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12 matrix 12 points antisymmetrical bump chord coefficients Conformal Mapping constant corresponding m terms cosine series cr cr cr gj curve diameter distortion elements entire mirror errors figure Fourier analysis Fourier matrix Fourier series gf gj gj gf I I I I I I I I I I LU LU imperfection integral equation inversion matrix KNIFE-EDGE TEST PATTERN Langley Research Center Legendre polynomials light intensity linearized LU LU LU LU LU UJ LU UJ LU mirror surface MIRROR WITH CENTRAL mr(x nine values O O I I o o o o o o odd function oooo I I I I LU optics perfect mirror pg pg pg rg primary mirror procedure quantitative rg cr rg pg rg rg rg ro rg shown simple mirror sketch solution symmetrical UJ LU LU unbroken parabolas vector zero