R

which expands as

16R5

where 1/R and k are the curvature and the conic constant.

In order to represent the shape of a segment mirror with respect to its vertex, we consider a local coordinate frame (x, y, z) whose origin is a point at the surface of the mirror, and at distance d from the Z-axis (Fig. 7.7). These coordinates are set such as

X = d, Y = 0, Z = ZOpt{X2 + Y2 = d2}, the (x, y) plane is tangent to the global surface at this origin, and the y- and Y-axis are parallel. Let us define the dimensionless quantities v = d/R, s = sin (arctan v), c = cos (arctan v),

where arctan v is the inclination angle of the (x, y) plane with respect to the (X, Y) plane. The coordinates of the two frames are linked together by

7.5.1 Off-Axis Segments of a Paraboloid Mirror

Now considering the case where the mirror is a paraboloid, i.e. k = -1 in eqs. (7.25), the system writes

Since Z = (X2 + Y2)/2R at the optical surface, after substitution we obtain

d2 1

— + sx + cz = — [(d + cx - sz)2 + y2], and since s/c = d/R = v, this becomes a quadratic equation in z,

The root satisfying the condition z{0,0} = 0 provides the optical shape zOpt of a segment in the local system. This is

0 0

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