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The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a coin graph ; more generally, intersection graphs of interior-disjoint geometric objects ...
This special line is the radical line of the two circles. Intersection of two circles with centers on the x-axis, their radical line is dark red. Special case = = = : In this case the origin is the center of the first circle and the second center lies on the x-axis (s. diagram).
Subtracting the two equations given above gives + (+) =. Since is a quadratic function of , the projection of the intersection onto the xz-plane is the section of an orthogonal parabola; it is only a section due to the fact that < <.
Intersection in two points. Methods for distinguishing these cases, and determining the coordinates for the points in the latter cases, are useful in a number of circumstances. For example, it is a common calculation to perform during ray tracing .
In geometry, an intersection curve is a curve that is common to two geometric objects. In the simplest case, the intersection of two non-parallel planes in Euclidean 3-space is a line . In general, an intersection curve consists of the common points of two transversally intersecting surfaces , meaning that at any common point the surface ...
D, the other point of intersection of the two circles, is the reflection of C across the line AB. If C = D (that is, there is a unique point of intersection of the two circles), then C is its own reflection and lies on the line AB (contrary to the assumption), and the two circles are internally tangential.
Consider the great circle that contains the side BC. This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as A is (conventionally) termed the pole of A and it is denoted by A'.
In general, the same inversion transforms the given line L and given circle C into two new circles, c 1 and c 2. Thus, the problem becomes that of finding a solution line tangent to the two inverted circles, which was solved above. There are four such lines, and re-inversion transforms them into the four solution circles of the Apollonius problem.