Parametric Equations

Parametric equations define relations as sets of equations. An image on a graph is said to be parametrized if the set of coordinates (x,y) on the image are represented as functions of a variable, usually t (parametric equations are usually used to represent the motion of an object at any given time t). From one input, we can find both coordinate points with our parametric equations.

Any equation can be parametrized and represented as a set of parametric equations. Usually, we parametrize using the following

where t is the set of real numbers. The variable t is called the parameter and the realtionship between the variables x, y, and t are called parametric equations. Conversely, given a pair of parametric equations, the set of points (f(t), g(t)) form a curve on the graph. Instead of worrying about two input variables (x and y), we have reduced the function to one input variable.

(1) Consider the quadratic equation

Parametrizing the curve, we would get the parametric equations

If this were a body in motion and we wanted to find the position at 3 seconds, we could plug in t = 3 and obtain our coordinate.

Equations can be parametrized in different ways. Taking our last example, we could use the following parametrizations

We should note that for different parametric equations of the same function, the (x,y) coordinate will vary, however, the graph will be exactly the same. Here is a graph of the parabola with the four pairs of parametric equations at t = 1.

The orientation of a parametrized curve is determined by the increasing values of the parameter. Sometimes the orientation is denoted by arrows drawn in the direction of the curve.

Finding the Original Function of Parametric Equations

It is beneficial to see how to find the original function given parametric equations to understand the connection.

(2) Let's start with the parametrized curves

Find a function y = f(x) whose graph gives the parametric equations.

Let's begin by solving x = 3t+2 for t.

Then, we plug this into the second equation given for y, which gives us

This is a quadratic equation which forms an upward opening parabola with vertex (2,0). We are not done yet, we cannot forget our domain.

Since the domain for our parameter is 0 ≤ t ≤ 5, we get a new inequality for the domain for x.

Lines and Segments

As we have seen, there are many ways to parametrize curves. For lines and segments, the most common way to parametrize a line segment L between points (a,b) and (c,d) is

(3) The line segment between the points (2,-5) and (-3, 4) would be parametrized as

At t = 0 we get our first point and at t = 1 we get our second point

If we want to parametrize a whole line, we do the same thing except let t go from negative to positive infinity


Circles and Ellipses

We can describe the motion of an object around a circle using parametric equations involving trigonometric equations. Circles and ellipses are parametrized using a pythagorean trig identity. We substitute x(t) for x, y(t) for y, and remember that cos2x + sin2x = 1. Recall that the unit circle can be written as x2 + y2 = 1. so we can parametrize the unit circle as {cos(t),sin(t)} with t going from [0,2Π]

(4) If we want to parametrize a circle centered at (-3,2) with radius 4, we can parametrize the circle as follows

This is the same idea with ellipses. To parametrize the ellipse

We would use

In general, the location of an object at time t depends on a number of things.

The object location at time t is given by:

This method of parametrization uses polar coordinates, which uses a different graphing system used mostly for circles and more complex curves.

Miscellaneous Curves

If we have curves that are piecewise functions or shapes, we can parametrize each piece seperately and then shift the parametrizations so each piece runs consecutively and there are no breaks.

(5) Given this image of a square formed by the following coordinates oriented clockwise

If we want to travel around a side per second, it would take 4 seconds. Our parameters are then 0 ≤t ≤ 4.

Since this is a piecewise function and each of our pieces are lines, we can use the formula for parametrizing lines and break it into four pairs of equations.

Our formula works only for the segment on the interval t:[0,1], so each segment is compensating to satisfy the formula.

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