New algorithm to find isoptic surfaces of polyhedral meshes

Highlights

• Fast and robust algorithm to find isoptic surface of a given mesh.

• It works also for concave meshes.

• A new, alternative definition of isoptic surface is provided.

Abstract

The isoptic surface of a three-dimensional shape is recently defined by Csima and Szirmai (2016) as the generalization of the well-known notion of isoptics of curves. In that paper, an algorithm has also been presented to determine isoptic surfaces of convex polyhedra. However, the computation of isoptic surfaces by that algorithm requires extending computational time and CAS resources (in Csima and Szirmai, 2016 Wolfram Mathematica Inc, 2015 was used), even for simple regular polyhedra. Moreover, the method cannot be extended to concave shapes. In this paper, we present a new searching algorithm to find points of the isoptic surface of a triangulated model in ${\mathbb{E}}^3$, which works for convex and concave polyhedral meshes as well. Alternative definition of the isoptic surface of a shape is also presented, and isoptic surfaces are computed based on this new approach as well.

Introduction

In CAGD systems there are only primitive tools available to determine camera positions to display and study three-dimensional objects from various directions. However, in certain applications, it can be important to find positions, from where a predefined portion of a given object can be seen.

In two dimensions we can use viewing angle to determine such points which led us to the well-known definition of isoptic curves of a given curve. For arbitrary given plane curve, the isoptic curve is the locus of those points of the plane from where the given curve can be seen under a predefined angle (of less than π). These points can be determined by drawing tangents to the given curve—the tangents have to meet at the given angle in these points (if there are more than two tangents from a point to the curve, then the widest angle is considered). Isoptics of several classical curves have been studied from the ancient Greek time to present (see, e.g. Yates, 1947 for an overview of these results).

In terms of recent applications, appropriate camera points or paths can be determined using isoptics. Although theoretical results can also be of great interest, the ultimate goal of the present paper is to provide a fast computation of isoptics that can be used as camera positions in a modeling software.

Extension of the two-dimensional principle to three-dimensional shapes is not trivial and not unique. One can try to find isoptic curves of the spatial object in a given “base” plane. Our previous work was such an extension of the isoptic curves to ${\mathbb{E}}^3$, described in Nagy and Kunkli (2013), together with a computation of the isoptic curve of a Bézier surface in a special case, that can be used as a camera path around the given Bézier surface (see Fig. 1).

If we consider isoptics as hypersurfaces of the given space, then one can also try to define and compute an isoptic surface in 3D, but then the given angle of the planar case must be substituted by an appropriate notion and measure. Bearing the applications in mind, we can consider this angle as the measure of visibility. In case of curves, it can also be measured by the assigned arc length of a unit circle around the viewpoint. Then the obvious generalization of this notion is the area of visibility in a unit sphere around the viewpoint, which was precisely defined in Csima and Szirmai (2013) and will be discussed in Section 2, together with the brief overview of the existing method. Unfortunately, that method works only for convex shapes and takes several minutes to calculate the isoptic surface. Our aim is to provide a fast and robust method to compute the isoptic surface, even for concave meshes. This new algorithm is presented in Section 3.

One can also generalize the 2D notion in an alternative way, by considering the isoptic point in the plane as the intersection of two tangent lines to the curve, i.e. the “widest view” of the curve from that point. With this approach, the 3D generalization of the angle can be the widest spatial view, that is the measure of the widest diameter of the projection of the shape onto the unit sphere around the viewpoint. This generalization will be briefly discussed in Section 4.