Parametric Curves

In parametric form of each coordinate point of the curve is shown as a function of one parameter. The parameter value specifies the coordinate vector of a point on the curve. For two-dimensional curve with parameter t coordinates of the point are:

(3.3)

Then the vector representation of the point on the curve is:

(3.4)

To obtain non-parametric form, it is necessary to exclude t from the two equations and take one in terms of x and y. The parametric form allows you to present closed and multi--valued curves. Derivative, i.e. tangent vector is:

(3.5)

The slope of the curve is:

(3.6)

Note that when x'(t) = 0 the slope is infinite. The parametric representation in this case has not the computational difficulties; it suffices to equate one component of the tangent vector to zero.

Since the point to the parametric curve is determined only by the parameter value, this form does not depend on the choice of coordinate system. Endpoints and the length of the curve define parameter range. It is often convenient to normalize the parameter of interest to a segment of the curve to 0 <= t <= 1. The independence of the parametric curve from the axes makes it easy to carry an affine transformation.

The simplest parametric representation is for a straight line. For two vectors P₁ and P₂ parametric form of a line segment between them is as follows:

(3.7)

Since P(t) is a vector, each component of it is a parametric representation x(t) and y(t) between P₁ and P₂:

(3.8)

In Fig.3.1 compares nonparametric and parametric representation of a circle for the first quadrant.

Fig.3.1. Nonparametric and Parametric Representation of a Circle for the First Quadrant

Non-parametric form:

(3.9)

is shown in Fig.3.1(a). Point on the arc corresponds to equal increments of x. In this case the arc consists of segments of varying length, and it produces very rough graphical representation of the circle. In addition, the calculation of the square root is a computationally expensive operation.

Standard parametric form of the unit circle is:

(3.10)

where the parameter θ is a geometric angle, measured counterclockwise from the positive half-axis. Fig.3.1(b) shows the arc was built on equal increments of the parameter in the range 0 <= θ <= π / 2. The points are located at equal distance along the circle and circle looks much better. The disadvantage of this representation is the complexity of calculating trigonometric functions.

Parametric representation of the curve is not unique, for example,

(3.11)

also represents the arc of the unit circle in the first quadrant. In Fig.3.1,c the result is shown for equal increments of t. It is better than explicit, but worse than the standard parametric representation. However, the latter equation is easier from the computational point of view and that is a compromise solution.

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