An ellipse is every point whose distances to two fixed points add up to the same number. Move the sliders and watch that one rule explain the shape, the reflections, and the orbits of planets.
a is the semi-major axis, b the semi-minor axis. The foci F₁ and F₂ sit at distance c from the centre, where c² = a² − b². Move P around the curve: r₁ + r₂ never changes.
The gardener's method: push two pins in at the foci, tie a loop of string around them, pull it tight with a pencil and go round. Because the string length is fixed, the pencil can only move along the ellipse. It uses the same a and b as above.
Every ray leaving one focus bounces off the ellipse straight through the other focus. That is how whispering galleries and kidney-stone lithotripters work. Drag the red source away from the focus to see the effect break.
Planets move on ellipses with the Sun at one focus (Kepler's 1st law). The line from the Sun sweeps equal areas in equal times (2nd law). Each coloured slice below takes the same time, so the planet races near the Sun and crawls far away.