4.7 Parametrization of implicitly defined functions

In mathematical terms, curves can be represented in a variety of ways. Two common methods include the explicit representation, where y is a function of x (y=f(x)), and the parametric representation, which provides more flexibility.

In parametric representation, we describe the x and y coordinates as functions of a third variable, usually denoted as t. This t parameter can be thought of as a sort of "time" variable; as t changes, the values of x and y change as well, tracing out the curve in the process. The pair of functions x(t) and y(t) that arise from this method are referred to as the parametric equations of the curve.

Validating this parametric representation involves substituting x(t) and y(t) back into the original equation in place of x and y. If the equation remains valid for all values of t in its domain, then we can say the parametric representation is correct.

Moving into more specific scenarios, consider a function f that is a function of x. In this situation, the equation y=f(x) can be parametrized. Instead of y being directly dependent on x, we express both y and x as functions of t, such that (x(t),\, y(t)) = (t,\, f(t)). In other words, we set x=t, and for each value of t, we find the corresponding y value by substituting t into function f.

Moreover, if function f is invertible, its inverse can also be parametrized. The parametric equations for the inverse function become (x(t),\, y(t)) = (f(t), t). Here, we set y=t and determine the corresponding x value for each t by substituting t into the inverse function.

In the context of curves in a plane, they can be parametrically represented by a pair of functions x(t) and y(t) that satisfy the original equation for all values of t in the domain. Functions and their inverses can be parametrized as (t,\, f(t)) and (f(t),\, t). For conic sections like parabolas, ellipses, and hyperbolas, specific formulas can be used for their parametrization. Ellipses and hyperbolas, in particular, make use of trigonometric functions in their parametrization.

Examples

In the mathematical study of conic sections, which include parabolas, ellipses, and hyperbolas, we can utilize specific formulas to achieve parametric representations. This allows us to express the x and y coordinates in terms of a third variable, typically denoted as t. In the case of ellipses and hyperbolas, trigonometric functions play an integral role in this parametrization.

For a parabola, the method for parametrization is similar to any equation that can be solved for either x or y. The most common form of a parabolic equation is y = x^{2}. In this form, the parabola opens upward, and the vertex of the parabola is at the origin (0,\, 0). If we can rearrange the equation to solve for x, we can represent the curve parametrically as (x(t),\, y(t)) = (f(t),\, t). This means that we find a function f such that when we input t, it gives us the x-coordinate. And then, for every such t, the y-coordinate is simply t. In other words, we're making t play the role that yusually does, and x is calculated as a function of this t.

We can solve our parabolic equation for y, the parametrization would be (x(t),\, y(t)) = (t,\, f(t)). In this scenario, we're letting t play the role that x usually does, and y is calculated as a function of this t. That is, for every t, the x-coordinate is simply t, and the y-coordinate is found by plugging t into our function f.

In both cases, as t varies, the pair of functions (x(t),\, y(t)) define the points on the parabola, thus giving us a way to trace out the curve. The choice between these two approaches usually depends on the form of the parabolic equation we're given and which variable it's easier to solve for.

An ellipse, a closed curve in which the sum of the distances from two points (foci) to every point on the curve is constant, can be parametrized using the trigonometric functions \cos and \sin. The specific functions used are x(t) = h + a \cos t and y (t) = k + b \sin t, where a and b are the semi-major and semi-minor axes respectively, h and k are the center of the ellipse, and t ranges from 0 to 2\pi. As t varies, these equations trace out the ellipse in the plane. The semi-major axis is the longest diameter of the ellipse, while the semi-minor axis is the shortest. These lengths determine the overall size and shape of the ellipse.

A hyperbola, a curve where the difference of distances from any point on the curve to two fixed points (foci) is constant, can also be parametrized using trigonometric functions.

Parametrizing a hyperbola involves representing all points on the hyperbola in terms of a third variable, typically denoted as t. This is achieved by using trigonometric functions, specifically secant (\sec) and tangent (\tan), which have properties that align with the characteristics of a hyperbola.

For a hyperbola that opens left and right, we use the functions x(t) = h + a \sec t and y(t) = k + b \tan t. Here, h and k denote the coordinates of the center of the hyperbola, the midpoint between its two foci. The terms a and b represent distances from the center of the hyperbola to the vertices (the points where the hyperbola intersects its major axis) and to the conjugate axis respectively. This conjugate axis is a line segment perpendicular to the major axis that passes through the center of the hyperbola. For a hyperbola that opens up and down, the functions switch to x(t) = h + a \tan t and y(t) = k + b \sec t.

In both cases, the variable t varies from 0 to 2\pi. As t changes in this range, the secant and tangent functions generate the wide range of values needed to trace out the open, ever-expanding shape of the hyperbola in each direction.

Examples