EllipseAsEnvelope

1. Ellipse as line-envelope

Let c be a circle centered at O and A a point inside 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 an ellipse.
- The ellipse has the circle as its auxiliary circle and touches it at the diametral points with line AO.
- The contact point E of line DE is the projection on it of the fourth harmonic G = F(A,C), where F the intersection of AB with DE.
- Analogous generation is valid also for hyperbolas. The only difference is on taking A to be outside the circle.
The proofs follow immediately from the discussion in Ellipse.html .

2. Generalization

Point E glides on circle B(BE). Point H is taken on AE (A fixed, inside the circle) such that AH/AE = k (constant).
Line HU is taken to be orthogonal at H to AE.
Line HU envelopes an ellipse.
The auxiliary circle of the ellipse is homothetic to the circle B(BE) w.r. to A and in ratio k to it.
The ellipses resulting for various k are all homothetic to each other.
The contact point U of line HU with the ellipse is the projection on HU of T, which is the harmonic fourth of K(A,S), where K is the intersection of HU with the line of centers AB.
The figure here is simply a homothetic image of the previous figure.

1. Ellipse as line-envelope

2. Generalization

See Also