In the interactive picture below, AB represents a ladder leaning against a wall. Imagine you're standing on the ladder at point C when the ladder starts to slide. What path does your foot trace? To find out, drag point A. Note that point D, the reflection of point C, is traced as well.
Now move the location of point C. Clear the traces by clicking the red 'X' in the lower right corner. Then drag point A again to see how the trace differs. You can change the length of the ladder by dragging an endpoint of the horizontal red segment.
How does the shape of the ellipse vary for different locations of point C?
What shape does point C trace when it is at the midpoint of AB?
What does point C trace when it lies directly on top of A or B?
Suppose you want to draw an ellipse with a 14-foot major axis and a 6-foot minor axis. How long a ladder do you need, and where should you place point C?
Can you prove algebraically that point C traces an ellipse?
The carpenter's trammel is a natural extension of the falling ladder. In the interactive picture below, ruler AC slides along the axes with A traveling on the x-axis and B moving along the y-axis.
Drag point A and observe the trace of point C and its reflection, point D. You can change the distance between A and B by dragging an endpoint of the red segment. You can also drag C to new locations, lengthening or shortening AC in the process. As before, the trace may be cleared by clicking the red "X" in the lower right corner.
How can you use the trammel to draw an ellipse with a 14-foot major axis and a 6-foot minor axis?