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Descartes' circle theorem (a.k.a. the kissing circle theorem) provides a quadratic equation satisfied by the radii of four mutually tangent circles. By solving this equation, one can determine the possible values for the radius of a fourth circle tangent to three given, mutually tangent circles.
24 Ιαν 2022 · Descartes circle theorem Theorem (Descartes circle theorem, 1643) If b 1;b 2;b 3;b 4 are bends of four mutually tangent circles, then (b 1 + b 2 + b 3 + b 4) 2 = 2(b2 1 + b 2 2 + b 2 3 + b 2 4): Example 0 0 1 1 b 1 = b 2 = 0, b 3 = b 4 = 1 (0 + 0 + 1 + 1)2 = 22 = 4 2(02 + 02 + 12 + 12) = 2(2) = 4 Edna Jones The Descartes circle theorem
Let C1, C2, and C3 be mutually tangent circles with radii r1, r2, and r3. Let us assume that the radius of a fourth circle tangent to the other three, the red circle in Figure 1, is r4.1 Descartes’s circle theorem asserts the following:
15 Απρ 2019 · Let us assume that the radius of a fourth circle tangent to the other three, the red circle in Figure 1, is \ (r_4\). 1 Descartes’s circle theorem asserts the following: Theorem. The radii\ (r_1\), \ (r_2\), \ (r_3\), \ (r_4\)of four mutually tangent circles satisfy.
Descartes’ Theorem: The formula we'll be using to calculate all of our circles' sizes. It states: if the curvature of the first three circles are labeled c1, c2 and c3, respectively, the Theorem states that the curvature of the circle tangent to all three, d, is d = c1 + c2 + c3 ± 2 (sqrt (c1 * c2 + c2 * c3 + c3 * c1)).
Descartes' Circle Theorem. Given four mutually tangent circles with curvatures a, b, c, and d as in Figure 2, the Descartes Circle Equation specifies that (a2 + b2 + c2 + d2) = (1/2) (a + b + c + d) 2, where the curvature of a circle is defined as the reciprocal of its radius. Figure 2. Mutually tangent circles.
Descartes’s Circle Theorem states that the radii of four mutually tangent circles r1, r2, r3, and r4 satisfy 1 r1 + 1 r2 + 1 r3 + 1 r4 2 = 2 1 r2 1 + 1 r2 2 + 1 r2 3 + 1 r2 4 . The radii are chosen to be negative if the corresponding circle encloses the others. In this way we preserve the relation d2 ij = (r i + r j)2 where d ij is the ...
