International Journal of Advances in Scientific Research and Engineering (ijasre) E-ISSN : 2454-8006 DOI: 10.31695/IJASRE.2021.33970 Volume 7, Issue 2 February - 2021 www.ijasre.net Page 1 Licensed Under Creative Commons Attribution CC BY-NC Coplanarity Condition for Photogrammetric Simultaneous and Self Calibration Block Adjustments Prof. Dr. Khalid L. A. El-Ashmawy Department of Civil Engineering College of Engineering and Islamic Architecture Umm Al-Qura University Makkah Saudi Arabia _______________________________________________________________________________________ ABSTRACT The well-known collinearity equations are widely employed for the determination of object space coordinates of points during the aerial and close-range photogrammetry applications. On the other hand, the coplanarity equation is applied for analytical relative orientation which is essential for sequential block adjustment procedure. This paper concentrates on deriving mathematical formulation based on coplanarity condition. Softwares utilizing the derived mathematical models have been developed and tested using mathematical photogrammetric data. The effects of random and lens distortion errors on simultaneous and self-calibration block adjustments using the derived mathematical models and collinearity equations have been studied. It was found that the derived mathematical models compensate for the lens distortion errors better than collinearity equations. Furthermore, the accuracy of the results of self-calibration block adjustment using the coplanarity equation is slightly better than those obtained by collinearity equations. Key Words: Collinearity Equations, Coplanarity Equation, Simultaneous Block Adjustments, Self Calibration Block Adjustments, Accuracy Analysis. ______________________________________________________________________________________________ 1. INTRODUCTION Analytical photogrammetry consists of mathematical modeling of the relationship between different systems ( e.g. photo and ground, photo and model, model and ground). To reach a solution, one may have to use a set of condition equations to establish first the relationship between the observed values and the unknown parameters. The condition equations most commonly used are: (i) Collinearity condition and (ii) coplanarity condition. Each condition equation has specific functions, scopes and limitations. The choice of the condition may, therefore, lead to different approaches to the solution of any specific problem. The condition of collinearity (Figure 1) is that an object point (P), its image point (p) and the perspective centre (O), must lie along the same line. Mathematically, this condition is expressed as [1]: 11 12 13 31 32 33 21 22 23 31 32 33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P O P O P O p P O P O P O P O P O P O p P O P O P O X X m Y Y m Z Z m x f X X m Y Y m Z Z m X X m Y Y m Z Z m y f X X m Y Y m Z Z m                              (1) where , p p x y are the corrected photo coordinates, , , P P P X Y Z are the object space coordinates of point P, , , O O O X Y Z are the object space coordinates of the perspective centre O, f is the calibrated focal length of the camera, and ij m (i=1,2,3; j=1,2,3) are the elements of the orientation matrix (M) of the photograph.