Conics by eccentricity

Idea

A point stays at a constant ratio of its distance to a focus and its distance to a directrix:

That number is the eccentricity. Below $1$ the curve is an ellipse, at $1$ a parabola, and above $1$ one hyperbola branch tied to that focus and directrix. The circle is only the limit $e\to 0$, and then the directrix ratio is hidden.

Interaction

• Mode: choose Eccentricity or Focus locus
• e: move the ratio from a near-circle ellipse, through a parabola at $e=1$, to one hyperbola branch
• Show focus / directrix: draw the focus, the directrix, and the point $P$. In the circle limit the directrix is left off, and the status hides the distance ratio
• Curve: in Focus locus, choose Ellipse, Parabola, or Hyperbola
• Animate point P: let $P$ travel on its own, or drag $P$ on the figure

Suggested order: start with a small e, near a circle, then $e=1$, then $e>1$ for the single branch. Switch Mode to Focus locus and compare Ellipse, Parabola, and Hyperbola. The hyperbola in that mode draws both branches.

What to notice

• The same ratio $e$ picks the family. Less than $1$, equal to $1$, and greater than $1$ are one rule crossing a threshold, not three definitions.
• As $e$ grows, the closed curve opens and then settles on the one branch that matches the focus and the directrix. Raising $e$ does not split the drawing into two branches.
• Focus locus reads the same curves another way. An ellipse keeps a constant sum of distances to two foci, a parabola keeps equal distances to the focus and the directrix, and a hyperbola keeps a constant absolute difference, with both branches drawn.

Related

• Conic envelope
• Focus locus
• Parabolic reflection

Further reading

• Conic section (Wikipedia)
• Eccentricity (mathematics) (Wikipedia)