On the tangent model for the density of lines and a Monte Carlo method for computing hypersurface area

1 Introduction

The problem of computing surface areas of geometric objects is of interest in many areas of applications. Studying and computing such measures is a central topic in integral geometry and geometric probability. The classical integral expression for the surface area from analytic geometry is often difficult to reduce analytically for dimensions n>3. These difficulties are also encountered, for instance, when employing quadrature methods. It is well known that Monte Carlo-type methods become the method of choice for estimating high dimensional integrals. The purpose of this article is to develop further a Monte Carlo-type method based on the Cauchy–Crofton formula (CCF) from integral geometry to compute hypersurface areas of compact convex bodies.

A interesting review of the early history of the CCF and its extensions is given in [3]. One can find the relevant theoretical treatment of integral geometry and geometric probability for example in [7, 9]. The CCF transforms the problem of finding the surface area into counting intersections of the surface with a set of uniformly distributed lines. The algorithm we employ is based on the CCF coupled with a comparison principle. More specifically, suppose we know the surface area S1 of a reference bounding object Σ1 containing in its interior the object with boundary Σ whose surface area S we wish to compute. Consider a random sample of a set of N lines from the set of lines that intersect Σ1. Let k1 and k be the total number of intersection points with Σ1 and Σ, respectively. Then S≈kk1⁢S1. The origins of this algorithm can be traced back to the work of W. M. Crofton (in 1867) and to E. Czuber (in 1884), who gave this algorithm for computing experimentally the perimeters of closed convex curves (see [3, p. 9]). It is advantageous to take Σ1 as a circumscribing hypersphere (circumsphere). This gives this algorithm some generality in that it allows one to take advantage of the well-known uniform sampling from spheres and balls. This algorithm was employed by Li, Wang, Martin and Bowyer in [4] to compute surface areas for constructive solid geometry models in 3D (n=3). In this study, we examine the implementation and application of this algorithm in dimensions n≥3. We note that this method may apply to compute surface areas of certain bodies which are not necessarily restricted to be convex – examples in 3D of this sort were examined in [4].

A key ingredient for applying this algorithm is to generate a set of uniformly distributed lines in Euclidean space 𝐄n. The theory for the density of lines in 𝐄n is given in Santalo [7]. For n=3, Li, Wang, Martin and Bowyer give in [4] two models for generating lines; the chord model and the tangent model. In Section 2, we visit briefly the problem of estimating surface areas in 3D. In Section 3, we develop the tangent model further to compute hypersurface areas for n=4 (or 4D), and we give an extension that applies in dimensions n≥5. In Section 4, we apply the results developed in Section 3 to compute hypersurface areas of Fermatoids – a class of compact convex hypersurfaces that are defined by Fermat varieties of even degree.

Before we present our results, we recall briefly a result on the surface areas of n-dimensional ellipsoids developed in [11]. We employ this result to verify our algorithm and computations. The hypersurfaces of n-dimensional ellipsoids are defined by the equation

where ai are constants. In [11], Tee gives a reduction of the hypersurface area integral to an abelian integral on [0,1], which is well suited for numerical evaluations. Let

For n≥3, the surface area of an n-dimensional ellipsoid is given by ([11, equation (93)])

In Table 2, we give, in the column labeled SA(1), estimates from (1) with the parameters ai for various ellipsoids in dimensions 3, 4, 5 and 8.

2 3D models

In [4], two models for generating uniformly distributed lines in Euclidean three space 𝐄3 are described; namely, the chord model and the tangent model. Also, a quasi-Monte Carlo method for computing surface areas in 3D for constructive solid geometry models was developed. The prefix “quasi” was used to indicate the use of low-discrepancy sequences, which were employed to improve the efficiency and accuracy of the estimates. We employ the low-discrepancy Halton sequences in all of our computations. To set the stage for further development in higher dimensions, we briefly discuss in this section the implementation of these models and fill in some additional details. For illustration, we compute the surface areas of some 3-dimensional ellipsoids with known surface areas (obtained from equation (1)).

2.1 The Chord model

In this model, a random line is defined as a line passing through two independently uniformly distributed points on a sphere SR2 of radius R in 𝐄3. Each random chord within the sphere can be associated with a random line which can be considered as its carrier. By considering the chord length distribution, Solomon [9] showed that the associated random lines have a uniform distribution. An alternative justification for this uniform distributivity is given in [4]. Now, two approaches to generate uniform distributions of points on SR2 were developed in [5, 8, 10, 6]. One method to obtain a uniform distribution of points within SR2 is to take

x=2⁢R⁢u13⁢w12⁢(1-w)12⁢cos⁡θ,y=2⁢R⁢u13⁢w12⁢(1-w)12⁢sin⁡θ,z=R⁢u13⁢(1-2⁢w),

where u and w are uniform on [0,1], and θ is uniform on [0,2⁢π]. Setting u=1, one obtains a set of uniformly distributed points on the surface of SR2. These formulas follow from results given by Tashiro in [10]. Tests of the CCF using this model on ellipsoids of various dimension are shown in Table 1. Moreover at this point, we also show in Table 1 results obtained for the chord model when extended and applied to compute surface areas of ellipsoids in 4D. This naive extension of the chord model from 3D to 4D does not work. Indeed, this method of generating chords fails in 2D as well, and perhaps the 3D case is an exception. In support of this empirical finding, and for comparison purposes with the tangent model (developed and discussed below), we examine the effect of scaling the radius of the circumsphere on the surface area estimates. We think of this invariance to perturbation of the radius of the circumsphere as a kind of a stability analysis. Figures a and b show the stability analysis in 3D and 4D, for both the chord model and the tangent model. It is clear that in 3D, both models are stable to scaling and give almost identical results and confirm further the findings in [4]. However, in the 4D case, we see that the chord model does not give good estimates and it is not stable to scaling of the circumsphere radius.

Table 1

Surface area estimates from equation (1), and the chord model for 3D and 4D ellipsoids.

Dimension	Parameters	SA(1)	Chord model
3	12, 34, 1	6.9715	6.9768
	98100, 99100, 1	12.3160	12.2949
4	12, 46, 56, 1	7.9904	10.7618
	94100, 96100, 98100, 1	18.0090	19.2201
	1720, 1820, 1920, 1	15.5850	17.8186

Figure 1

(a)

Stability analysis for 3D model.

(b)

Stability analysis for 4D model.

3 4D and higher dimensions

In this section, we present the extension of the tangent model to 4D, and to higher dimensions. One may proceed along the same lines in 4D as in the 3D case. That is, first pick a point P on the sphere Sr3; then pick two random coordinates for the point on the 3-sphere centered at P and solve for the other two coordinates. The results from this case are reported in the column labeled SA(a) of Table 2. This method is a bit awkward to extend as n increases.

A better alternative method is to first generate a point P on Sr3 using

where u and w are uniform on [0,1], and θ1,θ2 are uniform on [0,2⁢π]. Save for the factor R⁢u13, these formulas give the coordinates of points uniform on the surface of S3 given in [8]. Now the second step in the tangent model is to pick a line that is uniformly distributed on the hypersphere centered at P that is embedded within the hyperplane (a higher dimensional analogue of the unit disc centered at P we utilized in the 3D case). Therefore, all we need now is to find another suitable random point on this sphere to form the equation of a random line that we can use to find the intersections with Σ. The equation of the hyperplane H tangent to Sr3 at the point P=(x1,y1,z1,t1) is given by x1⁢x+y1⁢y+z1⁢z+t1⁢t=r2.

Now we resort to a different approach for sampling the uniform distribution on spheres (see [6]). Pick 𝐚=(a1,a2,a3,a4) from the standard normal distribution N⁢(0,1). Find the perpendicular line passing through 𝐚 in the direction of the vector OP. The foot of this line on H, is the projection 𝐛 of 𝐚 onto H, and is given by

Now let

Next, compute 𝐜=(c1,c2,c3,c4), the normalization of 𝐛, as

Now, it is easy to verify that x1⁢c1+y1⁢c2+z1⁢c3+t1⁢c4=r2D. Therefore, to bring 𝐜 back onto the sphere within the hyperplane, project again, and so we get 𝐝=(d1,d2,d3,d4) given by

It is easy to check that 𝐝 is on the unit sphere centered at P within the hyperplane. Set 𝐝=(x2,y2,z2,t2). This is the second point we need to find the equation of the random line defining a chord, and then we proceed as we did in the 3D case to find and count intersections. The results from this method are shown in Table 2, in the column labeled SA(c).

A variant of this method that we have also tested differs only in the way we obtain the point (x1,y1,z1,t1). In this case, this point is obtained using normal variates as well. Create a point (X,Y,Z,T) from N⁢(0,1). Then, to get it onto a sphere of radius r, we take

with similar expressions for y1,z1,t1, where r=w13 and w uniform on [0,R]. The results from this modification are shown in Table 2, under the column labeled SA(b). The advantage of this algorithm is that it can be extended to higher dimensions with a minimal effort. We present in Table 2 some results from this algorithm to estimate the hypersurface areas of ellipsoids in 5D and 8D. In instances where the ratio of the area of the circumsphere to the surface area of the n-ellipsoid is rather large, we note that the accuracy of the method is reduced, however, increasing the sample size helps. For example, the first result obtained for the 8D case in Table 2 is much more accurate with 106 samples than with 104 samples.

4 Surface areas of Fermatoids

In this section we present our estimates of hypersurface areas of Fermatoids. Methods to obtain such estimates in 2D were given in [2]. Consider the n-dimensional hyper-ellipsoid of degree 2⁢m, m=1,2,…, centered at the origin of a rectangular coordinate axes with semi-axes a1,…,an given by

The area of this (n-1)-dimensional surface is given by

For the n-ellipsoid (m=1) case, as was mentioned in the introduction, Tee [11, Section 4] gave a reduction of this n-dimensional surface area integral to an abelian integral on [0,1]. However, it appears that it is not an easy task to carry out a similar useful reduction for the hyper-ellipsoid case (m>1), and so this remains an open question at this point. Now if a1=⋯=an=1, we obtain the hypersurfaces that we call Fermatoids, as they are associated with the Fermat varieties ∑i=1nxi2⁢m=1, m>1. These power equations can be viewed as equations of n-dimensional l2⁢m unit spheres. Despite the symmetrical properties of Fermatoids, numerical quadrature methods could become tedious for n>3. As an alternative, we employ the Monte Carlo method developed above to compute estimates of hypersurface areas for a few Fermatoids. The results appear in Table 3 for Fermatoids of various dimensions and degrees.

5 Conclusion

In general, Monte Carlo type methods are the main tool for evaluating multidimensional integrals of high dimensions. In this article we presented a Monte Carlo type method based on the Cauchy–Crofton formula from integral geometry to compute hypersurface area integrals. For the practical implementation of this method, it is necessary to have a model to generate a set of uniformly distributed lines in Euclidean space 𝐄n. The chord model and the tangent model are two known models for generating such lines in 𝐄3 (see [4]). To our knowledge practical models for generating a uniform density of lines for dimensions n>3 are not available. In this study we extended the tangent model for dimensions n>3. We also found out that the chord model that works well for n=3, and which is known to fail for n=2, also fails for n=4. To test the performance of this Monte Carlo method, we carried out experiments to compute the hypersurface areas of n-ellipsoids. For most tests, our results agree very well with estimates for these hypersurface areas computed by another method given in the literature (see [11]).