Circle-Line Intersection
An (infinite) line determined by two points (x_1,y_1) and (x_2,y_2) may intersect a circle of radius r 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 in known as a secant line (Rhoad et al. 1984, p. 429).
Defining
gives the points of intersection as
where the function sgn^*(x) is defined as
The discriminant
| Delta=r^2d_r^2-D^2 |
(8)
|
therefore determines the incidence of the line and circle, as summarized in the following table.
See also
Circle, Circle-Circle Intersection, Line, Line-Line Intersection, Secant Line, Tangent LineExplore with Wolfram|Alpha
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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 IntersectionCite this as:
Weisstein, Eric W. "Circle-Line Intersection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Circle-LineIntersection.html