(Show Context)
Citation Context ... the polynomial. Garloff [15] extended the idea to the bivariate case. The bounds are found from the values of the polynomial at points of a regular grid subdividing the unit square. Gopalsamy et al. =-=[16]-=- proposed a method of evaluating compact geometric bounds for both univariate and bivariate polynomials by simply sampling the polynomial. The optimal sampling positions depend only on the degree of t... |

2 |
Implicit function algebra in set-theoretic geometric modelling
- Voiculescu
- 2001
(Show Context)
Citation Context ...01 25 y−512xy+1536x2y−2048x3y+1024x4y = 0 on [0, 1] × [0, 1]. This is an antisymmetric medium degree polynomial. The sixth example which differs only by a affine change of coordinates from the on=-=e in [34] is: 601 872 -=-9 − 3 x + 544x2 − 512x3 + 256x4 − 2728 2384 9 y + 3 xy − 768x2y + 5104 9 y2 − 2432 3 xy2 + 768x2y2 − 512y3 + 256y4 = 0 on [0, 1] × [0, 1]. This is an asymmetric medium degree polynomial. ... |

1 |
SCCI-Hybrid Methods for 2D-Curve Tracing, submitted for publication
- Ratschek, Rokne
(Show Context)
Citation Context ...metric medium degree polynomial. The curve (Figure 8) contains a single compnent with a self-intersection point. The ninth example which differs only by a affine change of coordinates from the one in =-=[24] is: 47.6 −-=- 220.8x + 476.8x2− 512x3 + 256x4 − 220.8y + 512xy −512x2y + 476.8y2 − 512xy2 + 512x2y2 − 512y3 + 256y4 = 0 on [0, 1] × [0, 1]. This is a symmetric medium degree polynomial. The curve (Figur... |