Curve Operations with Maple
Curve construction examples

This worksheet gives some examples of operations that can be done with Maple on curves
and vector operations.    See Section 2.4 for explanations of some of the words.    

Conchoids

The idea of a conchoid is to start with some curve    

                   rold(t) = ( x(t),y(t) )

We think of    r(t)   as a radius   vector from the origin.    The new curve is formed
by increasing the length of the  vector    r(t)   by  a fixed amount, say  1.  

This means that the new curve has    vector equation    

                            rnew(t) = rold(t) + rold(t)/| rold(t) |  .

The new curve is called a conchoid.    

Here are some examples that show how to build a conchoid efficiently using Maple.    Note
that all the data can be changed merely by editing the first input line.    

The example below shows how to construct the conchoid  based on the curve 

 
> x := 2+t; y := -2*t; a := -3; b := 3;
> radius := sqrt(x^2+y^2);
> plot({[x,y,t=a..b], [x*(1+1/radius),y*(1+1/radius),t=a..b] },scaling=constrained);

On speed

Here  are some commands that are pertinent to speed.     Recall that if    a curve is given
by   the vector 

                             r(t) = (x(t),y(t)),   

then   the speed is the scalar   sqrt(  x(t)^2 + y(t)^2  ).    It's another function of  t.

To  ``see'' the speed on a parametric curve, one way is to plot  points at equal time intervals
along the curve.   Here's how.   First we define a curve:




> x := t+cos(2*t);  y :=  1+sin(2*t);  a := 0; b := 2*Pi;
Now we can plot the curve, but let's do it with 50  points rather than as a continuous curve:

> plot( [x,y,t=a..b], style=point, numpoints=50,scaling=constrained);
Why aren't the points equally spaced?   Because the curve is not parametrized with constant
speed.

Let's try another curve:

> x := cos(t);  y :=  sin(t);  a := 0; b := 2*Pi;
> plot( [x,y,t=a..b], style=point, numpoints=50,scaling=constrained);
This time the plot points DO seem to be equally spaced.    

Given a curve, we can use Maple to find its length.    Here are some helpful commands.


> x := cos(t); y := sin(t); a:=0; b := 2*Pi;
> dx := diff(x,t); dy := diff(y,t);
> speed := sqrt( dx^2+dy^2);
> simplify(%);
> int(%,t=a..b);
>