Math

Cartesian coordinate system

Axis is a straight line with a given direction.

Three mutually perpendicular axes passing through point O form a rectangular coordinate system with origin at point O.

These axes are called coordinate axes:

  • x-axis aka abscissa usually represents horizontal coordinates
  • y-axis aka ordinate usually represents vertical coordinates
  • z-axis aka applicate usually represents depth coordinates

Cartesian coordinate system is an orthogonal coordinate system, since its coordinate axis are perpendicular.

A vector v that has its start at the origin and ends at some point P is called the radius vector of this point.

Let's take some segment as a unit of scale and put it on three axes in the positive direction, as a result we get three vectors: i, j, k.

Then any point in 3-dimension space is uniquely determined by their three cartesian coordinates (x, y, z), where x, y, z specify the signed distance from the origin along the x, y, and z-axes, respectively. In other words x, y, z are algebraic projections of a radius vector on the coordinate axes.

In 3D graphics, a point is also called a vertex.

point in the cartesian coordinate system

coordinate system orientation

There are two types of the coordinate system orientation.

For the right-handed coordinate system the right thumb points along the Z axis in the positive direction and the curl of the fingers represents a motion from the first or X axis to the second or Y axis. When viewed from the top or Z axis the system is counter-clockwise. For example, the default coordinate system in OpenGL is right-handed.

For the left-handed coordinate system the left thumb points along the Z axis in the positive direction and the curled fingers of the left hand represent a motion from the first or X axis to the second or Y axis. When viewed from the top or Z axis the system is clockwise. For example, the default coordinate system in the RenderMan interface is left-handed.