By Psang Dain Lin (auth.)
This ebook computes the 1st- and second-order by-product matrices of skew ray and optical course size, whereas additionally delivering an immense mathematical device for computerized optical layout. This publication includes 3 components. half One experiences the elemental theories of skew-ray tracing, paraxial optics and first aberrations – crucial studying that lays the root for the modeling paintings provided within the remainder of this e-book. half derives the Jacobian matrices of a ray and its optical direction size. even though this factor can also be addressed in different courses, they typically fail to contemplate all the variables of a non-axially symmetrical method. The modeling paintings therefore offers a higher framework for the research and layout of non-axially symmetrical platforms akin to prisms and head-up monitors. finally, half 3 proposes a computational scheme for deriving the Hessian matrices of a ray and its optical course size, delivering a good technique of making a choice on a suitable seek path whilst tuning the method variables within the approach layout process.
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14, the pose matrix 5 ¼ tranð1; 0; 0Þtranð0; 2; 0Þtranð0; 0; 3Þ ¼ tranð1; 2; 3Þ: A5 is given by 4 A 4 Fig. 5 15 Speciﬁcation of a Pose Matrix by Using Translation and Rotation Matrices g , can be speciﬁed in As discussed in the previous section, the pose matrix, h A various ways using rotation and translation matrices. To describe the procedure for g , several intermediate coordinate frames are required, say ðxyzÞ , obtaining h A 1 1 ¼ tranðtx ; 0; 0Þ, ðxyzÞ2 , ðxyzÞ3 … (see Fig. 15). As shown in Fig.
4 Basic Translation and Rotation Matrices 13 , not g j , and rotð j; ÀhÞ ¼ rotð j; hÞT . Note that the unit vector of Eq. 25) is h j Notably, this general rotation matrix enables each of the elementary rotation matrices to be individually obtained. For example, rotðx; hÞ is rotð j; hÞ with h into Eq. 25) yields the j ¼ ½ 1 0 0 0 T . Substituting these values of h j results shown in Eqs. 17). 25) provides an alternative method for specifying the orientation of a pose matrix. Unfortunately, however, the is not intuitively obvious when searching for a particular axis of rotation h j orientation.
25)). In other words, ‘i is obtained as T i ; hi Þ ni ‘i ¼ ½ ‘ix ‘iy ‘iz 0 ¼ rotðm 2 miy mix ð1 À Chi Þ À miz Shi m2ix ð1 À Chi Þ þ Chi 6 6 mix miy ð1 À Chi Þ þ miz Shi m2iy ð1 À Chi Þ þ Chi ¼6 6 4 mix m iz ð1 À Chi Þ À miy Shi miy miz ð1 À Chi Þ þ mix Shi 0 0 2 3 nix 6n 7 6 iy 7 6 7: 4 niz 5 0 miz mix ð1 À Chi Þ þ miy Shi miz miy ð1 À Chi Þ À mix Shi m2iz ð1 À Chi Þ þ Chi 0 3 0 7 07 7 7 05 1 ð2:25Þ Further simpliﬁcation of Eq. 25) is possible by utilizing Eq. , i ; p þ 2hi Þ‘iÀ1 : ‘i ¼ rotðm ð2:27Þ According to the refraction law of optics, the refracted unit directional vector ‘i i through an can be obtained by rotating the active unit normal vector ni about m angle p À hi (Fig.