Points, Vectors, and Coordinate Systems

Farhad Kamangar 2017

Coordinate System

In computer graphics a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point in 2 or 3-dimensional Euclidean space.

2-dimensional space:

Points:

a point, $p$, in 2-dimensional space is represented tuples of numbers: $P = \left[ {\matrix{ x \cr y \cr } } \right]$

Vectors:

a vector, $v$, in 2-dimensional space may represent direction, normal to a surface, force, etc, and is also represented by tuples of numbers: $\overrightarrow V = \left[ {\matrix{ x \cr y \cr } } \right]$

3-dimensional space:

The Cartesian coordinate systems in 3-dimensional space are either right-handed or left-handed

Points:

a point, $p$, in 3-dimensional space represents a location and is presented by tuples of numbers such as: $p=\left[ {\matrix{ {{x}} \cr {{y}} \cr {{z}} \cr } } \right]$

Vectors:

a vector, $v$, in 3-dimensional space may represent direction, normal to a surface, force, etc, and is also represented by tuples of numbers such as: $\overrightarrow v=\left[ {\matrix{ {{x}} \cr {{y}} \cr {{z}} \cr } } \right]$

Notes:

  • Since the vectors and points have the same representation, it is important to make a distinction between the concept of vector and point.
  • Adding a vector to another vector creates a new vector.
  • It is meaningless to add two points.
  • Adding a vector to a point creates another point.
  • Subtracting a point from another point creates a vector.