HyperbolaAsEnvelope

Hyperbola as line-envelope

Let c be a circle centered at B and A a point outside it.
Draw line AD to a variable point D of the circle and at D draw the orthogonal line DE to AD.
- Line DE envelopes a hyperbola.
- The hyperbola has the circle as its auxiliary circle and touches it at the diametral points with line AB. The foci are A and its symmetric w.r. to B.
- The contact point of line DE is the projection on it of the fourth harmonic G = F(A,C), where F the intersection of AB with DE.
- The asymptotes of the hyperbola form an angle equal to the angle of the tangents from point A.
- Analogous generation is valid also for ellipses. The only difference is on taking A to be inside the circle.
The proofs follow immediately from the discussion in Hyperbola.html .
See HyperbolaAsEnvelopeII.html for a generalization constructing hyperbolas homothetic to this one.

Hyperbola as line-envelope

See Also