Technical SectionStochastic algorithm for detecting intersection of implicit surfaces
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 , where is a d-dimensional position vector. An implicit function,defines a constraint condition imposed on , and thus defines an implicit surface, which is denoted by IF below.
Let us consider a fictitious particle whose position vector is . In the SSM we replace with a function of a “time” variable t, i.e. with . (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 IFA and IFB, which are defined with implicit functionsrespectively. The intersection can be detected by setting up and numerically solving the SDE (2) for one of IFA and IFB. Here, we choose to set up the SDE for IFA, which is obtained by replacing F with FA in , , , .
Numerical solutions of the SDE for IFA
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 LA∼1, we set tA=1/D and tmax=10/D. The parameters of the SDE (2) are chosen as Δt=0.01, K=1/Δt, and D=1. With these parameters, tA=1/D=1 becomes time for generating 100 sample points, and tmax=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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