On-Line Geometric Modeling Notes

In most applications in mathematics and engineering we work in the Cartesian coordinate system. This coordinate system can take two forms - the right-handed system, and the left-handed system. These two systems are shown in the following figure.

\includegraphics {figures/cartesian-coordinate-system}

In most applications, we use the right-handed system, and will continue to use this system in our notes.

In the computer graphics community, we also use the right-handed system, although it is drawn differently. We began drawing on two-dimensional screens - representing only $ x$ and $ y$ - and when it became necessary to represent depth, a $ z$ coordinate was added perpendicular to $ x$ and $ y$. This gives the following representation.

\includegraphics {figures/computer-graphics-coordinate-system}

Note that there are both left- and right-handed coordinate systems, depending on orientation of the $ z$-axis. These coordinate systems are identical to the Cartesian systems presented above, just oriented differently.

We also make extensive use of frames in computer graphics, which also represent coordinate systems. In three-dimensional space, frames consist of an origin and three linearly-independent vectors. These are used to describe local coordinate systems in order to define a model or the position and orientation of a camera. These frame-based coordinate systems may be left- or right-handed. A frame $ {\cal F} =( {\vec u} , {\vec v} , {\vec w} , {\bf O} )$ is illustrated in the figure below.

\includegraphics {figures/frame-example}

Using frames gives us some independence from fixed coordinates systems. In computer graphics, we will develop many frames of reference and freely convert the coordinates of a point between frames.

See the notes on Vector Spaces for additional information on linear independence and bases.

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Ken Joy