Parametric elipses.

Joshua DuMont


 

Derive the parametric equations for the locus of a point (x, y) on a line segment that is moved so that one end is on the x-axis and the other end is on the y-axis.

 

A way to parameterize this curve is by the angle t that the line segment makes with the x-axis. In the gray triangle sin(t)=or y=bsin(t). In the blue triangle, cos(t)=or x=acos(t).

A parameterization of the locus of the point then is:

x=acos(t), y=bsin(t)

This forms an ellipse, as t changes.

To see this, take =cos(t) , =sin(t) and square both sides of each equation.

=(t) ,=(t)Adding the left sides together and the right sides together yields:

+=(t)+(t) or just: +=1

Which is the familiar form for an ellipse.

In that form it may be that we recognize that the a value gives the horizontal distance from the center to the ellipse, and the b value gives the vertical distance from the center to the ellipse. This could be seen in the parametric form just as easily. The maximum y value happens when t=. Plugging this in we find the point (0,b). Similarly, the maximum x value happens when t=0. Plugging this in we get the point (a,0).

 

So, for different choices of lengths for a and b we get different ellipses.

 

 

We neednít have written the function in terms of the angle the segment makes with the x-axis. If we choose to parameterize by the angle the segment makes with the y-axis we get: x=asin(t), y=bcos(t). This turns out to trace the same graph. Using the same method as above, we can write this in Cartesian form as: +=1. The two parameterizations give the same graph, but donít follow the same path. The first gives the point (a,0) when t=0, and moves counterclockwise returning to the beginning point every time t is a multiple of 2π. The second gives the point (0,b) when t=0 and moves clockwise with the same period. A different choice for parameter will affect the direction travelled, the point at which we start and the period. Replacing t with 2t in either of our parameterizations, for instance, doesnít affect the process we used above to change to Cartesian form (the graph is the same). However, it does cause any point which previously happened at t=c to occur at t=. In the time it took previously to travel around the ellipse once we now travel around it twice.