ABSTRACT
In mathematics, a conic section, or just ‘conic’, is a curve obtained by intersecting a cone, or more precisely, a circular conical surface, with a plane. A conic section is therefore a restriction of a quadric1 surface to the plane. Conic sections, in all their various forms, were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.[39]
In all, there are three types of conics named the ellipse, the hyperbola and parabola. The circle can be considered as a fourth type, as it was by Apollonius of Tyana2, or as a kind of ellipse, thereby making it inclusive to the three original forms. The circle and the ellipse arise when the intersection of cone and plane is a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the conic formed is hyperbolic. In this case the plane will intersect both halves, or nappes, of the cone, producing two separate unbounded curves, though often one is ignored.[39]
The conic section is a key concept in abstract mathematics, but it is also seen with considerable regularity in the material world, with many practical applications in engineering, physics, and other fields. Furthermore the conic section may well hold the key to addressing an apparent weakness in the ‘X’-shaped pattern. Before that can be considered, however, we must first
understand that there are two types of possible behaviour in computational systems, those being linear and non-linear.
