“Circle-Circle Intersection”的概念、定义、习题解题方法及翻译-数学辞典

单词

Circle-Circle Intersection

释义

Circle-Circle Intersection

Let two Circles of Radii and and centered at and intersectin a Lens-shaped region. The equations of the two circles are

			(1)
			(2)

Combining (1) and (2) gives

(3)

Multiplying through and rearranging gives

(4)

Solving for

results in

(5)

The line connecting the cusps of the Lens therefore has half-length given by plugging

back in to obtain


			(6)

giving a length of


			(7)

This same formulation applies directly to the Sphere-Sphere Intersection problem.

To find the Area of the asymmetric ``Lens'' in which the Circles intersect,simply use the formula for the circular Segment of radius and triangular height

(8)

twice, one for each half of the ``Lens.'' Noting that the heights of the two segment triangles are

			(9)
			(10)

The result is



			(11)

The limiting cases of this expression can be checked to give 0 when

and

			(12)
			(13)

when

, as expected. In order for half the area of two Unit Disks (

) to overlap, set

in the above equation

(14)

and solve numerically, yielding

See also Lens, Segment, Sphere-Sphere Intersection

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