So far we have we have not yet been able to describe geometric shapes different from a circle, triangle or a line. The theory of parametric equations allows us to do describe shapes of various kinds. We restrict our attention to parametric curves.
A parametric curve #\orange{C}# is a figure in the plane which is described by two equations
\[\orange C \colon \phantom{x}\begin{cases}\blue{x}&=\blue{x(t)}\\ \green{y}&=\green{y(t)} \end{cases}\]
where #t# varies over a specified #\text{interval}#. These equations are called parametric equations. The curve #\orange{C}# consists of all points #\ivcc{\blue{x(t)}}{\green{y(t)}}# for #t# in the interval.
Example
The parametric equations \[\begin{array}{rcl}\blue{x(t)}&=&\blue{t^2}\\\green{y(t)}&=&\green{2-t},\end{array}\]with #t# in the interval #\ivcc{-4}{2}# define a parametric curve.
Example
In the figure you see the curve in the example defined by the equations \[\begin{array}{rcl}\blue{x(t)}&=&\blue{t^2}\\\green{y(t)}&=&\green{2-t},\end{array}\]
on the interval #[-4,2]#.
Two parametric curves with the same parametric equations can look very different if the intervals are different. In the example, the curve #\orange{C}# was defined by \[\begin{array}{rcl}\blue{x(t)}&=&\blue{t^2}\\\green{y(t)}&=&\green{2-t},\end{array}\] with #t# ranging from #-4# to #2#.
When we change the interval to #t# ranging from #0# to #6#, we get the following figure:
One can also define a parametric equation with more variables and using more equations. For example, the parametric equation given by a scaled version of \[\begin{array}{rcl}x(t)&=&\cos(2t)+t\\y(t)&=&t+t^2-4\\z(t)&=&\cos(2t) + \sin(3t) +0.2t,\end{array}\]where #t# ranges over the interval #\ivcc{-10}{10}# defines a curve in three dimensional space.
We will not study these kinds of curves in this course.
Orbit
Such parametric curves can be used to describe the orbit of a certain point #\orange P#. If such an orbit is described by parametrical equations \[\begin{cases}\blue x &= \blue{x(t)},\\ \green y & = \green{y(t)}. \end{cases}\] then one often writes #\orange{P_{t}}# for the 'location' of #\orange P# at time #t# - i.e. \[\orange{P_{t_0}}=\ivcc{\blue{x(t_0)}}{\green{y(t_0)}}\]
where #t_0# is a number in the interval. In the figure is an example where \[\ivcc{ \blue{x(t)}}{ \green{y(t)}} = \ivcc{ \blue{\sin(t) + \cos(t)}}{ \green{\cos(t)}}\]
On a parametric curve there is a notion of direction. Informally the direction of the curve is determined by the direction the curve has when #t# increases. One can reverse the direction of a parametric curve by "moving in the other direction". For example a parametric curve on an interval #[a,b]# \[\begin{cases}\blue{x}&=\blue{x(t)}\\ \green{y}&=\green{y(t)} \end{cases}\] can be reversed with the following curve: \[\begin{cases}\blue{x}&=\blue{x(-t)}\\ \green{y}&=\green{y(-t)} \end{cases}\]
on the interval #[-b, -a]#.
These notions can be made precise using the derivative. We will do so later.
The intersections of the curve with the axes of the #x,y#-plane can be computed. To compute the intersection of the curve with the #y#-axis, one should solve #\blue{x(t)}=0#. The intersections with the #x#-axis are computed by solving #\green{y(t)}=0#.
Parametric curves are very useful to describe motion in physics.
The curve
\[\orange P \colon \phantom{x}\begin{cases}\blue{x(t)}&=10 \cdot \cos( \theta ) t\\ \green{y(t)}&= 10 \cdot \sin(\theta) t - \frac{1}{2}g \cdot t^2 \end{cases}\]
describes the orbit of an object being thrown from the origin with a varying angle #\theta# and a speed of #10 \text{ } m/s# with a gravitational constant #g#. On most places on earth the gravitational constant is around #9.8 \text{ } m /s^2#.
It is possible to adjust the values in the picture by using the sliders. Note that the distance thrown is maximal when the angle is #45# degrees and that the value of #\theta# in the slider is in radians.
In the example we disregard friction for simplicity.
The maximal height of a parametric curve is attained whenever #\green{y(t)}# is maximal within the specified interval. If we want to know the maximal height of the curve in the example, we should find the maximal value of
\[\green{y(t)}= 10 \cdot \sin(\theta)\cdot t - \frac{1}{2}g \cdot t^2 \]
The time at which the maximum value is attained can be found by setting the derivative to zero:
\[\frac{\partial y(t)}{\partial t}=10\cdot \sin(\theta) - g\cdot t = 0\quad\text{ so }\quad t= \frac{10\cdot \sin(\theta)}{g}\]
The maximal height of the curve is therefore the height that is attained at time #t_h= \frac{10\cdot \sin(\theta)}{g}#, so we substitute this value:
\[\begin{array}{rcl}\green{y(t_h)}&=& 10 \cdot \sin(\theta)\cdot t_h - \frac{1}{2}g \cdot t_h^2\\ &=& 10\cdot \sin(\theta)\cdot \frac{10\cdot \sin(\theta)}{g} - \frac{1}{2}g\cdot \left(\frac{10\cdot \sin(\theta)}{g}\right)^2\\ &=& 100\cdot \frac{\sin^2(\theta)}{g}- 50\cdot \frac{\sin^2(\theta)}{g} \\ &=&50\cdot \frac{\sin^2(\theta)}{g} \end{array}\]
This gives a maximal height of # 50\cdot \frac{\sin^2(\theta)}{g}#.
Every graph of a function can be described with a parametric curve. In this sense, the theory of parametric curves is a broader theory than the theory of functions and graphs.
If #f(x)# is a function on a domain #\ivcc{a}{b}# then we can define a parametric curve #\orange{C}# as follows
\[\begin{array}{rcl}\blue{x(t)}&=&\blue{t}\\\green{y(t)}&=&\green{f(t)},\end{array}\] where #t# ranges over the interval #\ivcc{a}{b}#. This curve coincides with the graph of the function # f(x)#.
Using the definition of a parametric curve as above, it is actually not true that every graph of a function can be described with the parametric equation as in the example. There is a technicality that the domain of a function need not be an interval whereas the variable #t# in a parametric curve should always vary over an interval. The precise statement therefore is: Every graph of a function defined on an interval can be described with a parametric curve.
Another way to get rid of this technicality is relaxing the definition of a parametric equation such that #t# is also allowed to vary over any subset of the real numbers.
Not every parametric curve can be described as the graph of a single function. Take for example the unit circle, described with the parametric equation \[\begin{array}{rcl}\green{x(t)}&=&\green{\cos (t)}\\ \blue{y(t)}&=&\blue{\sin(t)}.\end{array}\]
This describes the circle with radius #1# centered around the origin.
Since a function can only have a single #y#-value corresponding to every #x#-value, we need at least two functions to describe the unit circle. These would be one for the top half and one for the bottom half of the circle.
Given parametric equations #\ivcc{\blue{x(t)}}{\green{y(t)}}# and an interval #\ivcc{a}{b}# for #t# it can be very useful to make a sketch of the curve. This can be done by picking some explicit values for #t# in #\ivcc{a}{b}# and substituting these in the parametric equations #\blue{x(t)}# and #\green{y(t)}#. After drawing these in the plane it will, in most cases, be clear what the corresponding curve should be.
Example
Consider the curve #\orange C# given by #\ivcc{ \blue{x(t)}}{ \green{y(t)} }= \ivcc{ \blue{\frac{3t}{1+t^3}}}{ \green{\frac{3t^2}{1+3t^3}}}# defined for #t \neq -1#. We picked some values for #t# and plotted the point #\orange{P_t}#. The dotted line is the curve #\orange C# itself.
Determine the maximal height #h# of the parametric curve given by equations
\[\begin{cases}
x(t) &= \left(t+1\right)^2+1, \\
y(t) &= 5\cdot \cos \left(\sqrt{t^2+4\cdot t+4}\right)+3.
\end{cases}\]
where #t# lies in #\left[ -9 , 9 \right] #.
The maximal height is #8#.
The maximal height is attained whenever #y(t) = 5\cdot \cos \left(\sqrt{t^2+4\cdot t+4}\right)+3# is maximal. The cosine is periodic and takes maximal value #1#. This happens whenever #\sqrt{t^2+4\cdot t+4} = 0#. Squaring the equation gives us #t^2+4\cdot t+4 = 0#. We solve this by factorizing. We get #\left(t+2\right)^2 = 0# and find solutions #\left[ t=-2 \right] #. These lie in the desired interval. Consequently, the maximal height is given by substituting one of these values in #y(t)#. We see that the height is given by #8#.
Parametric curves
