Generalized Solids of Revolution

1. Introduction

During evolution of geometric modeling, several modeling techniques were developed. In many cases, the goal was to provide not too complex technique for rapid modeling of non-trivial models. Some of them are well-known and widely spread. This includes such techniques as CSG, fractals, particle systems or sweeping. This paper deals with sweeping, or strictly speaking with special subset of sweeping - generalized solids of revolution.

Sweeping technique moves an object (called shape or cross-section) along arbitrary curve (called path) [Grat89]. A "tail" that profile leaves while is moving (or mathematically more precisely: union along all positions) creates resulting solid. The term "object" was used intentionally. It can be a three-dimen­sional solid, two-dimensional filled shape or any curve (or point, if you wish unusual way of visualizing a path). Shape and orientation of object can change during sweeping. Depending on swept object, result of sweeping can be in volumetric or boundary representation (or it is just a simple patch).

While general sweeping idea gives us wide range of possibilities, it is difficult to model. Therefore special cases of sweeping are more widely spread. The simplest variant of sweeping is extrude. Two-dimensional shape (usually represented by boundary curve) is swept along line and is not rotated or changed. More complex is path extrude. Path can be any curve and swept object, while not changing shape, can even stay unrotated or rotate to be perpendicular to path. Even more complex shapes can be created by lofting/skinning approach. Along path, there are several key positions where shape of swept object is defined. Between two key positions, shape is interpolated. Usual behavior is to rotate it to be perpendicular to path.

In all previous versions, closed path was an exception. Lathe (or solid of revolution) uses circular path to create three-dimensional objects. Two-dimensional shape is always perpendicular to path. Usual description of lathe objects uses axis instead of circular path. Center of path belongs to axis and plane created by path is perpendicular to axis.

One remark is needed at this place. Naming conventions of sweeping techniques are not consistent. One technique can have several names and there is some small variability what exactly the technique with arbitrary name allows. For example in [Smed02], above described techniques extrude, path extrude and lathe are noticed. In [Povr02], along with name lathe, also term surface of revolution is used. In this paper, the term solid of revolution is preferred for the technique earlier mentioned as lathe.

In a following text, only surface model is used (unless explicitly stated otherwise). Although solids of revolution can be defined to allow rotation by angle less than 360 degrees, in this paper only full 360 degrees rotation is discussed.