Difference between revisions of "Descartes' Circle Formula"
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Descartes' Circle Formula is a relation held between four mutually tangent circles. | Descartes' Circle Formula is a relation held between four mutually tangent circles. | ||
− | Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle A of radius <math>r_a</math> is externally tangent to circle B of radius <math>r_b</math>. Then the curvatures of the circles are simply the reciprocals of their radii, <math>\frac{1}{ | + | Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle A of radius <math>r_a</math> is externally tangent to circle B of radius <math>r_b</math>. Then the curvatures of the circles are simply the reciprocals of their radii, <math>\frac{1}{r_a}</math> and <math>\frac{1}{r_b}</math>. |
− | If circle <math>A</math> is internally tangent to circle <math>B</math>, however, a the curvature of circle <math>A</math> is still <math>\frac{1}{ | + | If circle <math>A</math> is internally tangent to circle <math>B</math>, however, a the curvature of circle <math>A</math> is still <math>\frac{1}{r_a}</math>, while the curvature of circle B is <math>-\frac{1}{r_b}</math>, the opposite of the reciprocal of its radius. |
<asy> | <asy> |
Revision as of 21:43, 1 February 2012
(based on wording of ARML 2010 Power)
Descartes' Circle Formula is a relation held between four mutually tangent circles.
Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle A of radius is externally tangent to circle B of radius
. Then the curvatures of the circles are simply the reciprocals of their radii,
and
.
If circle is internally tangent to circle
, however, a the curvature of circle
is still
, while the curvature of circle B is
, the opposite of the reciprocal of its radius.
In the above diagram, the curvature of circle is
while the curvature of circle
is
.
In the above diagram, the curvature of circle is still
while the curvature of circle
is
.
When four circles and
are pairwise tangent, with respective curvatures
and
, then the following equation holds:
.