Stochastic algorithm for detecting intersection of implicit surfaces

doi:10.1016/S0097-8493(00)00055-8

Computers & Graphics

Volume 24, Issue 4, August 2000, Pages 523-528

https://doi.org/10.1016/S0097-8493(00)00055-8 Get rights and content

Abstract

Recently, we proposed the stochastic sampling method (SSM), which can sample implicit surfaces with Monte Carlo simulation based on a stochastic differential equation. In this paper, we demonstrate that the SSM realizes an excellent intersection-detection algorithm for implicit surfaces. The algorithm is fast enough for practical use, reliable, and widely applicable to complicated surfaces topologically, mathematically, and/or in shape. Moreover, the algorithm is suitable to parallel calculation using multiple CPU powers, and it can accelerate the intersection detection to a great extent.

Introduction

Sampling and visualizing implicit surfaces, i.e. surfaces expressed with implicit functions, are becoming an interesting target of three-dimensional (3D) computer graphics. Implicit surfaces give compact and exact expressions to 3D objects, which are complicated topologically, mathematically, and/or in shape, compared with the commonly used polygon-based expression. Application of implicit surfaces to the ray-tracing rendering [1], visualization of equi-potential surfaces [2], etc., are also interesting topics.

In constructing a consistent 3D scene made of implicit surfaces, it becomes important to detect their intersection, i.e. collision. It is, however, not always a trivial problem to be solved. Fast and reliable sampling methods for implicit surfaces can make the intersection detection possible and easy for us.

Recently, we proposed a new stochastic method to numerically generate sample points on implicit surfaces, which we call the stochastic sampling method (SSM) below [3], [4]. The SSM is to perform Monte Carlo simulation confined on a sampled implicit surface. The Monte Carlo simulation is based on a stochastic differential equation (SDE) [5], [6]. The SSM can be categorized as sampling by physically based particle systems [7], [8], [9]. But it is different from the conventional methods of the same category in that it uses “stochastic” particles. The SSM is fast enough for practical use. It is widely and easily applicable, too. Moreover, it can generate a uniform distribution of sample points, and the uniformity can be mathematically proved explicitly. It should also be noted that the stochastic feature guarantees realization of equilibrium state even with single-particle-system calculation. It is also possible to perform multi-particle-system calculation completely in parallel for each particle.

In this paper, we demonstrate that the SSM realizes an excellent intersection-detection algorithm for implicit surfaces. The organization of this paper is as follows. In Section 2 we briefly review the prescription of the SSM. In Section 3 we describe the algorithm of intersection detection. In Section 4 we present real examples of intersection detection. In Section 5 we describe summaries of the achievements.

Section snippets

Stochastic sampling method

In this section we briefly review prescription of the SSM to sample implicit surfaces [3]. We consider a twice-differentiable scalar field $F(q)$ , where $q =(q_{1},q_{2},…,q_{d})$ is a d-dimensional position vector. An implicit function, $F(q)=0,$ defines a constraint condition imposed on $q$ , and thus defines an implicit surface, which is denoted by I_F below.

Let us consider a fictitious particle whose position vector is $q$ . In the SSM we replace $q$ with a function of a “time” variable t, i.e. with $q (t)$ . (Below the

Algorithm for intersection detection

In this section, we explain the algorithm of intersection detection for implicit surfaces based on the SSM. We consider to detect intersection between two implicit surfaces I_FA and I_FB, which are defined with implicit functions $F_{A} (q)=0, F_{B} (q)=0,$ respectively. The intersection can be detected by setting up and numerically solving the SDE (2) for one of I_FA and I_FB. Here, we choose to set up the SDE for I_FA, which is obtained by replacing F with F_A in , , , .

Numerical solutions of the SDE for I_FA

Examples

In this section we show examples of applying the above-mentioned intersection-detection algorithm to real implicit surfaces. Our numerical calculation here is performed with a Celeron 433 MHz PC and with C++ language. Since the implicit surfaces we take here are all with rough size L_A∼1, we set t_A=1/D and t_max=10/D. The parameters of the SDE (2) are chosen as Δt=0.01, K=1/Δt, and D=1. With these parameters, t_A=1/D=1 becomes time for generating 100 sample points, and t_max=10/D=10 becomes time

Conclusions

In this paper, we have proposed a new algorithm to detect intersection of implicit surfaces. The algorithm is based on the SSM [3], which we recently proposed. It uses fictitious stochastic particle(s), which perform Brownian motion confined on investigated implicit surfaces. The algorithm is fast and reliable, and it is easily and widely applicable to complicated surfaces topologically, mathematically, and/or in shape. It does not require initial conditions given strictly on investigated

Acknowledgements

Authors want to express their deep thanks to Akio Morisaki, Satoru Nakata and Kisou Shino for fruitful discussion.

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Technical SectionStochastic algorithm for detecting intersection of implicit surfaces

Abstract

Introduction

Section snippets

Stochastic sampling method

Algorithm for intersection detection

Examples

Conclusions

Acknowledgements

Computers & Graphics

A generalization of algebraic surface drawing

ACM Transactions on Graphics

Field functions for implicit surfaces

The Visual Computer

Technical Section
Stochastic algorithm for detecting intersection of implicit surfaces