## Thursday, October 10, 2013

### Exploring B-Splines in Python

This notebook is the first result of online exploration of material explaining B-Splines. As I get more familiar with them, I'll do other notebooks. I will document my sources and eventually add some notes about things that bothered me. This is only a reformulation with my own terms of the vast b-spline documentation around the web. I try to explain the pitfalls and complete the ellipsis that other websites do. However, I'm not a mathematician so I might get some things wrong. But I really want this to be as precise yet understandable as possible using both precise terms and common language. If you spot errors, misconceptions or cryptic parts, please do leave a comment.

Used Tools:

• Python 3
• Numpy
• Matplotlib
• IPython

I won't detail their installation but have a look at IPython's installation guide. I prefer running it inside a VirtualEnv. Once installed, run IPython with:

ipython3 notebook --pylab=inline


We will draw these types of curves:

# Introduction to B-Splines curves by a dummy

B-splines are mathematical objects which are very useful in computer aided design applications. They are a generalization of Bezier curves which were developped for the car industry in the mid 1900's. The user typically manipulates the vertices (control points) of a polyline and the b-spline will gracefully and smoothly interpolate those points. B-splines have nice curvature properties that make them very useful for design and industry and can be easily used in 2D or 3D (and maybe more...). They were further generalized into Non-Uniform Rational B-splines, yep: NURBs.

## Parametric curves

A b-spline curve is a parametric curve whose points' coordinates are determined by b-spline functions of a parameter usually called $t$ (for time - this comes from physics and motion description). More precisely: A $D$-Dimensionnal b-spline curve is a parametric curve whose $D$ component functions are b-spline functions.

Let's pretend $C(t)$ is a 2D B-Spline curve (each of its points has two coordinates) :

• eq. 1 : $C(t) = ( S_x(t), S_y(t) )$

$S_x$ and $S_y$ are the component functions of $C$ for the $x$ and $y$ coordinates respectively. In other words, $S_x(t)$ and $S_y(t)$ will yield the x and y coordinates of $C(t)$'s points. They are the actual b-spline functions.

# B-Spline Function

But $t$ can't define the shape of $C$ all alone. A b-spline curve is also defined by:

• $P$ : an $m$-sized list of control points. This defines which points the curve will interpolate. For example $P=[(2,3),(4,5),(4,7),(1,5)]$ be a list of $m=4$ vertices with two de Boor points each ($D=2$). Graphically, $P$ forms the control polygone or control cage of the curve.
• $V$ : a $k$-sized list of non-decreasing real values called knot vector. It's the key to b-splines but is quite abstract. Knot vectors are very intrically tied to...
• $n$ : the degree of the function. A higher degree means more derivability.

$S_x$ and $S_y$ are the same function $S$ which simply processes different de Boor points (different coordinates of $P$'s points). So for the 2D curve $C$:

• eq. 2 : $C_{P,V,n} (t) = ( S_{Px,V,n} (t), S_{Py,V,n} (t) )$

Note that $S$ is "configured" (subscripted) like $C$ except that for the $x$ component function it takes the $x$ coordinate of points in $P$, thus the $Px$ subscript, the same goes for the $y$ component/coordinate. This is exactly what I meant above by "processes different de Boor points". These two sentences are equivalent. I also didn't put $P$, $V$ and $n$ as parameters because when you evaluate $C$, you evaluate $S$ and you evaluate it for a varying $t$, while all other inputs remain constant. This expression makes sense when a curve is to be evaluated more often than its control points are modified. Now we just need to know the definition of a B-spline function.

The mathematical definition of a B-Spline of degree $n$ as found in literature [Wolfram,Wikipedia] is

• eq. 3 : $S_{n}(t) = \sum \limits_i^m p_i b_{i,n}(t)$

$b$ is the b-spline basis function, we're getting close! $p_i$ is the vertex of index i in $P$. This means eq. 3 is for one dimension! Let's express it for curve of $D$-dimensions.

• eq. 4 : $\forall d \in \{1,...,D\}, C_{P,V,n} = (S_{n,d_1}(t), ...,S_{n,d_D}(t) ) = (\sum \limits_i^m p_{i,d_1} b_{i,n}(t), ..., \sum \limits_i^m p_{i,d_D} b_{i,n}(t))$

Roughly, $S$ is a sum of monomials, making it a polynomial! :) There is one monomial for each index of a vertex and each monomial is the product of the de Boor point at that index and the basis function at $t$ for that index. This is repeated for $D$ dimensions. So computing b-splines this way is quite slow with many loops, etc... but the implementation is easy and there are ways to speed it up (like memoization [Wikipedia, StackOverflow]).

Okay, time to translate eq. 4 to Python even though we don't know what $b$ is exactly. I'll try to follow the same naming but keep in mind that Python uses 0-indexing while maths use 1-indexing, so we shall pay attention to the iterations and array sizes.

In [3]:
def C_factory(P, V=None, n=2):
""" Returns a b-spline curve C(t) configured with P, V and n.

Parameters
==========
- P (list of D-tuples of reals) : List of de Boor points of dimension D.
- n (int) : degree of the curve
- V (list of reals) : list of knots in increasing order (by definition).

Returns
=======
A D-dimensionnal B-Spline curve.
"""

# TODO: check that p_len is ok with the degree and > 0
m = len(P)    # the number of points in P
D = len(P[0]) # the dimension of a point (2D, 3D)

# TODO: check the validity of the input knot vector.
# TODO: create an initial Vector Point.

#############################################################################
# The following line will be detailed later.                                #
# We create the highest degree basis spline function, aka. our entry point. #
# Using the recursive formulation of b-splines, this b_n will call          #
# lower degree basis_functions. b_n is a function.                          #
#############################################################################
b_n = basis_factory(n)

def S(t, d):
""" The b-spline funtion, as defined in eq. 3. """
out = 0.
for i in range(m): #: Iterate over 0-indexed point indices
out += P[i][d]*b_n(t, i, V)
return out

def C(t):
""" The b-spline curve, as defined in eq. 4. """
out = [0.]*D           #: For each t we return a list of D coordinates
for d in range(D):     #: Iterate over 0-indexed dimension indices
out[d] = S(t,d)
return out

####################################################################
# "Enrich" the function with information about its "configuration" #
####################################################################
C.V = V                   #: The knot vector used by the function
C.spline = S              #: The spline function.
C.basis = b_n             #: The highest degree basis function. Useful to do some plotting.
C.min = V[0]              #: The domain of definition of the function, lower bound for t
C.max = V[-1]             #: The domain of definition of the function, upper bound for t
C.endpoint = C.max!=V[-1] #: Is the upper bound included in the domain.
return C


### Some python background

• C_factory is a high-order function: it is a function which returns a function. The returned C function is a closure, it inherits the namespace of its definition context and holds reference to those variables.
• In Python 3, the range(start, stop, step) function returns an iterator which iterates over integers like: $[0,X)$. This spares us much fiddling with "-1" adjustments all over the place. In Python 2, there it is called xrange(start, stop, step).

## What is the Knot Vector?

Right, good question. $V$ is a list whose length is $k$ (see eq. 6), made of reals $t_q$ (called knots) with $q \in [1, k]$ where $t_q \leq t_{q+1}$. In other words values must never decrease. The current $t$ must lay somewhere in $[t_1, t_k)$. During the calculations $t$ will be compared to all the knot spans and if it falls in those spans, then the basis function will have a certain influence on the final curve. It determines which vertices (de Boor points) will influence the curve at a given $t$.

The distribution of the knots in $V$ will determine if the curve meets the endpoints of the control polyline (it is then clamped), or not (unclamped - the domain of definition is shorter). It can be periodic or not if some knot spans repeat themselves, and uniform or not if the knot spans have a constant size, but more on this later when I wrap my head around it a bit more! Just remember that the distribution of knots will depend on the type of the knot vector. We will focus on clamped knot vectors which are a special case of b-splines behaving like Bezier curves.

### Clamped Knot Vector

A clamped knot vector looks like this:

• eq. 5 : $V = \{t_1 = ... = t_n, ..., t_{k-n-1}, t_{k-n} = ... = t_{k} \}$

Where, as a reminder, $k$ is the length of $V$, where $n$ is the degree of the curve. and the length of a knot vector is totally conditionned by the type of $V$, the degree of the curve and the number of control points.

The $t_1, ... ,t_n$ and $t_{k-n} = ... = t_k$ are called external knots, the others are internal knots.

• eq. 6 : $k = m + n + 2$

We can now implement a basic make_knot_vector function which only handles clamped knot vectors.

In [4]:
def make_knot_vector(n, m, style="clamped"):
"""
Create knot vectors for the requested vector type.

Parameters
==========
- n (int) : degree of the bspline curve that will use this knot vector
- m (int) : number of vertices in the control polygone
- style (str) : type of knot vector to output

Returns
=======
- A knot vector (tuple)
"""
if style != "clamped":
raise NotImplementedError

total_knots = m+n+2                           # length of the knot vector, this is exactly eq.6
outer_knots = n+1                             # number of outer knots at each of the vector.
inner_knots = total_knots - 2*(outer_knots)   # number of inner knots
# Now we translate eq. 5:
knots  = [0]*(outer_knots)
knots += [i for i in range(1, inner_knots)]
knots += [inner_knots]*(outer_knots)

return tuple(knots) # We convert to a tuple. Tuples are hashable, required later for memoization


# B-spline basis function

Basis functions are the functions that will be multiplied and summed in the B-Spline function.

## Definition

The b-spline basis function is defined as follows:

• eq. 7 : $b_{i,1}(x) = \left \{ \begin{matrix} 1 & \mathrm{if} \quad t_i \leq x < t_{i+1} \\0 & \mathrm{otherwise} \end{matrix} \right.$

• eq. 8 : $b_{i,k}(x) = \frac{x - t_i}{t_{i+k-1} - t_i} b_{i,k-1}(x) + \frac{t_{i+k} - x}{t_{i+k} - t_{i+1}} b_{i+1,k-1}(x).$

There is really not much to say about them right now. Sometime later we might try to understand them, but we're fine with this definition right now. However, I did get confused as some sites write eq. 7 this way:

• $b_{i,1}(x) = \left \{ \begin{matrix} 1 & \mathrm{if} \quad t_i \leq x \leq t_{i+1} \\0 & \mathrm{otherwise} \end{matrix} \right.$

This is wrong (the double $\leq$), at least as far as I could see : graphically, spikes appear on the curve at knot values. It makes sense since at knots the sum (eq. 3) will cumulate values from different intervals because $b_{i,1}$ will evaluate to 1 instead of 0 at internal knots.

## What is basis_factory(n) ?

It is a high-order function! It will return b-spline basis functions b_n of degree $n$ which will be used by S. We are ready to implement it. We need to translate eq. 7 and eq. 8 to Python. Again, we must be extra careful with indexing.

In [5]:
def basis_factory(degree):
""" Returns a basis_function for the given degree """
if degree == 0:

def basis_function(t, i, knots):
"""The basis function for degree = 0 as per eq. 7"""
t_this = knots[i]
t_next = knots[i+1]
out = 1. if (t>=t_this and t< t_next) else 0.
return out
else:

def basis_function(t, i, knots):
"""The basis function for degree > 0 as per eq. 8"""
out = 0.
t_this = knots[i]
t_next = knots[i+1]
t_precog  = knots[i+degree]
t_horizon = knots[i+degree+1]

top = (t-t_this)
bottom = (t_precog-t_this)

if bottom != 0:
out  = top/bottom * basis_factory(degree-1)(t, i, knots)

top = (t_horizon-t)
bottom = (t_horizon-t_next)
if bottom != 0:
out += top/bottom * basis_factory(degree-1)(t, i+1, knots)

return out

####################################################################
# "Enrich" the function with information about its "configuration" #
####################################################################
basis_function.lower = None if degree==0 else basis_factory(degree-1)
basis_function.degree = degree
return basis_function


# The puzzle is complete

We now have all that is needed to start drawing clamped b-splines!

## Constructing the spline curve

In [6]:
# We decide to draw a quadratic curve
n = 2
# Next we define the control points of our curve
P = [(3,-1), (2.5,3), (0, 1), (-2.5,3), (-3,-1)]
# Create the knot vector
V = make_knot_vector(n, len(P), "clamped")
# Create the Curve function
C = C_factory(P, V, n)


## Sampling the curve

$C(t)$ is ready to be sampled (evaluated)! That means that we are ready to feed it with values for $t$ and getting point coordinates out. In the case of clamped curves the domain of definition of $C$ - that is: the values for which $t$ is valid - is $[min(V), max(V))$. These bounds are available as C.min and C.max member attributes of C. Do NOT sample at the endpoint, the curve is not defined there in the case of clamped knot vectors. There are situations where it is possible to sample at C.max but this value will be different to $max(V)$.

There are many ways to sample a function. In general most effort is put in strategies the put more samples (decreasing sample steps) in curvy areas and less in straight lines. This can be done by analyzing the polyline angles for example. The best sampling visually is when each display point was tested for being a point of the curve.

These are complicated strategies that we might explore later. For now we will use a linear sampling of the definition interval.

In [7]:
# Regularly spaced samples
sampling = [t for t in np.linspace(C.min, C.max, 100, endpoint=C.endpoint)]
# Sample the curve!!!!
curvepts = [ C(s) for s in sampling ]

# Create a matplotlib figure
fig = plt.figure()
fig.set_figwidth(16)
# Draw the curve points
ax.scatter( *zip(*curvepts), marker="o", c=sampling, cmap="jet", alpha=0.5 )
# Draw the control cage.
ax.plot(*zip(*P), alpha=0.3)

Out[7]:
[<matplotlib.lines.Line2D at 0x7f46a6cf67d0>]


Mission complete! We can now draw clamped 2D Curves!

# Short discussion

## Degree

Compared to Bezier curves, a B-Spline of degree $n$ can interpolate a polyline of an almost arbitrary number of points without having to raise the degree. With Bezier curves, to interpolate a polyline of $m$ vertices you need a curve of $n=m-1$ degree. Or you do a "poly-bezier" curve: you connect a series of lower-degree bezier curves and put effort in keeping control points correctly placed to garantee the continuity of the curve at the "internal endpoints".

## Ugly endpoint

In our example, the endpoint is on the left (dark red). It is quite clear that the curve's end doesn't match that of the polyline. This is because in our basic sampling example, we didn't include the endpoint (when $t=max(V)$). And for a good reason : the curve is not defined there. However, we could include a sample $t_f$ very close to $max(V)$:

• $t_f=max(V)-\epsilon$

## Computation

In our example we sampled one short b-spline of low-degree and with a decent amount of samples and it went quickly. However, if you add more curves and sample them in a similar way, you will notice the slowness of the calculations. Memoization is interesting in our case for several reasons:

• the recursive definition of $b_n$ (eq. 8) and its implementation (see basis_function(n)) imply many requests to create lower-level basis functions.
• the fact that the same $S(t)$ is used for the different components (coordinates) of the curve points implies many calls to the basis_functions with the same exact input. So in a 2D case, for each point we save one full computation of $b_n$.
• if we consider that the curve will be more often sampled than modified (eg: adaptive sampling in a resolution-aware drawing system to get exact representation of the curve) then there are chances that the curve will be sampled many times for the same $t$.

This gives serious performance improvements at the expense of memory.

## Complete implementation with memoization

See the final listing for an implementation with memoization. The creation of the knot vector was moved inside the C_factory function. A helper function was created for drawing purpose. It includes a dirty fixed hack to approach the endpoint too.

# Full implementation

In [8]:
def memoize(f):
""" Memoization decorator for functions taking one or more arguments. """
class memodict(dict):
def __init__(self, f):
self.f = f
def __call__(self, *args):
return self[args]
def __missing__(self, key):
ret = self[key] = self.f(*key)
return ret
return memodict(f)

def C_factory(P, n=2, V_type="clamped"):
""" Returns a b-spline curve C(t) configured with P, V_type and n.
The knot vector will be created according to V_type

Parameters
==========
- P (list of D-tuples of reals) : List of de Boor points of dimension D.
- n (int) : degree of the curve
- V_type (str): name of the knit vector type to create.

Returns
=======
A D-dimensionnal B-Spline Curve.
"""

# TODO: check that p_len is ok with the degree and > 0
m = len(P)    # the number of points in P
D = len(P[0]) # the dimension of a point (2D, 3D)

# Create the knot vector
V = make_knot_vector(n, m, V_type)
# TODO: check the validity of the input knot vector.
# TODO: create an initial Vector Point.

#############################################################################
# The following line will be detailed later.                                #
# We create the highest degree basis spline function, aka. our entry point. #
# Using the recursive formulation of b-splines, this b_n will call          #
# lower degree basis_functions. b_n is a function.                          #
#############################################################################
b_n = basis_factory(n)

@memoize
def S(t, d):
""" The b-spline funtion, as defined in eq. 3. """
out = 0.
for i in range(m): #: Iterate over 0-indexed point indices
out += P[i][d]*b_n(t, i, V)
return out

def C(t):
""" The b-spline curve, as defined in eq. 4. """
out = [0.]*D           #: For each t we return a list of D coordinates
for d in range(D):     #: Iterate over 0-indexed dimension indices
out[d] = S(t,d)
return out

####################################################################
# "Enrich" the function with information about its "configuration" #
####################################################################
C.P = P                   #: The control polygone
C.V = V                   #: The knot vector used by the function
C.spline = S              #: The spline function.
C.basis = b_n             #: The highest degree basis function. Useful to do some plotting.
C.min = V[0]              #: The domain of definition of the function, lower bound for t
C.max = V[-1]             #: The domain of definition of the function, upper bound for t
C.endpoint = C.max!=V[-1] #: Is the upper bound included in the domain.
return C

def make_knot_vector(n, m, style="clamped"):
"""
Create knot vectors for the requested vector type.

Parameters
==========
- n (int) : degree of the bspline curve that will use this knot vector
- m (int) : number of vertices in the control polygone
- style (str) : type of knot vector to output

Returns
=======
- A knot vector (tuple)
"""
if style != "clamped":
raise NotImplementedError

total_knots = m+n+2                           # length of the knot vector, this is exactly eq.6
outer_knots = n+1                             # number of outer knots at each of the vector.
inner_knots = total_knots - 2*(outer_knots)   # number of inner knots
# Now we translate eq. 5:
knots  = [0]*(outer_knots)
knots += [i for i in range(1, inner_knots)]
knots += [inner_knots]*(outer_knots)

return tuple(knots) # We convert to a tuple. Tuples are hashable, required later for memoization

@memoize
def basis_factory(degree):
""" Returns a basis_function for the given degree """
if degree == 0:
@memoize
def basis_function(t, i, knots):
"""The basis function for degree = 0 as per eq. 7"""
t_this = knots[i]
t_next = knots[i+1]
out = 1. if (t>=t_this and t<t_next) else 0.
return out

else:
@memoize
def basis_function(t, i, knots):
"""The basis function for degree > 0 as per eq. 8"""
out = 0.
t_this = knots[i]
t_next = knots[i+1]
t_precog  = knots[i+degree]
t_horizon = knots[i+degree+1]

top = (t-t_this)
bottom = (t_precog-t_this)

if bottom != 0:
out  = top/bottom * basis_factory(degree-1)(t, i, knots)

top = (t_horizon-t)
bottom = (t_horizon-t_next)
if bottom != 0:
out += top/bottom * basis_factory(degree-1)(t, i+1, knots)

return out

####################################################################
# "Enrich" the function with information about its "configuration" #
####################################################################
basis_function.lower = None if degree==0 else basis_factory(degree-1)
basis_function.degree = degree
return basis_function

import matplotlib
def draw_bspline(C=None, P=None, n=None, V_type=None, endpoint_epsilon=0.00001):
"""Helper function to draw curves."""
if P and n and V_type:
C = C_factory(P, n, V_type)
if C:
# Use 2D or 3D
is3d = True if len(C.P[0])==3 else False
from mpl_toolkits.mplot3d import Axes3D

# Regularly spaced samples
sampling = [t for t in np.linspace(C.min, C.max, 100, endpoint=C.endpoint)]
# Hack to sample close to the endpoint
sampling.append(C.max - endpoint_epsilon)
# Sample the curve!!!!
curvepts = [ C(s) for s in sampling ]

# Create a matplotlib figure
fig = plt.figure()
fig.set_figwidth(12)
if is3d:
fig.set_figheight(10)
ax = fig.add_subplot(111, projection='3d')
else:

# Draw the curve points
ax.scatter( *zip(*curvepts), marker="o", c=sampling, cmap="jet", alpha=0.5 )
# Draw the control cage.
ax.plot(*zip(*C.P), alpha=0.3)
# Draw the knots
knotspos = [C(s) for s in C.V if s!= C.max]
knotspos.append( C(C.max - endpoint_epsilon) )
ax.scatter( *zip(*knotspos), marker="*", c=sampling, alpha=1, s=100 )

# Here we annotate the knots with their values
prev = None
occurences = 1
for i, curr in enumerate(C.V):
if curr == C.max:
kpos = C(curr-endpoint_epsilon)
else:
kpos = C(curr)
if curr == prev:
occurences += 1
else:
occurences = 1
kpos[0] -= 0.3*occurences
ax.text( *kpos, s="t="+str(curr), fontsize=12 )
prev = curr

In [9]:
# Next we define the control points of our 2D curve
P = [(3 , 1), (2.5, 4), (0, 1), (-2.5, 4), (-3, 0), (-2.5, -4), (0, -1), (2.5, -4), (3, -1)]
# Create the a quadratic curve function
draw_bspline(P=P, n=2, V_type="clamped")

# Next we define the control points of our 3D curve
P3D = [(3, 1, 8), (2.5, 3, 7), (0, 1, 6), (-2.5, 3, 5), (-3, 0, 4), (-2.5, -3, 3), (0, -1, 2), (2.5, -3, 1), (3, -1, 0)]
# Create the a quadratic curve function
draw_bspline(P=P3D, n=2, V_type="clamped")


# Revisions

I decided to add this section a bit late. At first I was planning to use git commits to document changes but it is not possible. The revision number is the gist revision number and I start at revision 11.

• r.14
• Fix details of range and xrange. This code is written for Python 3 and the info in that section was for Python 2.
• r.13
• More 1-indexing fixes in text, not in code.
• Corrections of some docstrings.
• Watch out for < and > in python code. For example "if d < myvar" won't interfer with the html layout when converting using nbconvert, but "if d <myvar" will open a new html tag and wreck the layout.
• r.12
• Ugh, I obviously got the gist numbering wrong. Fix to sync with gist.
• r.11
• Cosmetic fixes to markdown to get better rendering of mathjax elements and other elements. NBConvert renders to html with a different pipeline (pandoc) than IPython Notebook and uses a stricter version of markdown.
• Fixes to "What is The Knot Vector" section: there were 0-indexing vs 1-indexing inconsistencies. I think I got it right this time.

1. I just noticed that some Math Tex doesn't render well: eq. 8 is incomplete. I also got some indexing wrong (knot vector). Will update.

2. I've recently attempted to implement a B-Spline function in Fortran, and was forced to read many, many documents on B-Splines in order to understand how to actually calculate each step. You're explanation, which includes a full run-through from the original data points, obtaining the knot vector, calculating the basis functions and obtaining the interpolated data completes a missing link for me - almost all the documents I have read leave out the final step (i.e. calculating the interpolated curve).
Thanks, you've made my job much easier.

3. Hello Graham,

Thank you for your comment. I am glad that this post was helpful. It did take me quite some time to digest all the sites I read about b-splines so I'm really glad it helped!

Cheers!
Daniel

4. Hi Dani,
is it possible to export the spline's points?
i would like to create a spline wiht python and then use it in a CAD-programm (ICEM CFD) as it is not possible to draw splines in a controlled way in ICEM. they are always a bit random :)
So is there a possibility to get the interpolated points not just in matplotlib but write them in list in python or in a csv-file?
Are the points at all created in python or can I only see them in matplotlib? Is it possible to export the matplotlib-plot in a csv?
thanks for your help.

5. @nikkuen : yes it is possible, here's the hack (I'm using ">" to indicate indentation level as I can't get Blogger to accept preformatted code)

# First we modify draw_bspline so that it returns the sampled points.
def draw_bspline(C=None, P=None, n=None, V_type=None, endpoint_epsilon=0.00001):
> # ...
> if C:
>> # ...
>> return curvepts

Then, use it like this :

# We open a output file, loop over the points and write down each coordinate
sampledpts = draw_bspline(P=P, n=2, type='clamped'")
with open("bspline.csv", "w") as f:
> for point in sampledpts:
>> for coord in point:
>>> f.write(str(coord)+",")
>> f.write("\n")

There are other ways to write down the csv but this is the basic idea and you can customize the output to fit your needs. You can also have a look at python's "csv" module.

6. It works. Thanks a lot.
I have another question: Do you know if there is a way to calculate the area emebedded by a closed spline?

1. I think there is one simple way to approximately calculate the area of the closed spline.
The basic idea is this: approximate the closed spline as a polygon, and then calculate the area of the polygon.

In Python it is possible to approximate the closed spline as a polygon by using the Shapely Python package for computational geometry (see http://toblerity.org/shapely/manual.html). Thus, in order to solve this problem, the following steps could be performed: 1) compute a set of sampled points from the spline, 2) create a new shapely.Polygon object by using the set of points from the previous step , 3) get the area of the Polygon object, as this has an "area" attribute.

Of course, the error in the approximation (from closed spline to polygon) depends on the number of sampled points considered: the error decreases as the number of sampling points increases.

7. There certainly are ways but I haven't actually tried any! The best way would be to find a "closed form" solution, the bspline primitive, and implement it as a Python function that takes the same parameters as C but returns the area.
Another hackish way would be to tesselate the closed (discrete) spline and sum the sub-triangle areas.

8. Ear clipping (http://en.wikipedia.org/wiki/Polygon_triangulation) should be easy to do on the discrete bspline (sampledpts). One just must be careful to check that the third edge of the current triangle is really inside the polygon not outside. If it's inside, then its area can be summed and the triangle can then be discarded.

9. Hi Daniel! I stumbled upon your page while researching this question on stack overflow i asked: http://stackoverflow.com/questions/34803197/speed-up-a-cox-deboor-bspline-algorithm-in-numpy-scipy

I'm new to python, and your solution to the problem blew my mind. Your solution computes 100 times faster than mine which is what i was looking for! Now i have to take a moment and wrap my brain around memoization and decorators in general :) I noticed that in your code you only implemented a curve's open form. If you take a look at mine it'll show you how to calculate the knots appropriately.

10. Hi Eric! I'm happy this page helped! And I'll be looking into your implementation of closed form, I've been wanting to dig that subject (and extend this post) but never found time.

Concerning memoization: its really trading off memory to use less CPU but avoids actually calling the function. So depending on your use, you might have to watch the memory consumption. And this implementation only suits cases where the control points do not change. So if you plan on using it in, say, deformable body simulation, it... won't work well. Though for rigid transforms it will be ok.

I think there are ways to make the memoization more efficient AND the actual computations faster by vectorizing operations using numpy arrays, but the implementation then changes radically and the relation between the equations and their python version is less straightforward.

Regarding performance in Python, the general rule is : vectorization (because using native code) > iteration (because function calls are slow) > recursion. Unfortunately, not everything is vectorizable (and transforming recusive into iterative is tricky) but sometimes we can have good suprises!

Thanks for the comment and the closed form tip ;)

Daniel

11. hi,
when running code on my system I had this mistake.
I think it may be related to matplotlib version (1.4.0) ?

Traceback (most recent call last):
File "/usr/lib64/python3.4/site-packages/matplotlib/colors.py", line 367, in to_rgba
'length of rgba sequence should be either 3 or 4')
ValueError: length of rgba sequence should be either 3 or 4

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
File "/usr/lib64/python3.4/site-packages/matplotlib/colors.py", line 398, in to_rgba_array
return np.array([self.to_rgba(c, alpha)], dtype=np.float)
File "/usr/lib64/python3.4/site-packages/matplotlib/colors.py", line 375, in to_rgba
'to_rgba: Invalid rgba arg "%s"\n%s' % (str(arg), exc))
ValueError: to_rgba: Invalid rgba arg "[ 0. 0.07 0.14 0.21 0.28 0.35 0.42 0.49 0.56
0.63 0.7 0.77 0.84 0.91 0.98 1.05 1.12 1.19
1.26 1.33 1.4 1.47 1.54 1.61 1.68 1.75 1.82
1.89 1.96 2.03 2.1 2.17 2.24 2.31 2.38 2.45
2.52 2.59 2.66 2.73 2.8 2.87 2.94 3.01 3.08
3.15 3.22 3.29 3.36 3.43 3.5 3.57 3.64 3.71
3.78 3.85 3.92 3.99 4.06 4.13 4.2 4.27 4.34
4.41 4.48 4.55 4.62 4.69 4.76 4.83 4.9 4.97
5.04 5.11 5.18 5.25 5.32 5.39 5.46 5.53 5.6
5.67 5.74 5.81 5.88 5.95 6.02 6.09 6.16 6.23
6.3 6.37 6.44 6.51 6.58 6.65 6.72 6.79 6.86
6.93 6.99999]"
length of rgba sequence should be either 3 or 4

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
File "mybspline.py", line 204, in
draw_bspline(P=P, n=2, V_type="clamped")
File "mybspline.py", line 182, in draw_bspline
ax.scatter( *zip(*knotspos), marker="*", c=sampling, alpha=1, s=100 )
File "/usr/lib64/python3.4/site-packages/matplotlib/axes/_axes.py", line 3599, in scatter
colors = mcolors.colorConverter.to_rgba_array(c, alpha)
File "/usr/lib64/python3.4/site-packages/matplotlib/colors.py", line 402, in to_rgba_array
raise ValueError("Color array must be two-dimensional")
ValueError: Color array must be two-dimensional

12. Hi,

Indeed, there seems to be an error in my code. Try replacing:

ax.scatter( *zip(*knotspos), marker="*", c=sampling, alpha=1, s=100 )

with

ax.scatter( *zip(*knotspos), marker="*", c="#000000", alpha=1, s=100 )

or

ax.scatter( *zip(*knotspos), marker="*", c=None, alpha=1, s=100 )

off the top of my head, hope it solves it. If it does, I'll update the notebook, if it doesn't well, I'll give it a closer look later.

Daniel

13. This post is great. Leads you from control points to getting the curve.

Did you consider writing a follow-up to cover spline interpolation/approximation for a given data set

Many thanks

14. Hello Chygra! Thanks for your feedback! I'm happy this post was helpful! I'm indeed considering a follow up, but not in spline interpolation (yet), I don't have enough knowledge there. I'm planning to really dig down the b-spline structure, firstly by understanding how the knot vector works.

15. Hi Dani!

Found your post super helpful, I am not super familiar with Python so could you explain how the recursion in basis_factory works? I understand why you are doing it but..

Specifically these lines

if bottom != 0:
out = top/bottom * basis_factory(degree-1)(t, i, knots)

top = (t_horizon-t)
bottom = (t_horizon-t_next)
if bottom != 0:
out += top/bottom * basis_factory(degree-1)(t, i+1, knots)

return out

When you call basis_factory again you pass the value (degree-1) into it. What happens with the values (t, i, knots) and (t, i+1, knots)

Thanks so much!

16. Hello Rashad Ajward! I'm happy you found this post useful!

basis_factory creates basis functions of a certain degree : the thing it returns is really a function. So when you see basis_factory(degree-1) it creates a lower-degree function which is returned and immediately called with t, i+1 and knots.

Function which return functions are called higher-order functions.