Written by Toshimi Taki

Revised on February 29, 2004

Revised on February 29, 2004

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

His website is http://www.brayebrookobservatory.org/.

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

(1) "DomeSync" by John Oliver of University of Florida

This application is dome slit synchronization with equatorial telescope movement. You can find it at http://www.astro.ufl.edu/~oliver/DomeSync/.

(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 http://www.focusmax.org/.

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

4.4.1 Approximation of Trigonometric Functions

4.4.2 Approximation of Other Functions

5.1.1 Transformation Equations

5.1.2 Example Calculation

5.2.1 Equations

5.2.2 Example Calculation

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

5.3.5.1 Apparent Telescope Coordinates to True Telescope Coordinates

5.3.5.2 True Telescope Coordinates to Apparent Telescope Coordinates

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

5.5.1 Derivation of Equations

5.5.1.1 Relationship between Coordinate Systems

5.5.1.2 Basic Equations for Declination Drift Method

5.5.1.3 Challis' Method

5.5.1.4 Compensation of Atmospheric Refraction

5.5.2 Example Calculations

5.5.2.1 Two Star Declination Drift Method with Atmospheric Refraction Neglected

5.5.2.2 Challis' Method with Atmospheric Refraction Neglected

5.5.2.3 Two Star Drift Method with Atmospheric Refraction Compensated

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