# Difference between revisions of "Math:Coordinate Systems"

In mathematics, points in space can be represented using several different methods.

## Coordinates in 2D

### Cartesian

The Cartesian coordinate system is the most common system used to represent points in space. Generally, it's represented like this:

A point in this coordinate system takes the form $\left ( x, y \right )$; x is the distance on the horizontal axis, and y is the distance on the vertical axis. For example, point $\left ( 2, 3 \right )$ on the Cartesian plane is 2 units to the right and 3 units upward:

Each quadrant of the Cartesian plane, separated by the axes, is numbered:

### Polar

Polar coordinates aren't used as often as Cartesian, but they can still be very useful, especially in games. Where Cartesian coordinates work with rectangles, polar coordinates work with circles:

Points in the polar system are in the form of $\left ( r, \theta \right )$. r is the distance (radius) from the origin, and θ (theta) is the counter-clockwise angle with 0 being straight right. There are two main units for measuring the angle: degrees, of which there are 360 in a circle; and radians, of which there are 2π. Note that r cannot be negative.

## Conversion Between Systems

### Cartesian and Polar

To convert Cartesian to polar, we first need the distance from what will be the origin. This is done with an extension of the Pythagorean theorem: $d = \sqrt{\Delta x^2 + \Delta y^2}$. Next, to get the angle, we use $\arctan \frac{\Delta y}{\Delta x}$. However, arc tangent can only give an angle between -90 and 90 degrees; most languages should have an atan2 function, but if not, you can still get it by adding 180° to the arc tangent result if Δx is negative.

Here is a C++ example demonstrating how one might implement this conversion routine:

PointPolar CartesianToPolar(Point2f point)
{
PointPolar nPoint;
if (point.x != 0.0f || point.y != 0.0f)
{
nPoint.magnitude = point.GetLength();
nPoint.direction = atan2(point.y, point.x);
}

return nPoint;
}

Converting from polar to Cartesian is somewhat more simple: $\left ( x, y \right ) = \left ( r * \cos \theta , r * \sin \theta \right )$.

Point2f PolarToCartesian(PointPolar point)
{
Point2f nPoint;
nPoint.x = cos(point.direction) * point.magnitude;
nPoint.y = -sin(point.direction) * point.magnitude;

return nPoint;
}

### Cartesian and Spherical

PointSpherical CartesianToSpherical(Point3f point)
{
PointSpherical nPoint;
nPoint.magnitude = point.GetLength();

if (point.x != 0.0f || point.y != 0.0f)
{
nPoint.direction = atan2(point.y, point.x);
}

if (point.z != 0.0f)
{
nPoint.pitch = asin(point.z/magnitude);
}

return nPoint;
}
Point3f SphericalToCartesian(PointSpherical point)
{
float zPlaneScalar = cos(point.pitch);

Point3f nPoint;
nPoint.z = sin(point.pitch) * point.magnitude;
nPoint.x = cos(point.direction) * point.magnitude * zPlaneScalar;
nPoint.y = -sin(point.direction) * point.magnitude * zPlaneScalar;

return nPoint;
}

### Cartesian and Cylindrical

Conversion between cartesian and cylindrical systems are exactly the same as polar-cartesian conversions; though the new z coordinate exists, it will not be changed through the conversion.