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Nonimaging Optics Winston Episode 11

Tham khảo tài liệu 'nonimaging optics winston episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 11.6 Examples of Globally Optimized Concentrator Designs 289 the curve C at some point P on C. We also define the circle c that passes through the point P and is centered on the x-axis. The line tangent to c at point P is designated as T . The angle between T and T is designated as a. A constant-angle spiral is defined as any continuous curve C for which the angle a is a constant for all points on the curve. We define s to be the distance of a point P on a constant-angle spiral from the vertex of the cone. We consider the differential change ds in the distance s produced by a differential change df in the angle f. The constant-angle criterion leads to following relationship between ds and df ds - s tan a . 11.59 rdf The radial coordinate r can be expressed in the form r ro s - So sin b 11.60 where r0 and s0 are the r and s-coordinates respectively of the starting point on the spiral. Similarly the axial coordinate x can be expressed as x Xo s - So cos b 11.61 where x0 is the x-coordinate of the starting point on the spiral. Substitution of Eq. 11.60 into Eq. 11.59 gives ds tan a df. 11.62 r s - S0 sin b Integrating both sides of Eq. 11.62 we obtain an expression for s as a function of f s f s0 etan a sin b f-f0 - 11 11.63 sin b where f0 is the value of the angular coordinate f at the starting point of the spiral. Substitution of Eq. 11.63 into Eq. 11.60 gives the r-coordinate as a function of f r f r0etan a sin b Xf-f0 . 11.64 Substitution of Eq. 11.63 into Eq. 11.61 gives the x-coordinate as a function of f x f x0 r etan a sin bXf-f0 - 11. 11.65 tan b For the special case of b 0 Eqs. 11.64 and 11.65 become r f r 11.66 and x f x0 r0 tan a f - f0 . 11.67 For the special case of b 90 Eqs. 11.64 and 11.65 become r f r0 etan a xf-f 11.68 and x f x0. 11.69 For the special case of b 180 Eqs. 11.64 and 11.65 become 290 Chapter 11 Global Optimization of High-Performance Concentrators r f r 11.70 and x f x0 - r0tan a f - f0 . 11.71 We now combine segments of constant-angle