^ The paraxial segments are bent convex whilst the marginal ones are bent concave.

^ Compared to RS = R, the stresses for the curvature mode a20 are 2-times smaller. If a paraboloid mirror is at f/1, then um = 1/4.2, and from (7.39), RS ~ 1.0132R.

• Determination of the forces and moments at the contour: The conversion of the anm coefficients into the Anm coefficients which are basically used in this chapter is simply achieved by

The associated axial forces Fa,k and Fc,k generating the net shearing force and bending moment at the contour of the meniscus can be readily determined from the Clebsch-Seidel modes such as derived in Sect. 7.3.

• Linear segment number and global mirror f-ratio: The reasons for making some optical modes in a segment negligible come from elasticity: The Sphe 3 elastic mode requires a uniform load applied all over the segment which would introduce some complexity in practical applications of the stressing process. Next elasticity modes such as Coma 5 and Sphe 5 are not Clebsch-Seidel modes - in our definition of these modes (cf. the two diagonal lines of the matrix in Fig. 7.3) - since requiring prismatic and quadratic loadings respectively.

Hence, the reduction of Sphe 3 and Coma 5 modes to negligible values must be achieved by a convenient design of the segmentation. This involves an optimal setting of the linear segment number (N) and global mirror f-ratio (Q) which are fundamental parameters for the primary mirror of extremely large telescopes (ELTs).

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