The conicoids are rotational symmetry aspherics whose meridian sections belong to the conic family. These shapes are represented by
where R = 1/c is the radius of curvature and k the conic constant, also called the Schwarzschild constant, which characterizes the asphericity since k = 0 generates a sphere. This equation may be written
whose first terms of the expansion are
1 2 1 + k 4 (1 + k)2 6 5(1 + k)3 8 z = — r2 +——77- r4 + v ' r6 + v ' r8 + ■■■ ■ (1.38b)
We obtain for the general expansion of a conicoid z = T (2n-2)! n (1 + K)n-1 r2n (138c)
z 22"-1 (n!)2 R2— r , where the indeterminate form (1 + k)"-1 ^ 1 when n ^ 1 and k 1.
The conicoids are classed into families from the value of their conic constant k (see below and Fig. 1.20):
k < -1 hyperboloid k = -1 paraboloid -1 < k < 0 elongated ellipsoid or prolate ellipsoid k = 0 sphere k > 0 flattened ellipsoid or oblate ellipsoid
One may use the eccentricity e = ^J-k which is imaginary for a flattened ellipsoid.
Fig. 1.20 Conicoid sections of same curvature
A particular case is a biconicoid surface; the conic sections differ in the x- and y-directions. Denoting cx, cy, Kx, Ky the curvatures and conic constants, this surface is represented by
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