# Source code for kivy.geometry

```'''
Geometry utilities
==================

This module contains some helper functions for geometric calculations.
'''

__all__ = ('circumcircle', 'minimum_bounding_circle')

from kivy.vector import Vector

[docs]def circumcircle(a, b, c):
'''
Computes the circumcircle of a triangle defined by a, b, c.
See: http://en.wikipedia.org/wiki/Circumscribed_circle

:Parameters:
`a`: iterable containing at least 2 values (for x and y)
The 1st point of the triangle.
`b`: iterable containing at least 2 values (for x and y)
The 2nd point of the triangle.
`c`: iterable containing at least 2 values (for x and y)
The 3rd point of the triangle.

:Return:
A tuple that defines the circle :
* The first element in the returned tuple is the center as (x, y)
* The second is the radius (float)
'''
P = Vector(a, a)
Q = Vector(b, b)
R = Vector(c, c)

mPQ = (P + Q) * .5
mQR = (Q + R) * .5

numer = -(- mPQ.y * R.y + mPQ.y * Q.y + mQR.y * R.y - mQR.y * Q.y -
mPQ.x * R.x + mPQ.x * Q.x + mQR.x * R.x - mQR.x * Q.x)
denom = (-Q.x * R.y + P.x * R.y - P.x * Q.y +
Q.y * R.x - P.y * R.x + P.y * Q.x)

t = numer / denom

cx = -t * (Q.y - P.y) + mPQ.x
cy = t * (Q.x - P.x) + mPQ.y

return ((cx, cy), (P - (cx, cy)).length())

[docs]def minimum_bounding_circle(points):
'''
Returns the minimum bounding circle for a set of points.

For a description of the problem being solved, see the `Smallest Circle
Problem <http://en.wikipedia.org/wiki/Smallest_circle_problem>`_.

The function uses Applet's Algorithm, the runtime is ``O(h^3, *n)``,
where h is the number of points in the convex hull of the set of points.
**But** it runs in linear time in almost all real world cases.
See: http://tinyurl.com/6e4n5yb

:Parameters:
`points`: iterable
A list of points (2 tuple with x,y coordinates)

:Return:
A tuple that defines the circle:
* The first element in the returned tuple is the center (x, y)
* The second the radius (float)

'''
points = [Vector(p, p) for p in points]

if len(points) == 1:
return (points.x, points.y), 0.0

if len(points) == 2:
p1, p2 = points
return (p1 + p2) * .5, ((p1 - p2) * .5).length()

# determine a point P with the smallest y value
P = min(points, key=lambda p: p.y)

# find a point Q such that the angle of the line segment
# PQ with the x axis is minimal
def x_axis_angle(q):
if q == P:
return 1e10  # max val if the same, to skip
return abs((q - P).angle((1, 0)))
Q = min(points, key=x_axis_angle)

for p in points:
# find R such that angle PRQ is minimal
def angle_pq(r):
if r in (P, Q):
return 1e10  # max val if the same, to skip
return abs((r - P).angle(r - Q))
R = min(points, key=angle_pq)

# check for case 1 (angle PRQ is obtuse), the circle is determined
# by two points, P and Q. radius = |(P-Q)/2|, center = (P+Q)/2
if angle_pq(R) > 90.0:
return (P + Q) * .5, ((P - Q) * .5).length()

# if angle RPQ is obtuse, make P = R, and try again
if abs((R - P).angle(Q - P)) > 90:
P = R
continue

# if angle PQR is obtuse, make Q = R, and try again
if abs((P - Q).angle(R - Q)) > 90:
Q = R
continue

# all angles were acute..we just need the circle through the
# two points furthest apart!
break

# find the circumcenter for triangle given by P,Q,R
return circumcircle(P, Q, R)
```