### Geometry: Parametric curves

### Parametric curves

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.

Parametric curves are very useful to describe motion in physics.

Throwing a ball

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.

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)#.

Sketching a parametric curve 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.

The maximal height is attained whenever #y(t) = 6\cdot \cos \left(\sqrt{t^2+5\cdot t-6}\right)-1# is maximal. The cosine is periodic and takes maximal value #1#. This happens whenever #\sqrt{t^2+5\cdot t-6} = 0#. Squaring the equation gives us #t^2+5\cdot t-6 = 0#. We solve this by factorizing. We get #\left(t-1\right)\cdot \left(t+6\right) = 0# and find solutions #\left[ t=-6 , t=1 \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 #5#.

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