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Circle-Line Intersection

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CircleLineIntersection

An (infinite) line determined by two points and may intersect a circle of radius and center (0, 0) in two imaginary points (left figure), a degenerate single point (corresponding to the line being tangent to the circle; middle figure), or two real points (right figure).

In geometry, a line meeting a circle in exactly one point is known as a tangent line, while a line meeting a circle in exactly two points is known as a secant line (Rhoad et al. 1984, p. 429).

Defining

			(1)
			(2)
			(3)
			(4)

gives the points of intersection as

			(5)
			(6)

where the function is defined as

$sgn^*(x)={-1 for x<0; 1 otherwise.$

(7)

The upper signs in the two coordinate formulas are taken together, as are the lower signs.

The discriminant

(8)

therefore determines the incidence of the line and circle, as summarized in the following table.

	incidence
	no intersection
	tangent
	intersection

See also

Circle, Circle-Circle Intersection, Line, Line-Line Intersection, Secant Line, Tangent Line

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References

Rhoad, R.; Milauskas, G.; and Whipple, R. Geometry for Enjoyment and Challenge, rev. ed. Evanston, IL: McDougal, Littell & Company, 1984.

Referenced on Wolfram|Alpha

Circle-Line Intersection

Cite this as:

Weisstein, Eric W. "Circle-Line Intersection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Circle-LineIntersection.html

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