## Abstract

Fejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the *square* of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

## 1 Introduction

In [3] Fejes Tóth introduced inscribed triangulations approximating convex surfaces in *A*_{2} (Approximierbarkeit). By a triangulation we shall mean a geometric realization of a simplicial complex in Euclidean space homeomorphic to the surface, that is piecewise linear in ambient space. From now on we take a simplicial complex to mean the geometric realization.

Optimal triangulations with *m* vertices are triangulations which minimize the Hausdorff distance between the surface and the simplicial complex when this simplicial complex ranges over the space of triangulations with *m* vertices. We always assume that the vertices lie on the surface.

*X*and

*Y*in Euclidean space is defined as:

*x*−

*y*| denotes the standard Euclidean distance of

*x*and

*y*. The onesided Hausdorff distance from

*X*to

*Y*is given by

The inverse of the asymptotic value of the product of the number of vertices and the Hausdorff distance is referred to as *the approximation parameter (Approximierbarkeit) A*_{2}.

*K*is the Gaussian curvature, for the approximation parameter for convex surfaces in three dimensional Euclidean space. We refer to [1,4,5] for an introduction to geometry including the Gaussian curvature.

Fejes Tóth also claimed that the approximation of ruled surfaces embedded in three dimensional Euclidean space would be entirely different from the approximation of convex surfaces. In Section 12 of Chapter 5 of [3] he states the following:

*A*and

*B*. We inscribe

*A*and

*B*by regular

*m*-polygons

*T*

_{2m}is best described by its faces

The deviation *m*-polygon *A*, this implies that the order of magnitude of the deviation is 1*/m*^{2} and not 1*/m*.

Unfortunately this is incorrect. In fact we shall show that the order of magnitude of */m*, like in the convex case. Moreover in this particular case we can explicitly calculate

All calculations were performed by hand and verified using Mathematica, while producing the figures.

## 2 The triangulation of the hyperboloid

We prove that for the triangulation of the one-sheeted hyperboloid (with two circles of equal size as boundary) suggested by Fejes Tóth we have *d*_{H}(*T*_{m}*,* Σ) ∼ 1*/m*.

*x*

^{2}+

*y*

^{2}−

*z*

^{2}= 1) by

*.*

*t*, given by

*u*7→

*σ*(

*u,t*). We shall assume that

*u*∈ [−

*u*

_{max}

*, u*

_{max}], so that the two circles that form the boundary lie at a distance

*xy*-plane. In our parametrization we take

*t*∈ [0

*,*2

*π*].

The vertices of the triangulation discussed by Fejes Tóth are equally distributed along the lower and upper boundary and placed such that for every vertex on the lower boundary there is a vertex on the upper boundary that lies on the same ruling as the vertex on the lower boundary. Fejes Tóth assumes that *m* is even. The triangulation is characterized by the fact that these rulings connecting the vertices are edges of the triangulation.

The edges of the triangles in the triangulation fall into three different categories:

rulings that lie on the surface,

edges of a regular

*m/*2-gon approximating the upper or lower boundary,edges that connect ‘neighbouring’ vertices on the upper and lower boundary.

*z*-axis edges from the last category can be parametrized by

*s*= 4

*π/m*.

We now prove a lemma that refutes Fejes Tóth’s claim that *d*_{H} (*T*_{m}*,* Σ) ∼ 1*/m*^{2} for the triangulations described above:

*For the sequence of triangulations T*

_{m}

*of the one-sheeted hyperboloid*Σ

*bounded by two congruent circles suggested by Fejes Tóth, as described above, we have*

Proof. To determine *d*_{H}(Σ*, T*_{m}) we first give an upper bound. This bound is found by considering the triangulation and the surface restricted to horizontal planes. The hyperboloid restricted to a horizontal plane is a circle. The restriction of the triangulation is a (somewhat complicated) polygon, see Figure 4. The vertices of this polygon lie on the circle or are the restriction of an edge like (2), that is the intersection of a horizontal plane and the edge. For each horizontal plane we can determine the Hausdorff distance between the polygon and circle. This is straightforward because it is attained in the restriction of an edge like (2), due to the negative curvature of the surface. The maximum of all these pseudo-distances, defined below, bounds the Hausdorff distance between the hyperboloid and Fejes T´oth’s triangulation. This maximum is attained for *z* = 0, where the normal to the hyperboloid is horizontal so that here this pseudo-distance coincides with the Hausdorff distance between the hyperboloid and Fejes Tóth’s triangulation.

*d*(

*x,y*) = 0 does not imply

*x*=

*y*. Note that

*d*

_{hor}(

*v, w*) ≤ |

*v*−

*w*|, where |···| denotes the Euclidean norm. We are now able to calculate

*o*= (0

*,*0

*,*0). Note that

*s*= 4

*π/m*and we are interested in the limit of

*m*tending to infinity. On the other hand the

*z*-coordinate of the edge parametrized by (2) is

*z*= 0 and the normal to the hyperboloid is horizontal for

*z*= 0. This means that the point

*p*where the supremum is attained and the point

*p*both lie in the

*z*= 0 plane, by [2, Theorem 4.8.12]. This in turn implies that the right hand side of (4) equals the Hausdorff distance. We therefore find that

### Lemma 2.1

contradicts the assertion of Fejes Tóth.

## Acknowledgements

The authors are greatly indebted to Dror Atariah, Günther Rote and John Sullivan for discussion and suggestions. The authors also thank Jean-Daniel Boissonnat, Ramsay Dyer, David de Laat and Rien van de Weijgaert for discussion. This work has been supported in part by the European Union’s Seventh Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL Computational Geometry Learning) and ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions), the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement number 754411, and the Austrian Science Fund (FWF): Z00342_N31.

## REFERENCES

- [1]
Do Carmo, M. P.,

*Differential Geometry of Curves and Surfaces*, Prentice-Hall, 1976. - [2]
Federer, H., Curvature measures,

*Transactions of the America mathematical Society*, 93 (1959), 418–491. - [3]↑
Fejes Tóth, L.,

*Lagerungen in der Ebene, auf der Kugel und im Raum*, Berlin, Göttingen, Heidelberg: Springer, 1953. - [4]
O’Neill, B.,

*Elementary Differential Geometry*, Elsevier, 2006. - [5]
Spivak, M., A comprehensive introduction to differential geometry: Volume I to V,

*Publish or Perish*, 1999.