The Gardener's Ellipse
How can a gardener draw a perfectly elliptical contour for a flowerbed?
Three stakes and a rope are enough: two stakes are fixed at the position of the foci of the flowerbed, and the ends of the rope are attached to them.
At this point, the gardener can mark the contour of the elliptical flowerbed using the third stake to hold the rope, moving around the fixed stakes while ensuring that the rope is always taut.
Use the app below to create your virtual flowerbed using the gardener's method: the slider allows you to change the length of the rope, and the other slider sets the semi-distance of the foci (that is, half the distance between them).
Drag point P or the tip of the stake in the 3D View to create the flowerbed.
The 3D view can be rotated by dragging it with the mouse or using the predefined gestures for touch screens, and the buttons allow you to restore the 3D view to the original settings and delete the traces.
What do you notice about the distances from to and from to ?
Which conic section do you obtain when the two foci coincide?