Cartesian Coordinate transformation (3D and 2D)

The function performs both reverse and direct problem-solving based on dual quaternions between two Cartesian coordinate systems (3D and 2D)

%

% DQA_3d_transformation (Asymmetric similarity) with dual quaternions

% using scaled quaternions

% reverse and direct problem

% Model: xyz2=(2*w_r'*s)+(w_r'*Q_r*xyz1)

% function performs both reverse and direct problem-solving based on dual quaternions between two 3D Cartesian coordinate system

% The function can also 2D transformation between given coordinate systems.

% INPUT:

% n :number of control points

% xyz1 = nx3 start system coordinate matrix ([x1 y1 z1])[in meter]

% xyz1 contain [ n control points xyz1]

% | x1 y1 z1 |

% xyz1=| x2 y2 z2 |

% | ........ |

% | xn yn zn |

% xyz2 = nx3 target system coordinate matrix ([x2 y2 z2])[in meter]

% xyz2 contain [ n control points xyz2 ]

% | X1 Y1 Z1 |

% xyz2=| X2 Y2 Z2 |

% | ........ |

% | Xn Yn Zn |

% if there are new point to be transformed (revers problem+direct problem)

% xyz1 = mx3 start system coordinate matrix

% the new points are added under the control points in the xyz1 matrix.

% m :number of total points (control points+new points to be transformed)

% | x1 y1 z1 |==>control points

% | x2 y2 z2 |==>control points

% | .............. |==>control points

% xyz1=| xn yn zn |==>control points

% | xn+1 yn+1 zn+1 |==>new points

% | .............. |==>new points

% | xm ym zm |==>new points

% (m-n)= new points to be transformed (if any) ]

% FOR 2D TRANSFORMATIONS

% All of Z1 and Z2 coordinates must be entered as zero.

% P = nx1 weighted of points

% m-n : number of new points to be transformed

% m must be greater than or equal n (m>=n)

% if m=n there is no new point to be transformed (revers problem)

% if m>n there is new point to be transformed (revers problem+direct problem)

% OUTPUT:

% q =[2 x 8] matrix dual quaternion elements [r0 r1 r2 r3 s0 s1 s2 s3] [unitless]

% first row contained scaled quaternions

% second row contained unit quaternions

% L = scale factor [unitless]

% T = translation vector (Tx,Ty,Tz)[in meter]

% angles = (e,p,w) rotation angles (x,y,z) [in radian]

% res = residual of coordinates [in meter]

% so = standard error of transformation [in meter]

% Cqq = covariance matrix of scaled quaternions

% (r0,r1,r2,r3,s0,s1,s2,s3)respectively

% Cpar = covariance matrix of transformation parameters

% (L,e,p,w,tx,ty,tz,0)respectively

%

% xyz_new=(m-n)x3 matrix is the coordinates of the new points calculated in the target system. (if m>n)

% xyz_new=[] (if m=n)

%==========================================================================

%==========================================================================

% SAMPLE INPUT DATA For 3D Transformations

% n=4 number of control point

% m=5 number of row of xyz1

% m-n=1 number of new point to be converted (if any)

% xyz1=[ 6432.5800 7254.1200 200.6000

% 6354.3700 5724.5800 174.5700

% 7221.4400 6355.0800 254.5800

% 6433.5800 7255.1200 201.6000

% 6533.5800 7055.1200 211.6000 ] this row belong to new points to be converted.

% xyz2=[ 4208.8321 2111.9343 4182.9434

% 2034.5929 2073.9091 4924.8221

% 3397.0341 1919.6811 5773.1190

% 4207.7300 2110.9400 4182.1100 ]

%

% P=[1 2 3 4] weight

%======================================

% SAMPLE INPUT DATA For 2D Transformations

% n=4 number of control point

% m=5 number of row of xyz1

% m-n=1 number of new point to be converted (if any)

% xyz1=[ 9043.7400 5208.7900 0

% 9218.4200 4833.4900 0

% 9000.0000 5000.0000 0

% 9220.0200 5166.9100 0

% 9242.7000 5039.3800 0

% 9106.1700 5050.7100 0] this row belong to new points to be % converted.

% xyz2= [ 4618.7200 4068.8300 0

% 5579.4100 1115.6000 0

% 4103.9800 2553.3800 0

% 5893.3800 3597.0300 0

% 5946.7000 2626.7000 0 ]

% P=[1 1 1 1] weight

% ======================================================

% Please refer to:

%

% Bektas, S.(2023) An expanded dual quaternion algorithm for 3D Helmert %transformation and determination of the VCV matrix of the transformation’s %parameters. Journal of Spatial Science, DOI: 10.1080/14498596.2023.2274997

% Bektas, S.(2017) Adjustment Calculus,Ondokuz Mayis University press,

% ISBN 975-7636-54-1 Samsun,Turkey

%

%====================================================================

% written by Sebahattin BEKTAS January 2023

% [e-mail]