Focus locus

Parametric equations

A point $P$ travels on an ellipse. A segment is drawn from $P$ to each focus $F_1$ and $F_2$. The sum of those two distances is the constant $2a$; the figure does not print that number.

• $a$ is the semi-major axis. The eccentricity slider $e$ runs from $0.1$ to $0.95$, so the curve stays an ellipse and never becomes a circle or a parabola
• $b=\sqrt{a^2-c^2}$ and $c=a\cdot e$ is the distance from the center to each focus

Interaction

• Semi-major axis a: change the size of the ellipse; $P$ still travels on it, with $P{F}_1+P{F}_2=2a$
• Eccentricity e: flatten the ellipse. The foci move apart as $c=ae$
• Orbit speed ω: change how fast $P$ travels around the ellipse

What to notice

• The two segments are the distances from a point on the ellipse to the two foci, and their sum is $2a$.
• A larger $e$ makes the ellipse flatter and moves the foci toward the ends of the major axis.
• The foci and the orbit together are this ellipse. The slider never reaches a circle.

Related

• Conic envelope
• Parabolic reflection
• Conics by eccentricity

Further reading

• Focus (geometry) (Wikipedia)