Geometry Expressions

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Archimedes trammel is a simple device for drawing an ellipse. Another way to draw an ellipse uses a piece of string and two pins.

To match the ellipse drawn by a particular trammel, how long should the string be, and where should the pins go?

We model the trammel as a line segment AB with point A on the y-axis. We place a point C at the intersection of the segment and the x-axis.

We constrain the distances AB and AC and the location (effectively x-coordinate) t of C along the x-axis.

Picture 1

We find the locus of point B as t varies.

We measure the equation of the curve and observe that it is quadratic.

We copy the equation for later use.

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We now draw an ellipse and constrain its equation, pasting in the previously copied curve equation.

Picture 3

We can look at the coordinates of the foci, and measure the distance between a focus and a point at the intersection of the ellipse and the y-axis. By symmetry the string will be twice this length.

Picture 4