Differentiate synthetic curves based on its properties
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Curves can be described mathematically by nonparametric or parametric equations. Nonparametric equations can be explicit or implicit. For a nonparametric curve, the coordinates y and z of a point on the curve are expressed as two separate functions of the third coordinate x as the independent variable [see Equation (1)]. This curve representation is known as the nonparametric explicit form. If the coordinates x, y and z are related together by two functions [see Equation (2)], a nonparametric implicit form results.
(1)
(2)
There are three problems with describing curves using nonparametric equations:
(1) If the slope of a curve at a point is vertical or near vertical, its value becomes infinity or very large, a difficult condition to deal with both computationally and programming-wise. Other ill-defined mathematical conditions may result.
(2) Shapes of most engineering objects are intrinsically independent of any coordinate system. What determines the shape of an object is the relationship between its data points themselves and not between these points and some arbitrary coordinate system.
(3) If the curve is to be displayed as a series of point or straight-line segments, the computations involved could be extensive.
Parametric representation allows closed and multiple-valued functions to be easily defined and replaces the use of slopes with that of tangent vectors, as will be introduced shortly.
In parametric form, each point on a curve is expressed as a function of a parameter u. The parametric equation for a three-dimensional curve in space takes the following vector form:
(3)
Equation (3) implies that the coordinates of a point on the curve are the components of its position vector. It is a one-to-one mapping from the parametric space (Euclidean space E1 in u values) to the Cartesian space (E3 in x, y, z values), as shown in Figure 2.
Figure 2. Parametric representation of a three-dimensional curve
The tangent vector is defined as vector
(4)
The components of the tangent vector in the parametric space as
(5)
where x’(u), y’(u), and z’(u) are the first parametric derivatives (with respect to u) of the position vector components x’(u), y’(u), and z’(u) respectively. The slopes of the curve are given by the ratios of the components of the tangent vector:
(6)
2.2 Parametric representation of analytic curves
There are two categories of curves that can be represented parametrically: analytic curves and synthetic curves. Analytic curves are defined as those that can be described by analytic equations such as lines, circles, and conics. Synthetic curves are the ones that are described by a set of data points (control points) such as splines and Bezier curves.
Lines and circles are often expressed in analytic equations. They can also be expressed using parametric representation:
Lines
The parametric equation of the line becomes
(7)
In scalar form, this equation can be written as
(8)
Note that Equation (7), a line is expressed as a synthetic curve.
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