By Nikhat Parveen
A curve in the plane is said to be parameterized if the set of coordinates on the curve, (x, y), are represented as functions of a variable t, namely
x = f(t) y = g(t) where t R
where R is a set of real numbers. The variable t is called a parameter and the relations between x, y and t are called parametric equations. The set R is called the domain of f and g and it is the set of values t takes.
Let's explore the graph of the parametric equation given by
x = cos (t)
y = sin (t) for (0 ≤ t ≤ 2π)
One way to find the graph is to eliminate the parameter 't' by noting that
x2 + y2 = cos2t + sin2t = 1
Thus, the graph is contained in the unit circle x2 + y2 = 1. Geometrically, the parameter 't' can be interpreted as the angle swept out by the radial line from the origin to the point (x, y) = (cos t, sin t) on the unit circle as shown in the figure below.
(0 ≤ t ≤ 2π)
As t increases from 0 to 2π, the point traces the circle counterclockwise, starting at (1, 0) when t = 0 and completing one full revolution when t = 2π. One can obtain different parts of the circle by varying the interval over which the parameter varies. For example,
x = cos t, y = sin t for (0 ≤ t ≤ π)
represents just the upper semicircle as shown in the figure below:
(0 ≤ t ≤ π)
The direction in which the graph of a pair of parametric equations is traced as the parameter increases is called the direction of increasing parameter or sometimes the orientation is imposed on the curve by the equations. Thus, we make a distinction between a curve, which is a set of points, and a parametric curve, which is a curve with an orientation imposed on it by a set of parametric equations. For example, we saw in Example 1 that the circle represented parametrically by the figure shown above is traced counterclockwise as t increases and hence has counterclockwise orientation. The orientation of a parametric curve can be indicated by arrowheads.
To obtain parametric equations for the unit circle with clockwise orientation, we can replace t by −t in (1) and use the identities cos(−t) = cos t and sin(−t) = −sin t , which yields
x = cos t, y = −sint (0 ≤ t ≤ 2π)
Here, the circle is traced clockwise by a point
that starts at (1, 0) when t = 0 and completes
one full revolution when t = 2π.
[When working with the graphing calculator one cannot see the direction of orientation as it plots fast, so one might not be able to see the orientation of the graphs.]
Let's explore further by plotting the following parametric equation:
where 0 ≤ t ≤ 2π
We can explore the equation by setting different values for a and b. In particular, we can explore for when a = b, a < b, and a > b
Case i: a = b
a = b = 2 a = b = 20 a = b = -60 a = b = 100
click here for the animation file where 0 <a=b <200. Is the graph still circular when a = b = 200?
Here we see that the graph is the same for different values of a = b, except that it gets thicker ; however this graph can be further explained by considering the parameter 't' as time (measured as seconds, say) and the functions f and g as functions that describe the x and y position of an object as it moves in a plane. Then the parametric equations for example x = cos2t and y = sin2t for 0 ≤ t ≤ 2π describe an object moving twice around the unit circle. At t = 0, the object starts its journey, at t = π the object has made one revolution, and at t = 2π the object ends its journey. It takes the object π seconds to travel around the circle.
Also notice that the domain and range of the graphs remains fixed at [-1,1] for all values a = b.
Case ii: a < b and 0 ≤ t ≤ 2π
a = 2 b = 4 a = 3 b = 4 a = 1 b = 3 a = 4 b = 5
a = 2 b = 3 a = 5 b = 6 a = 6 b = 7 a = 10 b = 11
Click here for the animation when a = 2 (even) and b is odd and when a = 1 (odd) and b is even where 2 < b < 20
When a < b, we notice that the domain and range is equal and remains the same for all the graphs which is [-1 1].
We also notice that when 'a' is even and 'b' is even we have the graph of a closed loop. And the no. of loops is equal to the quotient when 'b' is divided by 'a', first I thought the no. of loops must be equal to the value of a but after noticing the graph for a = 1 and b = 3 with three loops I determine that the no. of loops must be equal to the quotient when (b/a)
When 'a' is even and 'b' is odd we see that the loops are open.
When 'a' is odd and 'b' is even we can see that the loops are closed.
Case iii: a > b and 0 ≤ t ≤ 2π
a = 2 b = 1 a = 3 b = 1 a = 4 b = 1 a = 5 b = 1
Looks like the graphs shifts between open and closed loop when b is constant and varies. Click here to see the animation when b = 1 and a varies between 2 and 10.
a = 4 b = 3 a = 5 b = 4 a = 6 b = 4 a = 7 b = 6
When a and b both changes the graphs look both open and close but loops are not simple any more they display many curvatures and gets complex with the increasing values of a and b. Click here to see the animation when a = 10 remains constant and b varies between b = 1 to b = 9
As can be seen, the domain and range for all the graphs when a > b lies between [-1, 1].
When 'a' is even and 'b' odd the loops are open.
When a and b both are odd then the graphs shows closed loops.
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The Parametric curves generated by our Parametric equations are related to parametric curves called Lissajous Curves.
Click the link below to visit the Lissajous Lab to view some of the amazing curves.