Parametric functions

Before elaborating the topic, let's first examine a case. Suppose there's a snail moving on the ground. We need to determine the trajectory of its motion mathematically. So a rectangular coordinates is established in order to describe the positions of the snail. There are two different viewports for us to view the trajectory of the motion. First, we can find an expression in terms of x and y that could represent the path of its motion. And then the trajectory of its motion could be determined by plotting the curve of all positions based on the expression. The question is the static viewport can only describe the path of its motion in a given domain, but not dynamically in all time. Moreover, the path must not be able to be expressed in a function in the rectangular coordinates, which makes it difficult to plot the non-function curve. How to address these issues?

In kinematics, a parameter t is introduced to represent the time, such that a set of functions of x and y in terms of t are established to trace out a path along the curve. So the trajectory of the motion could be determined dynamically with varying time. x and y are defined as functions of t, a group of unique output could be obtained which will transfer non-function to normal way of plotting. These functions are called parametric equations.

What are Parametric Equations

In mathematics, parametric equations are a set of quantities expressed as functions of a number of independent variables, known as parameters. In the above the case, instead of defining function y in terms of x. The quantities x, y could be defined as functions of an independent variable t such that x and y trace out the trajectory of the curve with varying t.

If x and y are continuous functions of t in the same domain, then the equations

x = f(t)

y = g(t)

are called parametric equations and t is called the parameter. The process of finding parametric equations of implicit equation that could represent a curve, a surface, etc. is called parametrization. The inverse process of parametrization is called implicitization. The graph of parametric equations is called a parametric curve or plane curve.

In Cartesian coordinate system, parametric equations provide a convenient way for describing curves that are not necessarily functions. Parametric equations that have a two-dimensional input and a three-dimensional output is a plane curve or a surface in three-dimensional space.

Examples of parametric equations

Line

A line is an one-dimensional object, but may exist in two, three, or higher dimension spaces. If it exists in more than two dimensions, it is more convenient to represent the line using parametric equations since it cannot be described by a single linear equation.

x = x_0 + pt

y = y_0 + qt

z = z_0 + rt

where (x_0,y_0,z_0) is any point on the line and (p,q,r ) is the direction vector that is parallel to the line.

Implicitization of the parametric equations for the line gives its implicit equations,

\dfrac{x-x_0}{p} = \dfrac{y-y_0}{q}=\dfrac{z-z_0}{r}

Circle

The equation of a circle in Cartesian coordinates can be given as

x^2+y^2 = r^2

The parametric equations could be given as

x = r\cos \theta

y = r\sin \theta

where \theta is the parameter, which represents the angle formed by the radius with positive x axis.

Ellipse

The curve of an ellipse could also be expressed in parametric form analogous to that of a circle.

\begin{aligned} x&= a\cos t \\ y&=b\sin t \end{aligned}

(12)

The parameter t is not the angle formed by the line joining the point P and center of the ellipse with the axis x, but determined geometrically by point A on the larger circle in the figure. A line segment joining point A on a circle with radius a has intersection with the circle with radius b at point B. A line that passes point B and is parallel to major axis has intersection with the ellipse at point P. Then Point P could be represented in parametric form using the trigonometric functions of the angle t formed by AB with major axis.

Parabola

Let's consider the general form of a parabola,

y^2 = 4ax

On the left hand side y^2 is a square. In order to produce a square on the right hand side, let x be a function of t such that

x = at^2

Substitute parametric function for x into the general form equation, then

y^2 = 4a\cdot at^2 = 4a^2t^2

Taking square root for both side yields

y = 2at

Therefore, the parametric form equations for the parabola is

x = at^2

y = 2at

Hyperbola

When a hyperbola is centered at the origin, the general form of equation for the hyperbola is

\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2} = 1

Using trigonometric idendity

\sec^2 \alpha -\tan^2\alpha = 1

We can convert the general equation to parametric ones.

Let

x = a\sec t

y = b\tan t

Substituting x, y into the general form equation shows the LHS is equal to 1. And the parametric equations for a hyperbola could be a pair of trigonometric functions. However, a question arises - is it possible to represent the parameter t with physical meaning?

Derivation of parametric equations for a hyperbola geometrically

In the figure are a hyperbola and its auxiliary circle centered at origin. P is an arbitrary point on the hyperbola with its perpendicular point to axis of symmetry at point A. AB is the tangent to the circle with tangent point at B, then

BC = a

Let t be the angle formed by radius BC with the positive x axis. Since \triangle ABC is a right triangle,

\cos t = \dfrac{BC}{AC} =\dfrac{a}{AC}

AC = a\cdot \dfrac{1}{\cos t} =a\sec t

which is the abscissa of the point P. Now we get parametric function for x,

x = a\sec t

Next we substitute x into general equation to solve for y.

\dfrac{(a\sec t)^2}{a^2}-\dfrac{y^2}{b^2} = 1

\sec^2 t-\dfrac{y^2}{b^2} = 1

Apply the trigonometric identity

\sec^2 \alpha -1 = \tan^2\alpha

Then,

y^2 = b^2 \tan^2 t

Taking square root gives the parametric function for ordinate of point P,

y = b\tan t

Now we have derived the parametric equations for a hyperbola. The parameter t is an angle formed by the radius of the auxiliary circle of the hyperbola width its value dependent on point P.

Converting the trigonometric functions to hyperbolic ones, gives hyperbolic version of parametric equations for a hyperbola.

x = a\cosh t

y = b \sinh t

Summary

Parametric equations provide a convenient way for describing curves that are not necessarily functions in rectangular coordinates.
There could be more than one way to parameterize a curve.
By removing the parameters, it's always possible to convert parametric equations back to implicit functions in rectangular coordinates.
Examples of parametric equations are given for popular geometric curves

Collected in the board: Conic sections

Steven Zheng posted 5 months ago