Representation of Spatial Curves

Surfaces are often depicted as a network of curves in the orthogonal cutting planes, with three-dimensional contours of parts. These sections are digitized physical model or design and selection of a mathematical curve that passes through all given points. We will discuss two such methods: cubic spline and parabolic interpolation.

Another approach consists in the fact that the mathematical description of curves is generated without the prior knowledge of the curve shape. Examples are Bezier curves and their generalization to B-splines. These methods differ in that the curve cannot pass through any given point. The control points determine only the direction of the bend.

Three-dimensional curves can be represented parametrically or non-parametrically. Explicit nonparametric representation has the form

(3.15)

Implicit nonparametric representation of the curve as the intersection of two surfaces is given by:

(3.16)

The general form of the parametric spatial curve can be written as:

(3.17)

In the above explicit non-parametric representation x can be regarded as a parameter, x = t. Then, this curve has the same parametric form

(3.18)

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