- Purpose
- Background
- Plotting and animating curves in the plane
- Defining parametric curves and vector valued functions simply in Maple

- Exercises

where and are functions defined on some interval . From our definition of a parametric curve, it should be clear that you can always associate a parametric curve with a vector-valued function by just considering the curve traced out by the head of the vector.

>with(CalcP7);The ParamPlot command produces an animated plot. To see the animation, execute the command and then click on the plot region below to make the controls appear in the Context Bar just above the worksheet window.

>ParamPlot([t,t^2],t=-2..2);the direction of the motion on the curve can be reversed by simply changing the first component from t to -t, as shown below.

>ParamPlot([-t,t^2],t=-2..2);The ParamPlot command is nice for visualization, but its output doesn't always show up in printouts. Toproduce a printable plot, you can use the VPlot command as shown below.

>VPlot([t^2,t^3-t],t=-1.5..1.5);

>f:=t->[2*cos(t),2*sin(t)];You can evaluate this function at any value of t in the usual way.

>f(0);This is how to access a single component. You would use f(t)[2] to get the second component.

>f(t)[1]

It is clear that this formula doesn't make sense if at some particular value of . If at that same value of , then it turns out the graph has a vertical tangent at that point. If both and are zero at some value of , then the curve often doesn't have a tangent line at that point. What you see instead is a sharp corner, called a cusp. An example of this appears in the second exercise.

While the concept of arc length is very useful for the theory of parametric curves, it turns out to be very difficult to compute in all but the simplest cases.

- Animate the following parametrization for
. Then animate the parametrization again after doubling the angle and negating the component for each trig function. Describe what effect these changes have on the animation.

- Plot the curve

for .- Find the formula for the slope of the parametric curve.
- Find values for two points on the graph where the slope does not exist or where there is a cusp.
- Calculate the coordinate location for the two points above and plot them on the graph along with the parametrization. You will find some of the commands below helpful, but you will need more commands than what is given.
>xp:=diff(r(t)[1],t); >solve({xp=0,yp=0},t); >r(0); >with(plots): >a:=VPlot(r(t),t=0..2*Pi): >b:=VPlot(r(0),t=0..2*Pi,style=point,symbolsize=30,color=black): >display(a,b,c);

- Find two different parametrizations, without just switching trig functions for and , and necessary interval for the ellipse

First show that the two parametrizations are the same shape by plotting them parametrically using`ParamPlot`(you may want to use scaling=constrained). Find the arclength of the ellipse for both parametrizations.

2016-01-26