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In this discussion, we will restrict ourselves to surfaces that are symmetric about a central axis. Although more exotic surfaces are used, such symmetric surfaces predominate in practical applications. The most general embodiment of this type of surface is the so-called even asphere. Its general form is specified by its surface sag, z, which represents the axial displacement of the surface with respect to the axial position of the vertex, located at the axis of symmetry. The surface sag of an even asphere is given by the following formula:

(5.1)

c = 1/R is the surface curvature (R is the radius); k is the conic constant; α_n is the even polynomial coefficient.

The curvature parameter, c, essentially describes the spherical radius of the surface. The conic constant, k, is a parameter that describes the shape of a conic surface. For k = 0, the surface is a sphere. More generally, the conic shapes are as set out in Table 5.1.

Table 5.1 Form of conic surfaces.

Conic constant	Surface description
k > 0	Oblate ellipsoid
k = 0	Sphere
−1 < k < 0	Prolate ellipsoid
k = −1	Paraboloid
k < −1	Hyperboloid

Without the further addition of the even polynomial coefficients, α_n, the surfaces are pure conics. Historically, the paraboloid, as a parabolic mirror shape, has found application as an objective in reflective telescopes. As will be seen subsequently, use of a parabolic mirror shape entirely eliminates spherical aberration for the infinite conjugate. The introduction of the even aspheric terms add further useful variables in optimisation of a design. However, this flexibility comes at the cost of an increase in manufacturing complexity and cost. Strictly speaking, at the first approximation, the terms, α₁ and α₂ are redundant for a general conic shape. Adding the conic term, k, to the surface prescription and optimising effectively allows local correction of the wavefront to the fourth order in r. In this context, the first two even polynomial terms are, to a significant degree, redundant.