Coordinate systems

coordinate system is a system that uses one or more numbers, called coordinates, to uniquely determine the position of a point or other geometric elements on 1D, 2D, and 3D dimensions.

Each of these numbers indicates the distance between the point and some fixed reference point, called the origin. The first number, known as the \(x\) value, indicates how far left or right the point is from the origin. The second number, known as the \(y\) value, indicates how far above or below the point is from the origin. The origin has a coordinate of \((0, 0)\).

Longitude and latitude are a special kind of coordinate system, called a spherical coordinate system since they identify points on a sphere or globe. However, there are hundreds of other coordinate systems used in different places around the world to identify locations on the earth. All of these coordinate systems place a grid of vertical and horizontal lines over a flat map of a portion of the earth.

A complete definition of a coordinate system requires the following:

  • the projection in 1, 2 or 3 dimensions;
  • the location of the origin;
  • the units that are used to measure the distance from the origin.

Common coordinate systems

  • Cartesian coordinate system
  • Polar coordinate system
  • Cylindrical coordinate systems
  • Spherical coordinate systems

Cartesian coordinate system

The term “cartesian coordinates” (also called rectangular coordinates) is used to specify the location of a point in the plane (two-dimensional), or in three-dimensional space.

In such a coordinate system you can calculate the distance between two points and perform operations like axis rotations without altering this value. The distance between any two points in rectangular coordinates can be found from the distance relationship.

The most common coordinate system for representing positions in space is one based on three perpendicular spatial axes generally designated \(x\), \(y\), and \(z\). The three axes intersect at the point called the origin \(O=(0,0,0)\).

Any point \(P\) may be represented by three signed numbers, usually written \((x, y, z)\) where the coordinate is the perpendicular distance from the plane formed by the other two axes.

Often positions are specified by a position vector \(\vec{r}\) which can be expressed in terms of the coordinate values and associated unit vectors:

\[\vec{r}=x\vec{i}+y\vec{j}+z\vec{k}\]

With above definitions of the positive x, y, and z-axis, the resulting coordinate system is called right-handed; if you curl the fingers of your right hand from the positive x-axis to the positive y-axis, the thumb of your right-hand points in the direction of the positive z-axis. Switching the locations of the positive x-axis and positive y-axis creates a left-handed coordinate system. The right-handed and left-handed coordinate systems represent two equally valid mathematical universes. The problem is that switching universes will change the sign on some formulas.

In addition to the three coordinate axes, we often refer to three coordinate planes. The xy-plane is the horizontal plane spanned by the x and y-axes. It is identical to the two-dimensional coordinate plane and contains the floor in the room analogy. Similarly, the xz-plane is the vertical plane spanned by the x and z-axes and contains the left wall in the room analogy. Lastly, the yz-plane is the vertical plane spanned by the y and the z-axis and contains the right wall in the room analogy.

Polar coordinate system

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

Where \(P=(r,\theta)\). The reference point (analogous to the origin of a cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinatepolar angle, or azimuth.

The polar coordinates \(r\) and \(\theta\) can be converted to the cartesian coordinates \(x\) and \(y\) by using the trigonometric functions sine and cosine:

\(x=\sin\theta\)

\(y=\cos\theta\)

The cartesian coordinates \(x\) and \(y\) can be converted to polar coordinates \(r\) and \(\theta\) with \(r\geq 0\) and \(\theta\) in the interval \((-\pi, \pi ]\) by: \(r=\sqrt{x^2+y^2}\)

Cylindrical coordinate systems

Spherical coordinate systems

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