Ellipse Between Two Circles

The locus of centers of the circles inscribed in an arbelos is an ellipse. Arbelos - a crescent shaped figure - is formed by two tangent circles, one inside the other. Relaxing the tangency condition we obtain what may be called a blunt arbelos - an intermediate shape obtained by morphing an arbelos into an annulus.

A third circle can be inscribed into the shape touching the bigger circle internally and the smaller one externally. As with the arbelos, the centers of such circles lie on an ellipse.

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Assuming the two circles have centers on the x-axis, one C(x₁, r₁) is centered at (x₁, 0) and has the radius r₁, the other - C(x₂, r₂) - has the center at (x₂, 0) and radius r₂. The diagram shows the case of x₂ > x₁ and r₂ > r₁.

Quite clearly the sum of distances of the center O of the inscribed circle to the centers of the two given circles is r₁ + r₂, implying that the locus of the centers is indeed an ellipse with foci at the centers of the given circles and the major axis r₁ + r₂.

The minor axis can be found by considering the extreme case of the inscribed circle with the center at the top point of the ellipse. This point along with the centers of C(x₁, r₁) and C(x₂, r₂) form an isosceles triangle with sides (r₁ + r₂) / 2 and base (x₂ - x₁), making the altitude equal to

h² = ((r₁ + r₂) / 2)² - ((x₂ - x₁) / 2)².

The minor axis of the ellipse is 2h.

Conic Sections > Ellipse

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