def example2():
"""
Plot the B-Splines bases of a given extended nodal partition
and order.
"""
p = np.array([0., 0., 0., 1., 2., 3., 4., 4., 5., 5., 5.])
k = 3
for base in fn.bSplineBases(p, k):
fn.drawVertexes(base)
def test1():
V = np.array([[0.5,0.5], [0.4,1.], [1.,1.2], [0.9,0.], [2., 0.1], [1.5, 0.6]])
k = 2
p = np.array([0., 0., 1./5., 2./5., 3./5., 4./5., 1., 1.])
#k = 3
#p = np.array([0., 0., 0., 1./4., 1./2., 3./4., 1., 1., 1.])
#print('pippo1 ' + str(fn.bSplineBaseFactory(p,k)(5,1)(0.9)))
#print('pippo2 ' + str(fn.bSplineBaseFactory(p,k)(5,1)(1)))
#return
for i in range(6):
N = fn.bSplineBaseFactory(p,k)(i)
P = np.empty((0,2), dtype=float);
for t in np.arange(0.,1.01,0.01):
P = np.vstack((P, np.array([t,N(t)])))
fn.drawVertexes(P)
def exerB_1():
"""
Given a fixed set of control vertexes, represent the B-Spline
curves for different nodal partitions and orders.
"""
V = np.array([[0.5,0.5], [0.4,1.], [1.,1.2], [0.9,0.], [2., 0.1], [1.5, 0.6]])
ks = [2, 3, 4, 5, 6]
ps = [
np.array([0., 0., 1./5., 2./5., 3./5., 4./5., 1., 1.]),
np.array([0., 0., 0., 1./4., 1./2., 3./4., 1., 1., 1.]),
np.array([0., 0., 0., 0., 1./3., 2./3., 1., 1., 1., 1.]),
np.array([0., 0., 0., 0., 0., 1./2., 1., 1., 1., 1., 1.]),
np.array([0., 0., 0., 0., 0., 0., 1., 1., 1., 1., 1., 1.]),
]
for p,k in zip(ps, ks):
fn.drawVertexes(fn.bSpline(V, k, p))
fn.drawVertexes(V, 'o')
def example3():
"""
Plot a B-Spline curve using a clamped and not clamped partition.
"""
V = np.array([[-0.,0.], [-0.,6.], [-1.,5.], [-3.,8.], [-1.,14.],[2.,14.],[4.,8.], [2.,5.], [1.,6.], [1.,0.]])
k = 4
pc = np.array([0., 0., 0., 0., 1./7., 2./7., 3./7., 4./7., 5./7., 6./7., 1., 1., 1., 1.])
pnc = np.array([-3./7., -2./7., -1./7., 0., 1./7., 2./7., 3./7., 4./7., 5./7., 6./7., 1., 8./7., 9./7., 10./7.])
fn.drawVertexes(V)
fn.drawVertexes(V,'o')
fn.drawVertexes(fn.bSpline(V, k, pc))
fn.drawVertexes(fn.bSpline(V, k, pnc))
def exer1_2():
"""
Draw a Bezier curve using De Casteljau algorithm.
"""
V = np.array([[0,0,0],[1,2,0],[3,2,3],[2,6,2]])
fn.drawVertexes(fn.bezier(V))
fn.drawVertexes(V)
fn.drawVertexes(V,'o')
def exer2_4():
"""
Given the control vertexes:
(1,0), (a,a), (0,1)
for different values of a find the weigths that
represents an arc of circumference.
"""
for a in [0.7, 1., 2., 3.]:
V = np.array([[1,0],[a,a],[0,1]])
W = np.array([1,fn.findArcWeigth(a),1])
centre, radius = fn.findCircle(a)
fn.drawVertexes(fn.circum(centre, radius))
fn.drawVertexes(V)
fn.drawVertexes(V,'o')
fn.drawVertexes(fn.bezierRational(V,W)[0])
def exer2_1():
"""
Plot a rational Bezier curve converting the control vertexes
in polinomial and using the same algorithms of the polinomial.
Test with two different weight sets.
"""
V = np.array([[0,0],[1,2],[3,1]])
W1 = np.array([1,1,1])
W2 = np.array([1,2,1])
fn.drawVertexes(V)
fn.drawVertexes(V,'o')
P1 = fn.bezierRational(V,W1)[0]
P2 = fn.bezierRational(V,W2)[0]
fn.drawVertexes(P1)
fn.drawVertexes(P2)
def exerB_6b():
"""
Plot a closed B-Spline curve, using a close control polygon.
"""
V = np.array([[0.,0.], [0.,3.], [1.,2.], [2., 3.], [2.,0.], [1.,1.], [0.,0.]])
k = 4
fn.drawVertexes(V, 'o')
fn.drawVertexes(V)
fn.drawVertexes(fn.bSplineClosed(V, k))
def example4():
"""
Plot a closed B-Spline curve.
"""
V = np.array([[0.,0.], [0.,6.], [-1.,5.], [-3.,8.], [-1.,14.],[2.,14.],[4.,8.], [2.,5.], [1.,6.], [1.,0.]])
k = 4
fn.drawVertexes(V, 'o')
fn.drawVertexes(V)
fn.drawVertexes(fn.bSplineClosed(V, k))
def example5():
"""
Plot a 3 dimensional B-Spline curve.
"""
V = np.array([[0.,0.,0.], [0.,6.,1.], [-1.,5.,2.], [-3.,8.,3.], [-1.,14.,4.],[2.,14.,5.],[4.,8.,6.], [2.,5.,7.], [1.,6.,8.], [1.,0.,9.]])
k = 4
p = np.array([0.,0.,0.,0.,1./7.,2./7.,3./7.,4./7.,5./7.,6./7.,1.,1.,1.,1.])
fn.drawVertexes(V, 'o')
fn.drawVertexes(V)
fn.drawVertexes(fn.bSpline(V, k, p))
def exer2_3():
"""
Represents the first quadrant of a circumference using
the rational Bezier curves with control vertexes:
(1,0), (1,1), (0,1)
and weights:
1, sqrt(2)/2, 1.
"""
V = np.array([[1,0],[1,1],[0,1]])
W = np.array([1,math.sqrt(2)/2,1])
fn.drawVertexes(V)
fn.drawVertexes(V,'o')
fn.drawVertexes(fn.bezierRational(V,W)[0])
def exer1_3():
"""
Calculate the Bezier curve of the parametric curve:
X(t)=1+t+t^2
Y(t)=t^3
for t in [0,1].
"""
X = sympy.Poly('1+t+t**2')
Y = sympy.Poly('t**3')
P = fn.mergePoly(X,Y)
#P = np.array([[1,0],[1,0],[1,0],[0,1]])
V = fn.exp2bernstein(P)
fn.drawVertexes(fn.bezier(V))
fn.drawVertexes(V)
fn.drawVertexes(V,'o')
def exerB_3():
"""
Represent the B-Spline for a given control vertexes set
with a multeplicity.
"""
V = np.array([[0.,2.], [1.,0.], [2.,1.], [2.,1.], [3., 0.], [4., 2.]])
k = 3
p= np.array([0., 0., 0., 1./4., 1./2., 3./4., 1., 1., 1.])
fn.drawVertexes(V, 'o')
fn.drawVertexes(V)
fn.drawVertexes(fn.bSpline(V, k, p))
def exerB_4():
"""
Represent the B-Splines for a given control vertexes set
with a multeplicity and different nodal partitions,
one with a multeplicity and one without.
"""
V = np.array([[1.,0.], [0.,1.], [2.,1.5], [2.,1.5], [4., 1.], [3., 0.]])
k = 4
ps = [
np.array([0., 0., 0., 0., 1./4., 3./4., 1., 1., 1., 1.]),
np.array([0., 0., 0., 0., 1./2., 1./2., 1., 1., 1., 1.]),
]
fn.drawVertexes(V, 'o')
fn.drawVertexes(V)
for p in ps:
fn.drawVertexes(fn.bSpline(V, k, p))
def exerB_2():
"""
Given a fixed set of control vertexes, represent the B-Spline
curves for different nodal partitions and orders.
"""
V = np.array([[0.,0.], [-0.4,-0.1], [-1.,0.2], [-0.1,1.], [1., 1.1], [1.1, 0.6], [1.2, 0.7], [1.3, 1.2], [2., 0.8], [2.5, 0.7]])
ks = [4, 4, 6, 6, 8]
ps = [
np.array([0., 0., 0., 0., 1., 2., 3., 4., 5., 6., 7., 8., 9., 10.]),
np.array([0., 0., 0., 0., 1., 1., 2., 2., 3., 3., 4., 4., 4., 4.]),
np.array([0., 0., 0., 0., 0., 0., 1., 2., 3., 4., 5., 5., 5., 5., 5., 5.]),
np.array([0., 0., 0., 0., 0., 0., 1., 2., 3., 3., 4., 4., 4., 4., 4., 4.]),
np.array([0., 0., 0., 0., 0., 0., 0., 0., 1., 2., 3., 3., 3., 3., 3., 3., 3., 3.]),
]
for p,k in zip(ps, ks):
fn.drawVertexes(fn.bSpline(V, k, p))
fn.drawVertexes(V, 'o')
fn.drawVertexes(V)
def exerB_5():
"""
Determine different behavior for B-Splines with same
control vertexes but different multeplicity.
"""
Vs = [
np.array([[0.,1.], [1.,0.], [2.,1.], [3., 0.], [4., 1.]]),
np.array([[0.,1.], [1.,0.], [2.,1.], [2.,1.], [3., 0.], [4., 1.]]),
np.array([[0.,1.], [1.,0.], [2.,1.], [2.,1.], [2.,1.], [3., 0.], [4., 1.]])
]
ks = [4,4,4]
ps = [
np.array([0., 0., 0., 0.,1./2., 1., 1., 1., 1.]),
np.array([0., 0., 0., 0., 1./4., 3./4.,1., 1., 1., 1.]),
np.array([0., 0., 0., 0., 1./2., 1./2., 1./2.,1., 1., 1., 1.])
]
fn.drawVertexes(Vs[0], 'o')
fn.drawVertexes(Vs[0])
for V,k,p in zip(Vs,ks,ps):
fn.drawVertexes(fn.bSpline(V, k, p))
def exer1_7():
"""
Given the control vertexes of a Bezier curve and a set of
extra vertexes, attach the extra vertexes to the original
ones with the chosen continuity (some vertexes will be created).
"""
V = np.array([[0,0,0],[1,2,0],[3,2,0],[6,-1,0]])
Va = fn.attachWithC(V, np.array([[2,-1,3]]),C=2,hOrig=1.,hAttach=1.)
fn.drawVertexes(fn.bezier(V))
fn.drawVertexes(V)
fn.drawVertexes(V,'o')
fn.drawVertexes(fn.bezier(Va))
fn.drawVertexes(Va)
fn.drawVertexes(Va,'o')
def exer1_6():
"""
Increase the grade of a Bezier curve defined by his
control vertexes.
"""
V = np.array([[0,0,0],[1,2,0],[3,2,3],[2,6,2]])
V1 = fn.increaseGrade(V)
V2 = fn.increaseGrade(V1)
V3 = fn.increaseGrade(V2)
fn.drawVertexes(fn.bezier(V))
fn.drawVertexes(V)
fn.drawVertexes(V,'o')
fn.drawVertexes(V1)
fn.drawVertexes(V1,'o')
fn.drawVertexes(V2)
fn.drawVertexes(V2,'o')
fn.drawVertexes(V3)
fn.drawVertexes(V3,'o')
def exer1_5():
"""
Show the effects of the increase of the multiplicity
of a control vertex of a Bezier curve.
"""
V1 = np.array([[0,0,0],[1,2,0],[3,2,3],[2,6,2]])
V2 = np.array([[0,0,0],[1,2,0],[3,2,3],[3,2,3],[2,6,2]])
V3 = np.array([[0,0,0],[1,2,0],[3,2,3],[3,2,3],[3,2,3],[2,6,2]])
V4 = np.array([[0,0,0],[1,2,0],[3,2,3],[3,2,3],[3,2,3],[3,2,3],[2,6,2]])
fn.drawVertexes(V1)
fn.drawVertexes(V1,'o')
fn.drawVertexes(fn.bezier(V1))
fn.drawVertexes(fn.bezier(V2))
fn.drawVertexes(fn.bezier(V3))
fn.drawVertexes(fn.bezier(V4))
def exer1_4():
"""
Split a bezier curve defined by control vertexes
in a specific point of the parametric domain.
"""
V = np.array([[0,0,0],[1,2,0],[3,2,3],[2,6,2]])
C1,C2 = fn.splitControlPoints(V, 0.6)
fn.drawVertexes(fn.bezier(C1))
fn.drawVertexes(C1)
fn.drawVertexes(C1,'o')
fn.drawVertexes(fn.bezier(C2))
fn.drawVertexes(C2)
fn.drawVertexes(C2,'o')
def exer2_2():
"""
Increase the grade of a rational Bezier curve converting the
control vertexes in polinomial and using the same algorithms
of the polinomial.
"""
V = np.array([[0,0,0],[1,2,0],[3,2,3],[2,6,2]])
W = np.array([1,1,1,1])
V1,W1 = fn.increaseGradeRational(V, W)
V2,W2 = fn.increaseGradeRational(V1, W1)
V3,W3 = fn.increaseGradeRational(V2, W2)
fn.drawVertexes(fn.bezierRational(V,W)[0])
fn.drawVertexes(V)
fn.drawVertexes(V,'o')
fn.drawVertexes(V1)
fn.drawVertexes(V1,'o')
fn.drawVertexes(V2)
fn.drawVertexes(V2,'o')
fn.drawVertexes(V3)
fn.drawVertexes(V3,'o')