# No duplicate points, regular grid.

nx = 5

ny = 4

x, y = np.meshgrid(np.linspace(0.0, 1.0, nx), np.linspace(0.0, 1.0, ny))

x = x.ravel()

y = y.ravel()

npoints = nx*ny

ntriangles = 2 * (nx-1) * (ny-1)

nedges = 3*nx*ny - 2*nx - 2*ny + 1

# Create delaunay triangulation.

triang = mtri.Triangulation(x, y)

# The tests in the remainder of this function should be passed by any

# triangulation that does not contain duplicate points.

# Points - floating point.

assert_array_almost_equal(triang.x, x)

assert_array_almost_equal(triang.y, y)

# Triangles - integers.

assert len(triang.triangles) == ntriangles

assert np.min(triang.triangles) == 0

assert np.max(triang.triangles) == npoints-1

# Edges - integers.

assert len(triang.edges) == nedges

assert np.min(triang.edges) == 0

assert np.max(triang.edges) == npoints-1

# Neighbors - integers.

# Check that neighbors calculated by C++ triangulation class are the same

# as those returned from delaunay routine.

neighbors = triang.neighbors

triang._neighbors = None

assert_array_equal(triang.neighbors, neighbors)

# Is each point used in at least one triangle?

# x[duplicate] == x[duplicate_of]

# y[duplicate] == y[duplicate_of]

npoints = 10

duplicate = 7

duplicate_of = 3

np.random.seed(23)

x = np.random.random((npoints))

y = np.random.random((npoints))

x[duplicate] = x[duplicate_of]

y[duplicate] = y[duplicate_of]

# Create delaunay triangulation.

triang = mtri.Triangulation(x, y)

# Duplicate points should be ignored, so the index of the duplicate points

# should not appear in any triangle.

assert_array_equal(np.unique(triang.triangles),

# Cannot triangulate points that are all in a straight line, but check

# that delaunay code fails gracefully.

x = np.linspace(0.0, 10.0, 11)

y = np.linspace(0.0, 10.0, 11)

with pytest.raises(RuntimeError):

mtri.Triangulation(x, y)

# Add an extra point not on the line and the triangulation is OK.

x = np.append(x, 2.0)

y = np.append(y, 8.0)

# Fails when mtri.Triangulation uses matplotlib.delaunay, works when using

# qhull.

tri_points = np.array([

[0.8660254037844384, -0.5000000000000004],

[0.7577722283113836, -0.5000000000000004],

[0.6495190528383288, -0.5000000000000003],

[0.5412658773652739, -0.5000000000000003],

[0.811898816047911, -0.40625000000000044],

[0.7036456405748561, -0.4062500000000004],

[0.5953924651018013, -0.40625000000000033]])

test_points = np.asarray([

[0.58, -0.46],

[0.65, -0.46],

[0.65, -0.42],

[0.7, -0.48],

[0.7, -0.44],

[0.75, -0.44],

[0.8, -0.48]])

# Utility function that indicates if a triangle defined by 3 points

# (xtri, ytri) contains the test point xy. Avoid calling with a point that

# lies on or very near to an edge of the triangle.

def tri_contains_point(xtri, ytri, xy):

tri_points = np.vstack((xtri, ytri)).T

return Path(tri_points).contains_point(xy)

# Utility function that returns how many triangles of the specified

# triangulation contain the test point xy. Avoid calling with a point that

# lies on or very near to an edge of any triangle in the triangulation.

def tris_contain_point(triang, xy):

count = 0

for tri in triang.triangles:

if tri_contains_point(triang.x[tri], triang.y[tri], xy):

count += 1

return count

# Using matplotlib.delaunay, an invalid triangulation is created with

# overlapping triangles; qhull is OK.

triang = mtri.Triangulation(tri_points[:, 0], tri_points[:, 1])

for test_point in test_points:

assert tris_contain_point(triang, test_point) == 1

# If ignore the first point of tri_points, matplotlib.delaunay throws a

# KeyError when calculating the convex hull; qhull is OK.

x = np.asarray([0, 0.5, 1, 0, 0.5, 1, 0, 0.5, 1, 0.75])

y = np.asarray([0, 0, 0, 0.5, 0.5, 0.5, 1, 1, 1, 0.75])

triangles = np.asarray([

[0, 1, 3], [1, 4, 3],

[1, 2, 4], [2, 5, 4],

[3, 4, 6], [4, 7, 6],

[4, 5, 9], [7, 4, 9], [8, 7, 9], [5, 8, 9]])

# Triangulation with same number of points and triangles.

triang = mtri.Triangulation(x, y, triangles)

Cpoints = x + 0.5*y

xmid = x[triang.triangles].mean(axis=1)

ymid = y[triang.triangles].mean(axis=1)

Cfaces = 0.5*xmid + ymid

plt.subplot(121)

plt.tripcolor(triang, Cpoints, edgecolors='k')

plt.title('point colors')

plt.subplot(122)

plt.tripcolor(triang, facecolors=Cfaces, edgecolors='k')

# Test that Triangulation does not modify triangles array passed to it.

triangles = np.array([[3, 2, 0], [3, 1, 0]], dtype=np.int32)

points = np.array([(0, 0), (0, 1.1), (1, 0), (1, 1)])

old_triangles = triangles.copy()

tri = mtri.Triangulation(points[:, 0], points[:, 1], triangles)

edges = tri.edges

# Test points within triangles of masked triangulation.

x, y = np.meshgrid(np.arange(4), np.arange(4))

x = x.ravel()

y = y.ravel()

triangles = [[0, 1, 4], [1, 5, 4], [1, 2, 5], [2, 6, 5], [2, 3, 6],

[3, 7, 6], [4, 5, 8], [5, 9, 8], [5, 6, 9], [6, 10, 9],

[6, 7, 10], [7, 11, 10], [8, 9, 12], [9, 13, 12], [9, 10, 13],

[10, 14, 13], [10, 11, 14], [11, 15, 14]]

mask = np.zeros(len(triangles))

mask[8:10] = 1

triang = mtri.Triangulation(x, y, triangles, mask)

trifinder = triang.get_trifinder()

xs = [0.25, 1.25, 2.25, 3.25]

ys = [0.25, 1.25, 2.25, 3.25]

xs, ys = np.meshgrid(xs, ys)

xs = xs.ravel()

ys = ys.ravel()

tris = trifinder(xs, ys)

assert_array_equal(tris, [0, 2, 4, -1, 6, -1, 10, -1,

12, 14, 16, -1, -1, -1, -1, -1])

tris = trifinder(xs-0.5, ys-0.5)

assert_array_equal(tris, [-1, -1, -1, -1, -1, 1, 3, 5,

-1, 7, -1, 11, -1, 13, 15, 17])

# Test points exactly on boundary edges of masked triangulation.

xs = [0.5, 1.5, 2.5, 0.5, 1.5, 2.5, 1.5, 1.5, 0.0, 1.0, 2.0, 3.0]

ys = [0.0, 0.0, 0.0, 3.0, 3.0, 3.0, 1.0, 2.0, 1.5, 1.5, 1.5, 1.5]

tris = trifinder(xs, ys)

assert_array_equal(tris, [0, 2, 4, 13, 15, 17, 3, 14, 6, 7, 10, 11])

# Test points exactly on boundary corners of masked triangulation.

xs = [0.0, 3.0]

ys = [0.0, 3.0]

tris = trifinder(xs, ys)

assert_array_equal(tris, [0, 17])

#

# Test triangles with horizontal colinear points. These are not valid

# triangulations, but we try to deal with the simplest violations.

#

# If +ve, triangulation is OK, if -ve triangulation invalid,

# if zero have colinear points but should pass tests anyway.

delta = 0.0

x = [1.5, 0, 1, 2, 3, 1.5, 1.5]

y = [-1, 0, 0, 0, 0, delta, 1]

triangles = [[0, 2, 1], [0, 3, 2], [0, 4, 3], [1, 2, 5], [2, 3, 5],

[3, 4, 5], [1, 5, 6], [4, 6, 5]]

triang = mtri.Triangulation(x, y, triangles)

trifinder = triang.get_trifinder()

xs = [-0.1, 0.4, 0.9, 1.4, 1.9, 2.4, 2.9]

ys = [-0.1, 0.1]

xs, ys = np.meshgrid(xs, ys)

tris = trifinder(xs, ys)

assert_array_equal(tris, [[-1, 0, 0, 1, 1, 2, -1],

[-1, 6, 6, 6, 7, 7, -1]])

#

# Test triangles with vertical colinear points. These are not valid

# triangulations, but we try to deal with the simplest violations.

#

# If +ve, triangulation is OK, if -ve triangulation invalid,

# if zero have colinear points but should pass tests anyway.

delta = 0.0

x = [-1, -delta, 0, 0, 0, 0, 1]

y = [1.5, 1.5, 0, 1, 2, 3, 1.5]

triangles = [[0, 1, 2], [0, 1, 5], [1, 2, 3], [1, 3, 4], [1, 4, 5],

[2, 6, 3], [3, 6, 4], [4, 6, 5]]

triang = mtri.Triangulation(x, y, triangles)

trifinder = triang.get_trifinder()

xs = [-0.1, 0.1]

ys = [-0.1, 0.4, 0.9, 1.4, 1.9, 2.4, 2.9]

xs, ys = np.meshgrid(xs, ys)

tris = trifinder(xs, ys)

assert_array_equal(tris, [[-1, -1], [0, 5], [0, 5], [0, 6], [1, 6], [1, 7],

[-1, -1]])

# Test that changing triangulation by setting a mask causes the trifinder

# to be reinitialised.

x = [0, 1, 0, 1]

y = [0, 0, 1, 1]

triangles = [[0, 1, 2], [1, 3, 2]]

triang = mtri.Triangulation(x, y, triangles)

trifinder = triang.get_trifinder()

xs = [-0.2, 0.2, 0.8, 1.2]

ys = [0.5, 0.5, 0.5, 0.5]

tris = trifinder(xs, ys)

assert_array_equal(tris, [-1, 0, 1, -1])

triang.set_mask([1, 0])

assert trifinder == triang.get_trifinder()

tris = trifinder(xs, ys)

# Test points within triangles of masked triangulation.

x, y = np.meshgrid(np.arange(4), np.arange(4))

x = x.ravel()

y = y.ravel()

z = 1.23*x - 4.79*y

triangles = [[0, 1, 4], [1, 5, 4], [1, 2, 5], [2, 6, 5], [2, 3, 6],

[3, 7, 6], [4, 5, 8], [5, 9, 8], [5, 6, 9], [6, 10, 9],

[6, 7, 10], [7, 11, 10], [8, 9, 12], [9, 13, 12], [9, 10, 13],

[10, 14, 13], [10, 11, 14], [11, 15, 14]]

mask = np.zeros(len(triangles))

mask[8:10] = 1

triang = mtri.Triangulation(x, y, triangles, mask)

linear_interp = mtri.LinearTriInterpolator(triang, z)

cubic_min_E = mtri.CubicTriInterpolator(triang, z)

cubic_geom = mtri.CubicTriInterpolator(triang, z, kind='geom')

xs = np.linspace(0.25, 2.75, 6)

ys = [0.25, 0.75, 2.25, 2.75]

xs, ys = np.meshgrid(xs, ys) # Testing arrays with array.ndim = 2

for interp in (linear_interp, cubic_min_E, cubic_geom):

zs = interp(xs, ys)

assert_array_almost_equal(zs, (1.23*xs - 4.79*ys))

# Test points outside triangulation.

xs = [-0.25, 1.25, 1.75, 3.25]

ys = xs

xs, ys = np.meshgrid(xs, ys)

for interp in (linear_interp, cubic_min_E, cubic_geom):

zs = linear_interp(xs, ys)

assert_array_equal(zs.mask, [[True]*4]*4)

# Test mixed configuration (outside / inside).

xs = np.linspace(0.25, 1.75, 6)

ys = [0.25, 0.75, 1.25, 1.75]

xs, ys = np.meshgrid(xs, ys)

for interp in (linear_interp, cubic_min_E, cubic_geom):

zs = interp(xs, ys)

matest.assert_array_almost_equal(zs, (1.23*xs - 4.79*ys))

mask = (xs >= 1) * (xs <= 2) * (ys >= 1) * (ys <= 2)

assert_array_equal(zs.mask, mask)

# 2nd order patch test: on a grid with an 'arbitrary shaped' triangle,

# patch test shall be exact for quadratic functions and cubic

# interpolator if *kind* = user

(a, b, c) = (1.23, -4.79, 0.6)

def quad(x, y):

return a*(x-0.5)**2 + b*(y-0.5)**2 + c*x*y

def gradient_quad(x, y):

return (2*a*(x-0.5) + c*y, 2*b*(y-0.5) + c*x)

x = np.array([0.2, 0.33367, 0.669, 0., 1., 1., 0.])

y = np.array([0.3, 0.80755, 0.4335, 0., 0., 1., 1.])

triangles = np.array([[0, 1, 2], [3, 0, 4], [4, 0, 2], [4, 2, 5],

[1, 5, 2], [6, 5, 1], [6, 1, 0], [6, 0, 3]])

triang = mtri.Triangulation(x, y, triangles)

z = quad(x, y)

dz = gradient_quad(x, y)

# test points for 2nd order patch test

xs = np.linspace(0., 1., 5)

ys = np.linspace(0., 1., 5)

xs, ys = np.meshgrid(xs, ys)

cubic_user = mtri.CubicTriInterpolator(triang, z, kind='user', dz=dz)

interp_zs = cubic_user(xs, ys)

assert_array_almost_equal(interp_zs, quad(xs, ys))

(interp_dzsdx, interp_dzsdy) = cubic_user.gradient(x, y)

(dzsdx, dzsdy) = gradient_quad(x, y)

assert_array_almost_equal(interp_dzsdx, dzsdx)

assert_array_almost_equal(interp_dzsdy, dzsdy)

# Cubic improvement: cubic interpolation shall perform better than linear

# on a sufficiently dense mesh for a quadratic function.

n = 11

x, y = np.meshgrid(np.linspace(0., 1., n+1), np.linspace(0., 1., n+1))

x = x.ravel()

y = y.ravel()

z = quad(x, y)

triang = mtri.Triangulation(x, y, triangles=meshgrid_triangles(n+1))

xs, ys = np.meshgrid(np.linspace(0.1, 0.9, 5), np.linspace(0.1, 0.9, 5))

xs = xs.ravel()

ys = ys.ravel()

linear_interp = mtri.LinearTriInterpolator(triang, z)

cubic_min_E = mtri.CubicTriInterpolator(triang, z)

cubic_geom = mtri.CubicTriInterpolator(triang, z, kind='geom')

zs = quad(xs, ys)

diff_lin = np.abs(linear_interp(xs, ys) - zs)

for interp in (cubic_min_E, cubic_geom):

diff_cubic = np.abs(interp(xs, ys) - zs)

assert(np.max(diff_lin) >= 10.*np.max(diff_cubic))

assert(np.dot(diff_lin, diff_lin) >=

# Below the 4 tests which demonstrate C1 continuity of the

# TriCubicInterpolator (testing the cubic shape functions on arbitrary

# triangle):

#

# 1) Testing continuity of function & derivatives at corner for all 9

# shape functions. Testing also function values at same location.

# 2) Testing C1 continuity along each edge (as gradient is polynomial of

# 2nd order, it is sufficient to test at the middle).

# 3) Testing C1 continuity at triangle barycenter (where the 3 subtriangles

# meet)

# 4) Testing C1 continuity at median 1/3 points (midside between 2

# subtriangles)

# Utility test function check_continuity

def check_continuity(interpolator, loc, values=None):

"""

n_star = 24 # Number of continuity points in a boundary of loc

epsilon = 1.e-10 # Distance for loc boundary

k = 100. # Continuity coefficient

(loc_x, loc_y) = loc

star_x = loc_x + epsilon*np.cos(np.linspace(0., 2*np.pi, n_star))

star_y = loc_y + epsilon*np.sin(np.linspace(0., 2*np.pi, n_star))

z = interpolator([loc_x], [loc_y])[0]

(dzx, dzy) = interpolator.gradient([loc_x], [loc_y])

if values is not None:

assert_array_almost_equal(z, values[0])

assert_array_almost_equal(dzx[0], values[1])

assert_array_almost_equal(dzy[0], values[2])

diff_z = interpolator(star_x, star_y) - z

(tab_dzx, tab_dzy) = interpolator.gradient(star_x, star_y)

diff_dzx = tab_dzx - dzx

diff_dzy = tab_dzy - dzy

assert_array_less(diff_z, epsilon*k)

assert_array_less(diff_dzx, epsilon*k)

assert_array_less(diff_dzy, epsilon*k)

# Drawing arbitrary triangle (a, b, c) inside a unit square.

(ax, ay) = (0.2, 0.3)

(bx, by) = (0.33367, 0.80755)

(cx, cy) = (0.669, 0.4335)

x = np.array([ax, bx, cx, 0., 1., 1., 0.])

y = np.array([ay, by, cy, 0., 0., 1., 1.])

triangles = np.array([[0, 1, 2], [3, 0, 4], [4, 0, 2], [4, 2, 5],

[1, 5, 2], [6, 5, 1], [6, 1, 0], [6, 0, 3]])

triang = mtri.Triangulation(x, y, triangles)

for idof in range(9):

z = np.zeros(7, dtype=np.float64)

dzx = np.zeros(7, dtype=np.float64)

dzy = np.zeros(7, dtype=np.float64)

values = np.zeros([3, 3], dtype=np.float64)

case = idof//3

values[case, idof % 3] = 1.0

if case == 0:

z[idof] = 1.0

elif case == 1:

dzx[idof % 3] = 1.0

elif case == 2:

dzy[idof % 3] = 1.0

interp = mtri.CubicTriInterpolator(triang, z, kind='user',

dz=(dzx, dzy))

# Test 1) Checking values and continuity at nodes

check_continuity(interp, (ax, ay), values[:, 0])

check_continuity(interp, (bx, by), values[:, 1])

check_continuity(interp, (cx, cy), values[:, 2])

# Test 2) Checking continuity at midside nodes

check_continuity(interp, ((ax+bx)*0.5, (ay+by)*0.5))

check_continuity(interp, ((ax+cx)*0.5, (ay+cy)*0.5))

check_continuity(interp, ((cx+bx)*0.5, (cy+by)*0.5))

# Test 3) Checking continuity at barycenter

check_continuity(interp, ((ax+bx+cx)/3., (ay+by+cy)/3.))

# Test 4) Checking continuity at median 1/3-point

check_continuity(interp, ((4.*ax+bx+cx)/6., (4.*ay+by+cy)/6.))

check_continuity(interp, ((ax+4.*bx+cx)/6., (ay+4.*by+cy)/6.))

# Now 3 basic tests of the Sparse CG solver, used for

# TriCubicInterpolator with *kind* = 'min_E'

# 1) A commonly used test involves a 2d Poisson matrix.

def poisson_sparse_matrix(n, m):

"""

l = m*n

rows = np.concatenate([

np.arange(l, dtype=np.int32),

np.arange(l-1, dtype=np.int32), np.arange(1, l, dtype=np.int32),

np.arange(l-n, dtype=np.int32), np.arange(n, l, dtype=np.int32)])

cols = np.concatenate([

np.arange(l, dtype=np.int32),

np.arange(1, l, dtype=np.int32), np.arange(l-1, dtype=np.int32),

np.arange(n, l, dtype=np.int32), np.arange(l-n, dtype=np.int32)])

vals = np.concatenate([

4*np.ones(l, dtype=np.float64),

-np.ones(l-1, dtype=np.float64), -np.ones(l-1, dtype=np.float64),

-np.ones(l-n, dtype=np.float64), -np.ones(l-n, dtype=np.float64)])

# In fact +1 and -1 diags have some zeros

vals[l:2*l-1][m-1::m] = 0.

vals[2*l-1:3*l-2][m-1::m] = 0.

return vals, rows, cols, (n*m, n*m)

# Instantiating a sparse Poisson matrix of size 48 x 48:

(n, m) = (12, 4)

mat = mtri.triinterpolate._Sparse_Matrix_coo(*poisson_sparse_matrix(n, m))

mat.compress_csc()

mat_dense = mat.to_dense()

# Testing a sparse solve for all 48 basis vector

for itest in range(n*m):

b = np.zeros(n*m, dtype=np.float64)

b[itest] = 1.

x, _ = mtri.triinterpolate._cg(A=mat, b=b, x0=np.zeros(n*m),

tol=1.e-10)

assert_array_almost_equal(np.dot(mat_dense, x), b)

# 2) Same matrix with inserting 2 rows - cols with null diag terms

# (but still linked with the rest of the matrix by extra-diag terms)

(i_zero, j_zero) = (12, 49)

vals, rows, cols, _ = poisson_sparse_matrix(n, m)

rows = rows + 1*(rows >= i_zero) + 1*(rows >= j_zero)

cols = cols + 1*(cols >= i_zero) + 1*(cols >= j_zero)

# adding extra-diag terms

rows = np.concatenate([rows, [i_zero, i_zero-1, j_zero, j_zero-1]])

cols = np.concatenate([cols, [i_zero-1, i_zero, j_zero-1, j_zero]])

vals = np.concatenate([vals, [1., 1., 1., 1.]])

mat = mtri.triinterpolate._Sparse_Matrix_coo(vals, rows, cols,

(n*m + 2, n*m + 2))

mat.compress_csc()

mat_dense = mat.to_dense()

# Testing a sparse solve for all 50 basis vec

for itest in range(n*m + 2):

b = np.zeros(n*m + 2, dtype=np.float64)

b[itest] = 1.

x, _ = mtri.triinterpolate._cg(A=mat, b=b, x0=np.ones(n*m + 2),

tol=1.e-10)

assert_array_almost_equal(np.dot(mat_dense, x), b)

# 3) Now a simple test that summation of duplicate (i.e. with same rows,

# same cols) entries occurs when compressed.

vals = np.ones(17, dtype=np.float64)

rows = np.array([0, 1, 2, 0, 0, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1],

dtype=np.int32)

cols = np.array([0, 1, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2],

dtype=np.int32)

dim = (3, 3)

mat = mtri.triinterpolate._Sparse_Matrix_coo(vals, rows, cols, dim)

mat.compress_csc()

mat_dense = mat.to_dense()

assert_array_almost_equal(mat_dense, np.array([

# Tests to check computation of weights for _DOF_estimator_geom:

# The weight sum per triangle can be 1. (in case all angles < 90 degrees)

# or (2*w_i) where w_i = 1-alpha_i/np.pi is the weight of apex i ; alpha_i

# is the apex angle > 90 degrees.

(ax, ay) = (0., 1.687)

x = np.array([ax, 0.5*ax, 0., 1.])

y = np.array([ay, -ay, 0., 0.])

z = np.zeros(4, dtype=np.float64)

triangles = [[0, 2, 3], [1, 3, 2]]

sum_w = np.zeros([4, 2]) # 4 possibilities ; 2 triangles

for theta in np.linspace(0., 2*np.pi, 14): # rotating the figure...

x_rot = np.cos(theta)*x + np.sin(theta)*y

y_rot = -np.sin(theta)*x + np.cos(theta)*y

triang = mtri.Triangulation(x_rot, y_rot, triangles)

cubic_geom = mtri.CubicTriInterpolator(triang, z, kind='geom')

dof_estimator = mtri.triinterpolate._DOF_estimator_geom(cubic_geom)

weights = dof_estimator.compute_geom_weights()

# Testing for the 4 possibilities...

sum_w[0, :] = np.sum(weights, 1) - 1

for itri in range(3):

sum_w[itri+1, :] = np.sum(weights, 1) - 2*weights[:, itri]

assert_array_almost_equal(np.min(np.abs(sum_w), axis=0),

# Tests interpolating inside a triangulation with horizontal colinear

# points (refer also to the tests :func:`test_trifinder` ).

#

# These are not valid triangulations, but we try to deal with the

# simplest violations (i. e. those handled by default TriFinder).

#

# Note that the LinearTriInterpolator and the CubicTriInterpolator with

# kind='min_E' or 'geom' still pass a linear patch test.

# We also test interpolation inside a flat triangle, by forcing

# *tri_index* in a call to :meth:`_interpolate_multikeys`.

# If +ve, triangulation is OK, if -ve triangulation invalid,

# if zero have colinear points but should pass tests anyway.

delta = 0.

x0 = np.array([1.5, 0, 1, 2, 3, 1.5, 1.5])

y0 = np.array([-1, 0, 0, 0, 0, delta, 1])

# We test different affine transformations of the initial figure ; to

# avoid issues related to round-off errors we only use integer

# coefficients (otherwise the Triangulation might become invalid even with

# delta == 0).

transformations = [[1, 0], [0, 1], [1, 1], [1, 2], [-2, -1], [-2, 1]]

for transformation in transformations:

x_rot = transformation[0]*x0 + transformation[1]*y0

y_rot = -transformation[1]*x0 + transformation[0]*y0

(x, y) = (x_rot, y_rot)

z = 1.23*x - 4.79*y

triangles = [[0, 2, 1], [0, 3, 2], [0, 4, 3], [1, 2, 5], [2, 3, 5],

[3, 4, 5], [1, 5, 6], [4, 6, 5]]

triang = mtri.Triangulation(x, y, triangles)

xs = np.linspace(np.min(triang.x), np.max(triang.x), 20)

ys = np.linspace(np.min(triang.y), np.max(triang.y), 20)

xs, ys = np.meshgrid(xs, ys)

xs = xs.ravel()

ys = ys.ravel()

mask_out = (triang.get_trifinder()(xs, ys) == -1)

zs_target = np.ma.array(1.23*xs - 4.79*ys, mask=mask_out)

linear_interp = mtri.LinearTriInterpolator(triang, z)

cubic_min_E = mtri.CubicTriInterpolator(triang, z)

cubic_geom = mtri.CubicTriInterpolator(triang, z, kind='geom')

for interp in (linear_interp, cubic_min_E, cubic_geom):

zs = interp(xs, ys)

assert_array_almost_equal(zs_target, zs)

# Testing interpolation inside the flat triangle number 4: [2, 3, 5]

# by imposing *tri_index* in a call to :meth:`_interpolate_multikeys`

itri = 4

pt1 = triang.triangles[itri, 0]

pt2 = triang.triangles[itri, 1]

xs = np.linspace(triang.x[pt1], triang.x[pt2], 10)

ys = np.linspace(triang.y[pt1], triang.y[pt2], 10)

zs_target = 1.23*xs - 4.79*ys

for interp in (linear_interp, cubic_min_E, cubic_geom):

zs, = interp._interpolate_multikeys(

xs, ys, tri_index=itri*np.ones(10, dtype=np.int32))

# 1) Testing that the interpolation scheme is invariant by rotation of the

# whole figure.

# Note: This test is non-trivial for a CubicTriInterpolator with

# kind='min_E'. It does fail for a non-isotropic stiffness matrix E of

# :class:`_ReducedHCT_Element` (tested with E=np.diag([1., 1., 1.])), and

# provides a good test for :meth:`get_Kff_and_Ff`of the same class.

#

# 2) Also testing that the interpolation scheme is invariant by expansion

# of the whole figure along one axis.

n_angles = 20

n_radii = 10

min_radius = 0.15

def z(x, y):

r1 = np.sqrt((0.5-x)**2 + (0.5-y)**2)

theta1 = np.arctan2(0.5-x, 0.5-y)

r2 = np.sqrt((-x-0.2)**2 + (-y-0.2)**2)

theta2 = np.arctan2(-x-0.2, -y-0.2)

z = -(2*(np.exp((r1/10)**2)-1)*30. * np.cos(7.*theta1) +

(np.exp((r2/10)**2)-1)*30. * np.cos(11.*theta2) +

0.7*(x**2 + y**2))

return (np.max(z)-z)/(np.max(z)-np.min(z))

# First create the x and y coordinates of the points.

radii = np.linspace(min_radius, 0.95, n_radii)

angles = np.linspace(0 + n_angles, 2*np.pi + n_angles,

n_angles, endpoint=False)

angles = np.repeat(angles[..., np.newaxis], n_radii, axis=1)

angles[:, 1::2] += np.pi/n_angles

x0 = (radii*np.cos(angles)).flatten()

y0 = (radii*np.sin(angles)).flatten()

triang0 = mtri.Triangulation(x0, y0) # Delaunay triangulation

z0 = z(x0, y0)

# Then create the test points

xs0 = np.linspace(-1., 1., 23)

ys0 = np.linspace(-1., 1., 23)

xs0, ys0 = np.meshgrid(xs0, ys0)

xs0 = xs0.ravel()

ys0 = ys0.ravel()

interp_z0 = {}

for i_angle in range(2):

# Rotating everything

theta = 2*np.pi / n_angles * i_angle

x = np.cos(theta)*x0 + np.sin(theta)*y0

y = -np.sin(theta)*x0 + np.cos(theta)*y0

xs = np.cos(theta)*xs0 + np.sin(theta)*ys0

ys = -np.sin(theta)*xs0 + np.cos(theta)*ys0

triang = mtri.Triangulation(x, y, triang0.triangles)

linear_interp = mtri.LinearTriInterpolator(triang, z0)

cubic_min_E = mtri.CubicTriInterpolator(triang, z0)

cubic_geom = mtri.CubicTriInterpolator(triang, z0, kind='geom')

dic_interp = {'lin': linear_interp,

'min_E': cubic_min_E,

'geom': cubic_geom}

# Testing that the interpolation is invariant by rotation...

for interp_key in ['lin', 'min_E', 'geom']:

interp = dic_interp[interp_key]

if i_angle == 0:

interp_z0[interp_key] = interp(xs0, ys0) # storage

else:

interpz = interp(xs, ys)

matest.assert_array_almost_equal(interpz,

interp_z0[interp_key])

scale_factor = 987654.3210

for scaled_axis in ('x', 'y'):

# Scaling everything (expansion along scaled_axis)

if scaled_axis == 'x':

x = scale_factor * x0

y = y0

xs = scale_factor * xs0

ys = ys0

else:

x = x0

y = scale_factor * y0

xs = xs0

ys = scale_factor * ys0

triang = mtri.Triangulation(x, y, triang0.triangles)

linear_interp = mtri.LinearTriInterpolator(triang, z0)

cubic_min_E = mtri.CubicTriInterpolator(triang, z0)

cubic_geom = mtri.CubicTriInterpolator(triang, z0, kind='geom')

dic_interp = {'lin': linear_interp,

'min_E': cubic_min_E,

'geom': cubic_geom}

# Testing that the interpolation is invariant by expansion along

# 1 axis...

for interp_key in ['lin', 'min_E', 'geom']:

interpz = dic_interp[interp_key](xs, ys)

# Image comparison based on example tricontour_smooth_user.

n_angles = 20

n_radii = 10

min_radius = 0.15

def z(x, y):

r1 = np.sqrt((0.5-x)**2 + (0.5-y)**2)

theta1 = np.arctan2(0.5-x, 0.5-y)

r2 = np.sqrt((-x-0.2)**2 + (-y-0.2)**2)

theta2 = np.arctan2(-x-0.2, -y-0.2)

z = -(2*(np.exp((r1/10)**2)-1)*30. * np.cos(7.*theta1) +

(np.exp((r2/10)**2)-1)*30. * np.cos(11.*theta2) +

0.7*(x**2 + y**2))

return (np.max(z)-z)/(np.max(z)-np.min(z))

# First create the x and y coordinates of the points.

radii = np.linspace(min_radius, 0.95, n_radii)

angles = np.linspace(0 + n_angles, 2*np.pi + n_angles,

n_angles, endpoint=False)

angles = np.repeat(angles[..., np.newaxis], n_radii, axis=1)

angles[:, 1::2] += np.pi/n_angles

x0 = (radii*np.cos(angles)).flatten()

y0 = (radii*np.sin(angles)).flatten()

triang0 = mtri.Triangulation(x0, y0) # Delaunay triangulation

z0 = z(x0, y0)

xmid = x0[triang0.triangles].mean(axis=1)

ymid = y0[triang0.triangles].mean(axis=1)

mask = np.where(xmid*xmid + ymid*ymid < min_radius*min_radius, 1, 0)

triang0.set_mask(mask)

# Then the plot

refiner = mtri.UniformTriRefiner(triang0)

tri_refi, z_test_refi = refiner.refine_field(z0, subdiv=4)

levels = np.arange(0., 1., 0.025)

plt.triplot(triang0, lw=0.5, color='0.5')

# Image comparison based on example trigradient_demo.

def dipole_potential(x, y):

""" An electric dipole potential V """

r_sq = x**2 + y**2

theta = np.arctan2(y, x)

z = np.cos(theta)/r_sq

return (np.max(z)-z) / (np.max(z)-np.min(z))

# Creating a Triangulation

n_angles = 30

n_radii = 10

min_radius = 0.2

radii = np.linspace(min_radius, 0.95, n_radii)

angles = np.linspace(0, 2*np.pi, n_angles, endpoint=False)

angles = np.repeat(angles[..., np.newaxis], n_radii, axis=1)