### Abstract

Very few characteristic functions, or equations, are reported so far for fractals. Such functions, called Rvachev functions in function-based modeling, are zero on the boundary, negative for inside points and positive for outside points. This paper proposes Rvachev functions for some classical fractals. These functions are convergent series, which are bounded with interval arithmetic and interval analysis in finite time. This permits to extend the Recursive Space Subdivision (RSS) method, which is classical in Computer Graphics (CG) and Interval Analysis, to fractal geometric sets. The newly proposed fractal functions can also be composed with classical Rvachev functions today routinely used in Constructive Solid Geometry (CSG) trees of CG or function-based modeling.

Original language | English (US) |
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Journal | Fractals |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

### Fingerprint

### Keywords

- Fractal
- Function Representation
- Function-Based Modeling
- Interval Arithmetic

### ASJC Scopus subject areas

- Modeling and Simulation
- Geometry and Topology
- Applied Mathematics

### Cite this

*Fractals*. https://doi.org/10.1142/S0218348X18500597

**EQUATIONS and INTERVAL COMPUTATIONS for SOME FRACTALS.** / Fang, Lincong; Michelucci, Dominique; Foufou, Sebti.

Research output: Contribution to journal › Article

*Fractals*. https://doi.org/10.1142/S0218348X18500597

}

TY - JOUR

T1 - EQUATIONS and INTERVAL COMPUTATIONS for SOME FRACTALS

AU - Fang, Lincong

AU - Michelucci, Dominique

AU - Foufou, Sebti

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Very few characteristic functions, or equations, are reported so far for fractals. Such functions, called Rvachev functions in function-based modeling, are zero on the boundary, negative for inside points and positive for outside points. This paper proposes Rvachev functions for some classical fractals. These functions are convergent series, which are bounded with interval arithmetic and interval analysis in finite time. This permits to extend the Recursive Space Subdivision (RSS) method, which is classical in Computer Graphics (CG) and Interval Analysis, to fractal geometric sets. The newly proposed fractal functions can also be composed with classical Rvachev functions today routinely used in Constructive Solid Geometry (CSG) trees of CG or function-based modeling.

AB - Very few characteristic functions, or equations, are reported so far for fractals. Such functions, called Rvachev functions in function-based modeling, are zero on the boundary, negative for inside points and positive for outside points. This paper proposes Rvachev functions for some classical fractals. These functions are convergent series, which are bounded with interval arithmetic and interval analysis in finite time. This permits to extend the Recursive Space Subdivision (RSS) method, which is classical in Computer Graphics (CG) and Interval Analysis, to fractal geometric sets. The newly proposed fractal functions can also be composed with classical Rvachev functions today routinely used in Constructive Solid Geometry (CSG) trees of CG or function-based modeling.

KW - Fractal

KW - Function Representation

KW - Function-Based Modeling

KW - Interval Arithmetic

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U2 - 10.1142/S0218348X18500597

DO - 10.1142/S0218348X18500597

M3 - Article

AN - SCOPUS:85052674399

JO - Fractals

JF - Fractals

SN - 0218-348X

ER -