**Herong's Notes on Physics** - Version 3.21, by Dr. Herong Yang

What Is Coordinate System

This section provides an introduction of coordinate system, which is a mathematical model associated with a frame of reference that converts every location in the frame into set of numeric value or values.

**What Is a Coordinate System?**
A coordinate system is a mathematical model associated with a frame of reference
that converts every location in the frame into a set of numeric value or values.

For example, to describe the location of the mercury surface level in a thermometer, we can use a frame of reference using 2 specific mercury surface levels:

- Freezing Point Level - The mercury surface level when the thermometer is placed in icy water.
- Boiling Point Level - The mercury surface level when the thermometer is placed in boiling water.

Then we can create a coordinate system for this frame of reference as below:

- Define that the temperature is 0 degree when the mercury surface reaches the Freezing Point Level.
- Define that the temperature is 100 degrees when the mercury surface reaches the Boiling Point Level.
- Divide the distance between the Freezing Point Level and the Boiling Point Level into 100 equal scales.
- Define that the temperature is 1 degree higher each time the mercury surface climbs up 1 scale.

Now we can describe any location of the mercury surface level using a numeric number. For example, if you put the thermometer in air conditioned room, the mercury surface level may stays at the 20th scale location, this location can be described now as 20 degrees.

You can create multiple coordinate systems for a single frame of reference. In that case, the same location will be described with different numbers in different coordinate systems.

For example, there are 3 coordinate systems exist today to describe a temperature: the Celsius system, the Fahrenheit system and the Kelvin system. The freezing point is described as 3 different numbers:

- The freezing point is 0 degree in the Celsius system.
- The freezing point is 32 degree in the Fahrenheit system.
- The freezing point is 273.15 degree in the Kelvin system.

*Last update: 2014.*

Table of Contents

►Introducion of Frame of Reference

Frame of Reference with 2 Objects

2-Dimensional Cartesian Coordinate System

3-Dimensional Cartesian Coordinate System

1 Frame of Reference with 2 Coordinate Systems

Introduction of Special Relativity

Time Dilation in Special Relativity

Length Contraction in Special Relativity

The Relativity of Simultaneity