Tracing a Hyperbola, from Focus and Directrix
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here.
This animation shows how to trace out a hyperbola using the focus-directrix
definition.
(See notes below.)
Given a straight line L (the directrix), a point F (a focus) not lying
on that line, and a number e > 1, the hyperbola of eccentricity e is
the set of all points P for which the ratio of the distance PF to the
distance PL is e.
In this animation, the vertical line at the left side
of the frame is the directrix and the black point to its right is the focus. As the
curve is traced out, the green line segment is always the perpendicular from a moving point
on the curve to the directrix; its length is therefore the distance from the moving point to
the directrix. The bright blue segment connecting the moving point
on the curve to the focus is always the same length as the other bright blue segment
connecting the moving point to the directrix; this is evident from the circle having
the two blue segments as radii. Together with a piece of the directrix, the green
segment and one of the blue segments form a right triangle—which changes size as
the curve is traced out. However, the right triangles corresponding to different points
on the curve are always similar to each other, showing that the ratio of the blue segment
to the green segment is always the same number > 1.
Every hyperbola has two branches. For clarity, we picture only the right branch of this hyperbola in this
animation. We have placed the directrix just one unit to the left
of the focus; the eccentricity is 2/√3.
(10/29/09)