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Two circles may intersect in two imaginary points, a single degenerate point, or two distinct points. The intersections of two circles determine a line known as the radical ...

The points of tangency t_1 and t_2 for the four lines tangent to two circles with centers x_1 and x_2 and radii r_1 and r_2 are given by solving the simultaneous equations ...

An ellipse intersects a circle in 0, 1, 2, 3, or 4 points. The points of intersection of a circle of center (x_0,y_0) and radius r with an ellipse of semi-major and ...

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 ...

Taking the locus of midpoints from a fixed point to a circle of radius r results in a circle of radius r/2. This follows trivially from r(theta) = [-x; 0]+1/2([rcostheta; ...

A circle is the set of points in a plane that are equidistant from a given point O. The distance r from the center is called the radius, and the point O is called the center. ...

A circle bundle pi:E->M is a fiber bundle whose fibers pi^(-1)(x) are circles. It may also have the structure of a principal bundle if there is an action of SO(2) that ...

Consider a unit circle and a radiant point located at (mu,0). There are four different regimes of caustics, illustrated above. For radiant point at mu=infty, the catacaustic ...

An arrangement of overlapping circles which cover the entire plane. A lower bound for a covering using equivalent circles is 2pi/sqrt(27) (Williams 1979, p. 51).