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.