Constructive solid geometries

In order to create micromagnetic problem input sources, such as MIF files for
OOMMF or to describe boundaries for magpar's finite element meshes, we need to define the basic geometry of the problem.

A basic geometry is one described by a simple mathematical equation, and these are generally accepted by three-dimensional graphics modellers to be primitives -- constructive solid geometries (CSGs) (see figure F.1). The primitives can be considered to be the set of spheres, torii, cylinders, cuboids, pyramids and cones, as well as the associated two-dimensional shapes. By adding or removing conditions for satisfying the equations, the shape can be modified. For example, if one considers a sphere where the volume conditions are defined by:

$$x^2 + y^2 + z^2 \leq 1$$ (F.1)

where $x$, $y$ and $z$ are between $-1$ and $1$ then it is straightforward to modify this such that the conditions in equation F.1 are met and $z \leq 0$, then a half-sphere is produced. This could be called a Boolean shape because the resultant geometry shows a ``negative'' cuboid has been cut away from the sphere. A section of a sphere could be created as though two cuboids have been subtracted from it by, in addition to satisfying the above equations, satisfying $z \geq -0.5$. If this is satisfied, and the conditions for a cone:

$$x^2 + y^2 \leq \left({z+1 \over 2}\right)^2$$ (F.2)

are also met, then a section of a cone results and so forth. By continuing in this fashion, it is possible to use a handful of primitives to build more complex objects and subsequently arrays of these objects (see figure F.1).

Figure F.1: Simple constructive solid geometries as described by the equations in 3.3. From left to right: sphere, half-sphere, quarter-sphere and quarter-cone