ROTRO2EQ converts a 3x3 rotation matrix into the corresponding euler angles Inputs: M(1,3) a string of 3 characters from the set {'x','y','z'} e.g. "zxz" or "zyx" or, equivalently, a vector whose elements are 1, 2 or 3 RO(3,3) 3x3 rotation matrix Outputs: E(3,1) 3 euler angles in the range +-pi. A positive rotation is clockwise if looking along the axis away from the origin. The string M specifies the axes (fixed in space) about which the rotations of an object are performed. You cannot have the same axis in adjacent positions and so there are 12 possibilities. Common ones are "ZXZ" and "ZYX". If you want the axes to move with the object, you should reverse the ordering of both "m" and "e". There is some reduncancy in euler angle values: (i) If m(1)==m(3) then e=[a b c] and e=[a+-pi -b c+-pi] are equivalent. The output of this routine will always have b>=0; (ii) If m(1)~=m(3) then e=[a b c] and e=[a+-pi pi-b c+-pi] are equivalent. The output of this routine will always have |b|<=pi/2

- atan2sc ATAN2SC sin and cosine of atan(y/x) [S,C,R,T]=(Y,X)

- rectifyhomog RECTIFYHOMOG Apply rectifying homographies to an image set

0001 function e=rotro2eu(m,ro) 0002 %ROTRO2EQ converts a 3x3 rotation matrix into the corresponding euler angles 0003 % Inputs: 0004 % 0005 % M(1,3) a string of 3 characters from the set {'x','y','z'} e.g. "zxz" or "zyx" 0006 % or, equivalently, a vector whose elements are 1, 2 or 3 0007 % RO(3,3) 3x3 rotation matrix 0008 % 0009 % Outputs: 0010 % 0011 % E(3,1) 3 euler angles in the range +-pi. A positive rotation is clockwise 0012 % if looking along the axis away from the origin. 0013 % 0014 % The string M specifies the axes (fixed in space) about which the rotations of 0015 % an object are performed. You cannot have the same axis in adjacent positions 0016 % and so there are 12 possibilities. Common ones are "ZXZ" and "ZYX". 0017 % If you want the axes to move with the object, you should reverse the 0018 % ordering of both "m" and "e". 0019 % 0020 % There is some reduncancy in euler angle values: 0021 % (i) If m(1)==m(3) then e=[a b c] and e=[a+-pi -b c+-pi] are equivalent. 0022 % The output of this routine will always have b>=0; 0023 % (ii) If m(1)~=m(3) then e=[a b c] and e=[a+-pi pi-b c+-pi] are equivalent. 0024 % The output of this routine will always have |b|<=pi/2 0025 0026 % 0027 % Copyright (C) Mike Brookes 2007 0028 % Version: $Id: rotro2eu.m 2187 2012-07-20 13:45:35Z dmb $ 0029 % 0030 % VOICEBOX is a MATLAB toolbox for speech processing. 0031 % Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html 0032 % 0033 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0034 % This program is free software; you can redistribute it and/or modify 0035 % it under the terms of the GNU General Public License as published by 0036 % the Free Software Foundation; either version 2 of the License, or 0037 % (at your option) any later version. 0038 % 0039 % This program is distributed in the hope that it will be useful, 0040 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0041 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0042 % GNU General Public License for more details. 0043 % 0044 % You can obtain a copy of the GNU General Public License from 0045 % http://www.gnu.org/copyleft/gpl.html or by writing to 0046 % Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA. 0047 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0048 0049 e=zeros(3,1); 0050 if ischar(m) 0051 m=lower(m)-'w'; 0052 end 0053 if numel(m)~=3 | any(abs(m-2)>1), error('Euler axis must be an x,y or z triplet'); end 0054 u=m(1); 0055 v=m(2); 0056 w=m(3); 0057 if sum(m==v)>1, error('Consecutive Euler axes must differ'); end 0058 % first we rotate around w to null element (v,u) with respect to element (!vw,u) of rotation matrix 0059 g=2*mod(u-v,3)-3; % +1 if v follows u or -1 if u follows v 0060 h=2*mod(v-w,3)-3; % +1 if w follows v or -1 if v follows w 0061 [s,c,r,e(3)]=atan2sc(h*ro(v,u),ro(6-v-w,u)); 0062 r2=ro; 0063 ix=1+mod(w+(0:1),3); 0064 r2(ix,:)=[c s; -s c]*ro(ix,:); 0065 % next we rotate around v to null element (!uv,u) with repect to element (u,u) of rotation matrix 0066 e(2)=atan2(-g*r2(6-u-v,u),r2(u,u)); 0067 % finally we rotate around u to null element (v,!uv) with respect to element (!uv,!uv) = element (v,v) 0068 e(1)=atan2(-g*r2(v,6-u-v),r2(v,v)); 0069 if (u==w && e(2)<0) || (u~=w && abs(e(2))>pi/2) % remove redundancy 0070 mk=u~=w; 0071 e(2)=(2*mk-1)*e(2); 0072 e=e-((2*(e>0)-1) .* [1; mk; 1])*pi; 0073 end 0074

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