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A parametric representation of a function expresses the functional relationship between several variables by means of auxiliary variable parameters.
In the case of two variables x and y, the expression F(x, y) = 0 can be geometrically interpreted as the equation of a plane curve. Any variable t that determines the position of a point (x, y) on the curve can be a parameter—for example, arc length measured positively or negatively from some point on the curve taken as the origin, or time for a specified motion of a point that describes the curve. The variables x and y are expressed as functions of this parameter:
x = Φ(t) y = Ψ(t)
These functions yield a parametric representation of the functional relationship between x and y, and equations (*) are said to be parametric equations of the corresponding curve. Thus, the equation x2 + y2 = 1 has a parametric representation x = cos t, y = sin t, where 0 ≤ t < 2π; these equations are called the parametric equations of the circle. The equation x2 — y2 = 1 can be represented parametrically by x = (1 + t2)/2t, y = (1 — t2)/2t, where t ≠ 0, or by x = cosec t, y = cot t, where — π < t < π and t ≠ 0; these equations are parametric equations of the hyperbola. If the parameter t can be chosen so that the functions (*) are rational, the curve is said to be unicursal; the hyperbola, for example, is a unicursal curve.
Parametric representations of space curves—that is, representations by equations of the form x = Φ (t),y = Ψ(t), z = χ(t)—are of particular importance. A line in space has the parametric presentation x = a + mt, y = b+ nt, z = c + pt. The parametric equations for a circular helix are x = a cos t, y = a sin t, z = ct.
In the case of three variables x, y, and z whose relationship is expressed by F(x, y, z) = 0 (one of the variables, for example z, may be considered an implicit function of the other two), the geometric figure is a surface. To determine the position of a point on the surface, we require two parameters u and v —for example, longitude and latitude on the surface of the globe. Thus, the parametric representation has the form x = Φ(u, v),y =Ψ(u, v), z = χ(u, v)- For example, the surface x2 + y2 = (z2 + l)2 has the parametric equations x = (u2 — 1) cos v, y= (u2 + 1) sin v, and z + u.
The chief advantages of parametric representations are: (1) such representations permit us to study implicit functions in cases where it is difficult to write the functions in explicit form without using parameters, and (2) through parametric representations multiple-valued functions can be expressed as single-valued functions. Parametric representations have been particularly intensively studied for analytic functions. The parametric representation of analytic functions by single-valued analytic functions is the subject of the theory of uniformization.