# Algorithm Implementation/Geometry/Tangents between two circles

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This code finds the set of common tangents between two circles.

## Java[edit]

```
import java.util.Arrays;
public class CircleTangents {
/**
* Finds tangent segments between two given circles.
*
* Returns an empty, or 2x4, or 4x4 array of doubles representing
* the two exterior and two interior tangent segments (in that order).
* If some tangents don't exist, they aren't present in the output.
* Each segment is represent by a 4-tuple x1,y1,x2,y2.
*
* Exterior tangents exist iff one of the circles doesn't contain
* the other. Interior tangents exist iff circles don't intersect.
*
* In the limiting case when circles touch from outside/inside, there are
* no interior/exterior tangents, respectively, but just one common
* tangent line (which isn't returned at all, or returned as two very
* close or equal points by this code, depending on roundoff -- sorry!)
*
* Java 6 (1.6) required, for Arrays.copyOf()
*/
public static double[][] getTangents(double x1, double y1, double r1, double x2, double y2, double r2) {
double d_sq = (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2);
if (d_sq <= (r1-r2)*(r1-r2)) return new double[0][4];
double d = Math.sqrt(d_sq);
double vx = (x2 - x1) / d;
double vy = (y2 - y1) / d;
double[][] res = new double[4][4];
int i = 0;
// Let A, B be the centers, and C, D be points at which the tangent
// touches first and second circle, and n be the normal vector to it.
//
// We have the system:
// n * n = 1 (n is a unit vector)
// C = A + r1 * n
// D = B +/- r2 * n
// n * CD = 0 (common orthogonality)
//
// n * CD = n * (AB +/- r2*n - r1*n) = AB*n - (r1 -/+ r2) = 0, <=>
// AB * n = (r1 -/+ r2), <=>
// v * n = (r1 -/+ r2) / d, where v = AB/|AB| = AB/d
// This is a linear equation in unknown vector n.
for (int sign1 = +1; sign1 >= -1; sign1 -= 2) {
double c = (r1 - sign1 * r2) / d;
// Now we're just intersecting a line with a circle: v*n=c, n*n=1
if (c*c > 1.0) continue;
double h = Math.sqrt(Math.max(0.0, 1.0 - c*c));
for (int sign2 = +1; sign2 >= -1; sign2 -= 2) {
double nx = vx * c - sign2 * h * vy;
double ny = vy * c + sign2 * h * vx;
double[] a = res[i++];
a[0] = x1 + r1 * nx;
a[1] = y1 + r1 * ny;
a[2] = x2 + sign1 * r2 * nx;
a[3] = y2 + sign1 * r2 * ny;
}
}
return Arrays.copyOf(res, i);
}
}
```

## External links[edit]

- Circle-Circle Tangents on MathWorld
- Visualizer for the above code is available. (Requires Java. Source is included in the jar archive.)