#!/usr/bin/env python3
+# This example is inspired from a math question in the French baccalauréat 2014,
+# consisting in computing the intersection of a plane with a line.
+
from linpy import *
x, y, z = symbols('x y z')
-DF = Eq(x, y) & Eq(z, 6 - 2*x)
-P = Eq(x + y - 2*z, 0)
+plane = Eq(x, y) & Eq(z, 6 - 2*x)
+line = Eq(x + y - 2*z, 0)
-print('DF ∩ P =', DF & P)
+if __name__ == '__main__':
+ print('plane: ', plane)
+ print('line: ', line)
+ print('intersection:', plane & line)
+++ /dev/null
-#!/usr/bin/env python3
-
-import matplotlib.pyplot as plt
-
-from matplotlib import pylab
-from mpl_toolkits.mplot3d import Axes3D
-
-from linpy import *
-
-x, y, z = symbols('x y z')
-
-fig = plt.figure(facecolor='white')
-
-diam_plot = fig.add_subplot(2, 2, 1, aspect='equal')
-diam_plot.set_title('Diamond')
-diam = Ge(y, x - 1) & Le(y, x + 1) & Ge(y, -x - 1) & Le(y, -x + 1)
-diam.plot(diam_plot, fill=True, edgecolor='red', facecolor='yellow')
-
-cham_plot = fig.add_subplot(2, 2, 2, projection='3d', aspect='equal')
-cham_plot.set_title('Chamfered cube')
-cham = Le(0, x) & Le(x, 3) & Le(0, y) & Le(y, 3) & Le(0, z) & Le(z, 3) & \
- Le(z - 2, x) & Le(x, z + 2) & Le(1 - z, x) & Le(x, 5 - z) & \
- Le(z - 2, y) & Le(y, z + 2) & Le(1 - z, y) & Le(y, 5 - z) & \
- Le(y - 2, x) & Le(x, y + 2) & Le(1 - y, x) & Le(x, 5 - y)
-cham.plot(cham_plot, facecolors=(1, 0, 0, 0.75))
-
-rhom_plot = fig.add_subplot(2, 2, 3, projection='3d', aspect='equal')
-rhom_plot.set_title('Rhombicuboctahedron')
-rhom = cham & \
- Le(x + y + z, 7) & Ge(-2, -x - y - z) & \
- Le(-1, x + y - z) & Le(x + y - z, 4) & \
- Le(-1, x - y + z) & Le(x - y + z, 4) & \
- Le(-1, -x + y + z) & Le(-x + y + z, 4)
-rhom.plot(rhom_plot, facecolors=(0, 1, 0, 0.75))
-
-cubo_plot = fig.add_subplot(2, 2, 4, projection='3d', aspect='equal')
-cubo_plot.set_title('Truncated cuboctahedron')
-cubo = Le(0, x) & Le(x, 5) & Le(0, y) & Le(y, 5) & Le(0, z) & Le(z, 5) & \
- Le(x -4, y) & Le(y, x + 4) & Le(-x + 1, y) & Le(y, -x + 9) & \
- Le(y -4, z) & Le(z, y + 4) & Le(-y + 1, z) & Le(z, -y + 9) & \
- Le(z -4, x) & Le(x, z + 4) & Le(-z + 1, x) & Le(x, -z + 9) & \
- Le(3, x + y + z) & Le(x + y + z, 12) & \
- Le(-2, x - y + z) & Le(x - y + z, 7) & \
- Le(-2, -x + y + z) & Le(-x + y + z, 7) & \
- Le(-2, x + y - z) & Le(x + y - z, 7)
-cubo.plot(cubo_plot, facecolors=(0, 0, 1, 0.75))
-
-pylab.show()
#!/usr/bin/env python3
+# Plot a Menger sponge.
+#
+# The construction of a Menger sponge can be described as follows:
+#
+# 1. Begin with a cube.
+# 2. Divide every face of the cube into 9 squares, like a Rubik's Cube. This
+# will sub-divide the cube into 27 smaller cubes.
+# 3. Remove the smaller cube in the middle of each face, and remove the smaller
+# cube in the very center of the larger cube, leaving 20 smaller cubes. This
+# is a level-1 Menger sponge (resembling a Void Cube).
+# 4. Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue
+# to iterate.
+
import argparse
import matplotlib.pyplot as plt
from linpy import *
+
x, y, z = symbols('x y z')
_x, _y, _z = x.asdummy(), y.asdummy(), z.asdummy()
if __name__ == '__main__':
parser = argparse.ArgumentParser(
description='Compute a Menger sponge.')
- parser.add_argument('-n', '--iterations', type=int, default=1,
- help='number of iterations (default: 1)')
+ parser.add_argument('-n', '--iterations', type=int, default=2,
+ help='number of iterations (default: 2)')
parser.add_argument('-c', '--cut', action='store_true', default=False,
help='cut the sponge')
args = parser.parse_args()
#!/usr/bin/env python3
+# This is an implementation of the algorithm described in
+#
+# [ACI10] C. Ancourt, F. Coelho and F. Irigoin, A modular static analysis
+# approach to affine loop invariants detection (2010), pp. 3 - 16, NSAD 2010.
+#
+# to compute the transitive closure of an affine transformer. A refined version
+# of this algorithm is implemented in PIPS.
+
from linpy import *
if __name__ == '__main__':
- i, iprime, j, jprime = symbols("i i' j j'")
- transformer = Transformer(Eq(iprime, i + 2) & Eq(jprime, j + 1),
- [i, j], [iprime, jprime])
+ i0, i, j0, j = symbols('i0 i j0 j')
+ transformer = Transformer(Eq(i, i0 + 2) & Eq(j, j0 + 1),
+ [i0, j0], [i, j])
print('T =', transformer.polyhedron)
print('T* =', transformer.star().polyhedron)
--- /dev/null
+#!/usr/bin/env python3
+
+# This program plots several 2D and 3D polyhedra on the same figure,
+# illustrating some of the possible plot options.
+
+import matplotlib.pyplot as plt
+
+from matplotlib import pylab
+from mpl_toolkits.mplot3d import Axes3D
+
+from linpy import *
+
+
+x, y, z = symbols('x y z')
+
+diam = Ge(y, x - 1) & Le(y, x + 1) & Ge(y, -x - 1) & Le(y, -x + 1)
+
+cham = Le(0, x, 3) & Le(0, y, 3) & Le(0, z, 3) & \
+ Le(z - 2, x, z + 2) & Le(1 - z, x, 5 - z) & \
+ Le(z - 2, y, z + 2) & Le(1 - z, y, 5 - z) & \
+ Le(y - 2, x, y + 2) & Le(1 - y, x, 5 - y)
+
+rhom = cham & \
+ Le(x + y + z, 7) & Ge(-2, -x - y - z) & \
+ Le(-1, x + y - z, 4) & Le(-1, x - y + z, 4) & Le(-1, -x + y + z, 4)
+
+cubo = Le(0, x, 5) & Le(0, y, 5) & Le(0, z, 5) & \
+ Le(x -4, y, x + 4) & Le(-x + 1, y, -x + 9) & \
+ Le(y -4, z, y + 4) & Le(-y + 1, z, -y + 9) & \
+ Le(z -4, x, z + 4) & Le(-z + 1, x, -z + 9) & \
+ Le(3, x + y + z, 12) & Le(-2, x - y + z, 7) & \
+ Le(-2, -x + y + z, 7) & Le(-2, x + y - z, 7)
+
+
+if __name__ == '__main__':
+
+ fig = plt.figure(facecolor='white')
+
+ diam_plot = fig.add_subplot(2, 2, 1, aspect='equal')
+ diam_plot.set_title('Diamond')
+ diam.plot(diam_plot, fill=True, edgecolor='red', facecolor='yellow')
+
+ cham_plot = fig.add_subplot(2, 2, 2, projection='3d', aspect='equal')
+ cham_plot.set_title('Chamfered cube')
+ cham.plot(cham_plot, facecolors=(1, 0, 0, 0.75))
+
+ rhom_plot = fig.add_subplot(2, 2, 3, projection='3d', aspect='equal')
+ rhom_plot.set_title('Rhombicuboctahedron')
+ rhom.plot(rhom_plot, facecolors=(0, 1, 0, 0.75))
+
+ cubo_plot = fig.add_subplot(2, 2, 4, projection='3d', aspect='equal')
+ cubo_plot.set_title('Truncated cuboctahedron')
+ cubo.plot(cubo_plot, facecolors=(0, 0, 1, 0.75))
+
+ pylab.show()
#!/usr/bin/env python3
-from linpy import *
-import matplotlib.pyplot as plt
-from matplotlib import pylab
-
-a, x, y, z = symbols('a x y z')
-
-sq1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
-sq2 = Le(2, x) & Le(x, 4) & Le(2, y) & Le(y, 4)
-sq3 = Le(0, x) & Le(x, 3) & Le(0, y) & Le(y, 3)
-sq4 = Le(1, x) & Le(x, 2) & Le(1, y) & Le(y, 2)
-sq5 = Le(1, x) & Le(x, 2) & Le(1, y)
-sq6 = Le(1, x) & Le(x, 2) & Le(1, y) & Le(y, 3)
-sq7 = Le(0, x) & Le(x, 2) & Le(0, y) & Eq(z, 2) & Le(a, 3)
-p = Le(2*x+1, y) & Le(-2*x-1, y) & Le(y, 1)
-
-universe = Polyhedron([])
-q = sq1 - sq2
-e = Empty
-
-print('sq1 =', sq1) #print correct square
-print('sq2 =', sq2) #print correct square
-print('sq3 =', sq3) #print correct square
-print('sq4 =', sq4) #print correct square
-print('universe =', universe) #print correct square
-print()
-print('¬sq1 =', ~sq1) #test complement
-print()
-print('sq1 + sq1 =', sq1 + sq2) #test addition
-print('sq1 + sq2 =', Polyhedron(sq1 + sq2)) #test addition
-print()
-print('universe + universe =', universe + universe)#test addition
-print('universe - universe =', universe - universe) #test subtraction
-print()
-print('sq2 - sq1 =', sq2 - sq1) #test subtraction
-print('sq2 - sq1 =', Polyhedron(sq2 - sq1)) #test subtraction
-print('sq1 - sq1 =', Polyhedron(sq1 - sq1)) #test subtraction
-print()
-print('sq1 ∩ sq2 =', sq1 & sq2) #test intersection
-print('sq1 ∪ sq2 =', sq1 | sq2) #test union
-print()
-print('sq1 ⊔ sq2 =', Polyhedron(sq1 | sq2)) # test convex union
-print()
-print('check if sq1 and sq2 disjoint:', sq1.isdisjoint(sq2)) #should return false
-print()
-print('sq1 disjoint:', sq1.disjoint()) #make disjoint
-print('sq2 disjoint:', sq2.disjoint()) #make disjoint
-print()
-print('is square 1 universe?:', sq1.isuniverse()) #test if square is universe
-print('is u universe?:', universe.isuniverse()) #test if square is universe
-print()
-print('is sq1 a subset of sq2?:', sq1.issubset(sq2)) #test issubset()
-print('is sq4 less than sq3?:', sq4.__lt__(sq3)) # test lt(), must be a strict subset
-print()
-print('lexographic min of sq1:', sq1.lexmin()) #test lexmin()
-print('lexographic max of sq1:', sq1.lexmax()) #test lexmin()
-print()
-print('lexographic min of sq2:', sq2.lexmin()) #test lexmax()
-print('lexographic max of sq2:', sq2.lexmax()) #test lexmax()
-print()
-print('Polyhedral hull of sq1 + sq2 is:', q.aspolyhedron()) #test polyhedral hull
-print()
-print('is sq1 bounded?', sq1.isbounded()) #bounded should return True
-print('is sq5 bounded?', sq5.isbounded()) #unbounded should return False
-print()
-print('sq6:', sq6)
-print('sample Polyhedron from sq6:', sq6.sample())
-print()
-print('sq7 with out constraints involving y and a', sq7.project([a, z, x, y]))
-print()
-print('the verticies for s are:', p.vertices())
-
-
-# plotting the intersection of two squares
-square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
-square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
-
-fig = plt.figure()
-plot = fig.add_subplot(1, 1, 1, aspect='equal')
-square1.plot(plot, facecolor='red', alpha=0.3)
-square2.plot(plot, facecolor='blue', alpha=0.3)
-
-squares = Polyhedron(square1 + square2)
-squares.plot(plot, facecolor='blue', alpha=0.3)
-
-pylab.show()
+# This is the code example used in the tutorial. It shows how to define and
+# manipulate polyhedra.
+
+import code
+
+
+class InteractiveConsole(code.InteractiveConsole):
+ def push(self, line=''):
+ if line:
+ print('>>>', line)
+ return super().push(line)
+ else:
+ print()
+
+
+if __name__ == '__main__':
+
+ shell = InteractiveConsole()
+
+ shell.push('from linpy import *')
+ shell.push("x, y = symbols('x y')")
+ shell.push()
+
+ shell.push('square1 = Le(0, x, 2) & Le(0, y, 2)')
+ shell.push('square1')
+ shell.push()
+
+ shell.push("square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')")
+ shell.push('square2')
+ shell.push()
+
+ shell.push('inter = square1.intersection(square2)')
+ shell.push('inter')
+ shell.push()
+
+ shell.push('hull = square1.convex_union(square2)')
+ shell.push('hull')
+ shell.push()
+
+ shell.push('square1.project([y])')
+ shell.push()
+
+ shell.push('inter <= square1')
+ shell.push('inter == Empty')
+ shell.push()
+
+ shell.push('union = square1 | square2')
+ shell.push('union')
+ shell.push('union <= hull')
+ shell.push()
+
+ shell.push('diff = square1 - square2')
+ shell.push('diff')
+ shell.push('~square1')
#!/usr/bin/env python3
+# In geometry, the tesseract is the four-dimensional analog of the cube; the
+# tesseract is to the cube as the cube is to the square. Just as the surface of
+# the cube consists of 6 square faces, the hypersurface of the tesseract
+# consists of 8 cubical cells.
+
from linpy import *
+
x, y, z, t = symbols('x y z t')
-tesseract = \
- Le(0, x) & Le(x, 1) & \
- Le(0, y) & Le(y, 1) & \
- Le(0, z) & Le(z, 1) & \
- Le(0, t) & Le(t, 1)
+tesseract = Le(0, x, 1) & Le(0, y, 1) & Le(0, z, 1) & Le(0, t, 1)
def faces(polyhedron):
for points in polyhedron.faces():
face = points[0].aspolyhedron()
- face = face.union(*[point.aspolyhedron() for point in points[1:]])
- face = face.aspolyhedron()
+ face = face.convex_union(*[point.aspolyhedron() for point in points[1:]])
yield face
-print('Faces of tesseract\n\n {}\n\nare:\n'.format(tesseract))
-for face in faces(tesseract):
- assert(len(face.vertices()) == 8)
- print(' {}'.format(face))
+if __name__ == '__main__':
+ print('Faces of tesseract\n\n {}\n\nare:\n'.format(tesseract))
+ for face in faces(tesseract):
+ assert(len(face.vertices()) == 8)
+ print(' {}'.format(face))
projects / linpy.git / commitdiff