Curved lines and curved surfaces were originally approximated using straight line segments. This will work visually if left to degenerate to the pixel level but it comes at a high storage price. Splines are now employed, usually of the cubic variety, to approximate curves. The essential parameters are the endpoints, called knots, and the a,b,c,d coefficients of the cubic equation. The tangent slopes at the knots are also required, as well as a weighting coefficient to ensure smooth transition from one cubic segment to the next.

Curved surfaces are usually represented by either polygon mesh or by bicubic patches. Polygon meshes need to adhere to these constraints to be effective:

• Efficient storage
• Ease of calculation
• Deal with shared edges and shared vertices
• Closure

Bicubic patches are usually surfaces bound by three or four cubic curve segments and provide more accurate rendering. Although they are slightly more complicated entities than polygons, there are far fewer required to cover the same surface. This is the current method of choice for surface rendering if there are no legacy issues to deal with.

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