Apollonisches Berührungsproblem - Kreise

Tangent to three circles

Three objects in a plane are given. Find all circles that have one point in common wich each of these given objects. The objects are generalized circles, hence including points and straight lines. Wikipedia gives a good overview.

For the cases of three points and three circles the Excel workbook presented here shows the algebraic solution and its 2D graphical representation.

Algebraic solution - PPP (three points)

When three points are given, the solution circle - defined by the X and Y coordinate of its center (x_s,y_s) and its radius r_s, results from the condition that the distance from the center (xs,ys) to each given point (x_i,y_i) be equal to r_s. The Excel workbook contains the solution of the equation system for the general case. The result is the circle circumscring the triangle between the three points.

The center could also be plotted as the crossing of the three perpendicular bisectors over each triangle side. However, this plot is not executed in the workbook.

Special cases result when the three points are collinear. The solution is then given by that line - a circle of infinite radius.

When two points are identical, an infinite numer of solutions result. The centers of these circles are situatied on the perpendicular bisector between two distinguished points.

Algebraic solution - CCC (three circles)

When three circles are given, the insertion of the relationships of x_s and y_s from  r_s in any initial condition yiealds a quadratic equation in r_s and hence one or two possible solutions. By excluding negative values for r_s the solution set is effectively confined to unique solutions.

When three circles are given, depending upon their constellations, there are up to 8 combinations of possibilities for a solution to include or exclude any of the given circles. When the circles are disjoint every possibility corresponds to a different initial condition. When the circles intersect the possibilities are more variable and it is necessary to consider both results from the quadratic equation for r_s.

Excel workbook

Sheet PPP

Shows how the solution is derived and plots a graphical representation for three user-defined points. Also plotted is the triangle and the bisector points.

Sheet CCC

Shows how the solution is derived and two graphical representations.

The left graph plos the solution for the given circles and the given set of inclusion and exclusion conditions for the solution ccircle.

The right graph shows all solutions together.

Special cases

If the solution is a straight line tangent to all circles, the calculation cannot find it.

The calculation neither treats any case involing a given straight line. Such cases belong to category "L" and will be treated apart in later versions.