# Ellipse & Hyperbola Construction

- Author:
- Michael Horvath

This activity demonstrates two methods of constructing the ellipse and hyperbola.

The definition for the first method is:

- Points
*A*and*B*are the foci of the ellipse/hyperbola. - Line
*a*passes through the intersections of circles*c*and*d*. - The ellipse/hyperbola is the locus of point
*G*, which is the intersection of lines*a*and*b*. - Move point
*D*to trace the locus of point*G*. - Move points
*A*and*B*to change the shape of the ellipse/hyperbola. Move point*A*inside the circle centered on point*B*to change the curve to an ellipse. Move point*A*outside the circle to change the curve to a hyperbola.

Here's a second method, this time using three parameters: vectors

*A*and*B*, and the angle*θ*(all colored blue). The angle*θ*is the angle between vector*T*and the x-axis. Also shown is how to construct the curves' foci and directrixes. The parameters and plotted points for the constructions are the same as those used in the standard parametric equations for each curve. For comparison, the parametric equation for an ellipse is:```
x = a * cos(θ)
y = b * sin(θ)
```

And, the parametric formula for a hyperbola is:
```
x = a * sec(θ)
y = b * tan(θ)
```

The steps to achieve the same plotted points, *P*_{1}and*P*_{2}, in the graph as the parametric equations are:- Simply enter the lengths of vectors
*A*and*B*, as well as the value for*θ*, so that they match the equations above. - Drag vectors
*A*and*B*to change the shape of the curve(s). - Drag vector
*T*to trace the loci of points*P*_{1}and*P*_{2}. - The blue circle is the circle of unit radius, and is used to calculate the value of
*θ*, only. - Check the "Show components" and "Show construction" options for each curve to see the curve components and constructions.