Astronomical Computing

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Matrix Method for Coodinates Transformation

Written by Toshimi Taki
Revised on February 29, 2004


Coordinates transformation is a basic part of astronomical calculation and spherical trigonometry has been long used for astronomical calculation in amateur astronomy. Spherical trigonometry equations can be a little bit difficult for amateurs to understand.

In the last two decades, development of personal computers has brought about a change in the way astronomical calculations are carried out. In my opinion, spherical trigonometry is not appropriate to astronomical calculation using personal computers. I recommend the matrix method for coordinates transformation, because of its simplicity and ease of generalization in writing computer programs.

In this monograph, I describe coordinates transformation using the matrix method. I also extend the method to some specific applications.

You will find the following applications with numerical examples.
(1) Transformation from equatorial coordinates to altaimuth coordinates
(2) Mounting fabrication errors
(3) Telescope pointing algorithm
(4) Polar axis misalignment determination
(5) Dome slit synchronization


Mr. Christopher J. R. Lord of Brayebrook Observatory in U.K. encouraged me to write this monograph. He provided valuable information about declination drift method for dertemination of polar axis misalignment. He also lead me to tackle dome slit synchronization equation.
His website is

Mr. Martin Cibulski sent valuable comments on exact solution for apparent telescope coordinate in section 5.3.4.


Following applications have been developed with the help of my monograph.

(1) "DomeSync" by John Oliver of University of Florida
This application is dome slit synchronization with equatorial telescope movement. You can find it at

(2) "PoleAlignMax" by Larry Weber and Steve Brady
This application is a polar alignment software designed for GoTo scopes with CCDs. You can find it at

Download Document Click here to download "Matrix Method for Coordinates Transformation (950KB) in pdf file".

Table of Contents

1. Introduction

2. References

3. Notations
3.1 Note
3.2 Symbols

4. Basic Equations of Coordinates Transformation in Matrix Method
4.1 Polar Coordinates and Rectangular Coordinates
4.2 Coordinate Transformation
4.2.1 New Coordinate System Rotated around Z-axis
4.2.2 New Coordinate System Rotated around X-axis
4.2.3 New Coordinate System Rotated around Y-axis
4.3 Obtaining Polar Coordinates from Direction Cosine
4.4 Notes on Approximation
4.4.1 Approximation of Trigonometric Functions
4.4.2 Approximation of Other Functions

5. Applications
5.1 Transformation from Equatorial Coordinates to Altazimuth Coordinates
5.1.1 Transformation Equations
5.1.2 Example Calculation
5.2 Angular Separation
5.2.1 Equations
5.2.2 Example Calculation
5.3 Compensation of Mounting Fabrication Errors
5.3.1 Telescope Coordinates
5.3.2 Fabrication Errors of Mount
5.3.3 Derivation of Equations
5.3.4 Apparent Telescope Coordinates without Approximation <--- Revised
5.3.5 Example Calculations Apparent Telescope Coordinates to True Telescope Coordinates True Telescope Coordinates to Apparent Telescope Coordinates
5.4 Equations for Pointing Telescope
5.4.1 Introduction
5.4.2 Transformation Matrix
5.4.3 Derivation of Transformation Matrix
5.4.4 Example Calculation
5.4.5 Comment on Accuracy of the Pointing Method <--- New
5.5 Polar Axis Misalignment Determination
5.5.1 Derivation of Equations Relationship between Coordinate Systems Basic Equations for Declination Drift Method Challis' Method Compensation of Atmospheric Refraction
5.5.2 Example Calculations Two Star Declination Drift Method with Atmospheric Refraction Neglected Challis' Method with Atmospheric Refraction Neglected Two Star Drift Method with Atmospheric Refraction Compensated
5.6 Dome Slit Synchronization
5.6.1 Object in First Quadrant
5.6.2 Object in Second Quadrant
5.6.3 Object in Third Quadrant
5.6.4 Object in Fourth Quadrant
5.6.5 Intersection

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