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There is one important attribute of conic surfaces that lies in their mathematical definition. To illustrate this, a section of an ellipsoid, i.e. an ellipse, is shown in Figure 5.1. An ellipse is defined by its two foci and has the property that a line drawn from one focus to any point on the ellipse and thence to the other focus has the same total length regardless of which point on the ellipse was included.

The ellipsoid is defined by its two foci, F₁ and F₂. In this instance, the shape of the ellipsoid is defined by its semi-major distance, a, and its semi-minor distance, b. As suggested, the key point about the ellipsoid shape sketched in Figure 5.1 is that the aggregate distance F₁P + PF₂ is always constant. By virtue of Fermat's principle, this inevitably implies that, since the optical path is the same in all cases, F₁ and F₂, from an optical perspective, represent perfect focal points with no aberration whatsoever generated by reflection from the ellipsoidal surface. In describing the ellipsoid above, it is useful to express it in terms of polar coordinates defined with respect to the focal points. If we label the distance F₁P as d, then this distance may be expressed in the following way in terms of the polar angle, θ:

Figure 5.1 Ellipsoid of revolution.

(5.2)

The parameter, ε, is the so-called eccentricity of the ellipse and is related to the conic parameter, k. In addition, the parameter, d₀ is related to the base radius, R, as defined in the conic section formula in Eq. (5.1). The connection between the parameters is as set out in Eq. (5.3):

(5.3)

From the perspective of image formation, the two focal points, F₁ and F₂ represent the ideal object and image locations for this conic section. If x₁ in Figure 5.1 represents the object distance u, i.e. the distance from the object to the nearest surface vertex, then it is also possible to calculate the distance, v, to the other focal point. These distances are presented below in the form of Eq. (5.2):

(5.4)

From the above, it is easy to calculate the conjugate parameter for this conjugate pair:

In fact, object and image conjugates are reversible, so the full solution for the conic constant is as in Eq. (5.5):

(5.5)

Thus, it is straightforward to demonstrate that for a conic section, there exists one pair of conjugates for which perfect image formation is possible. Of course, the most well known of these is where k = −1, which defines the paraboloidal shape. From Eq. (5.5), the corresponding conjugate parameter is −1 and relates to the infinite conjugate. This forms the basis of the paraboloidal mirror used widely (at the infinite conjugate) in reflecting telescopes and other imaging systems.

As for the spherical mirror, the effective focal length of the mirror remains the same as for the paraxial relationship:

(5.6)

More generally, the spherical aberration produced by a conic mirror is of a similar form as for the spherical mirror but with an offset:

(5.7)