Vector Geometry

Euclidean space

Fig. 1 Euclid (fl 300 BCE) [de Ribera, C1630]

The field of computer graphics is almost soley concerned with the manipulation of objects in Euclidean space. Attributed to the Greek mathematician Euclid, Euclidean space is a representation of physical space where the position of a point in the space can be described by the signed distance along perpendicular real numbers lines called axes (singular: axis).

An \(n\)-dimensional Euclidean space is defined by \(n\) perpendicular real axes and is referred to as \(\mathbb{R}^n\). For example, consider the diagram of \(\mathbb{R}^3\) in Fig. 2. Here we have a representation of a 3-dimensional Euclidean space defined by the 3 axes labelled \(x\), \(y\) and \(z\). This representation uses the right-hand rule so-called because if we use the thumb on our right hand to represent the \(x\) axis, the index finger for the \(y\) axis and the middle finger for the \(z\) axis then holding out the right hand palm up with the thumb and index finger at right-angles and the middle finger pointing up then we have the axis configuration shown in Fig. 2. Doing similar with the left hand gives the left-hand rule where the \(x\) axis is pointing in the opposite direction than in the right-hand rule.

Fig. 2 Euclidean space axes using the right-hand rule.

Co-ordinate systems

Fig. 3 René Descartes (1596 - 1650) [Hals, C1700]

A co-ordinate system is a system that uses a sequence of numbers to determine the position of an object in Euclidean space. The numbers are known as co-ordinates and are represented in an ordered set of numbers enclosed in parenthesis known as a tuple. The most common example of a co-ordinate system is the Cartesian co-ordinate system, named after French mathematician and philosopher René Descartes), which uses perpendicular number lines to define points in the space.

Fig. 4 The co-ordinates of a point in \(\mathbb{R}^3\) are represented as the 3-tuple \((x, y, z)\).

Consider Fig. 4 where three number lines called axes (singular: axis) are labelled \(x\), \(y\) and \(z\). The position of a point in this space is defined by the co-ordinates \((x, y, z)\) where \(x\), \(y\) and \(z\) are the distances along the axes from zero. Since these values are real numbers a three-dimensional space is often denoted by \(\mathbb{R}^3\).