bezier.curve module

Helper for Bézier Curves.

See Curve-Curve Intersection for examples using the Curve class to find intersections.

class bezier.curve.Curve(nodes, degree, start=0.0, end=1.0, root=None, _copy=True)

Bases: bezier._base.Base

Represents a Bézier curve.

We take the traditional definition: a Bézier curve is a mapping from \(s \in \left[0, 1\right]\) to convex combinations of points \(v_0, v_1, \ldots, v_n\) in some vector space:

\[B(s) = \sum_{j = 0}^n \binom{n}{j} s^j (1 - s)^{n - j} \cdot v_j\]
../_images/curve_constructor.png
>>> import bezier
>>> nodes = np.array([
...     [0.0  , 0.0],
...     [0.625, 0.5],
...     [1.0  , 0.5],
... ])
>>> curve = bezier.Curve.from_nodes(nodes)
>>> curve
<Curve (degree=2, dimension=2)>
Parameters:
  • nodes (numpy.ndarray) – The nodes in the curve. The rows represent each node while the columns are the dimension of the ambient space.
  • degree (int) – The degree of the curve. This is assumed to correctly correspond to the number of nodes. Use from_nodes() if the degree has not yet been computed.
  • start (Optional [ float ]) – The beginning of the sub-interval that this curve represents.
  • end (Optional [ float ]) – The end of the sub-interval that this curve represents.
  • root (Optional [ Curve ]) – The root curve that contains this current curve.
  • _copy (bool) – Flag indicating if the nodes should be copied before being stored. Defaults to True since callers may freely mutate nodes after passing in.
classmethod from_nodes(nodes, start=0.0, end=1.0, root=None, _copy=True)

Create a Curve from nodes.

Computes the degree based on the shape of nodes.

Parameters:
  • nodes (numpy.ndarray) – The nodes in the curve. The rows represent each node while the columns are the dimension of the ambient space.
  • start (Optional [ float ]) – The beginning of the sub-interval that this curve represents.
  • end (Optional [ float ]) – The end of the sub-interval that this curve represents.
  • root (Optional [ Curve ]) – The root curve that contains this current curve.
  • _copy (bool) – Flag indicating if the nodes should be copied before being stored. Defaults to True since callers may freely mutate nodes after passing in.
Returns:

The constructed curve.

Return type:

Curve

__repr__()

Representation of current object.

Returns:Object representation.
Return type:str
length

float: The length of the current curve.

start

float: Start of sub-interval this curve represents.

This value is used to track the current curve in the re-parameterization / subdivision process. The curve is still defined on the unit interval, but this value illustrates how this curve relates to a “parent” curve. For example:

>>> nodes = np.array([
...     [0.0, 0.0],
...     [1.0, 2.0],
... ])
>>> curve = bezier.Curve(nodes, 1)
>>> curve
<Curve (degree=1, dimension=2)>
>>> left, right = curve.subdivide()
>>> left
<Curve (degree=1, dimension=2, start=0, end=0.5)>
>>> right
<Curve (degree=1, dimension=2, start=0.5, end=1)>
>>> _, mid_right = left.subdivide()
>>> mid_right
<Curve (degree=1, dimension=2, start=0.25, end=0.5)>
>>> mid_right.nodes
array([[ 0.25, 0.5 ],
       [ 0.5 , 1.  ]])
end

float: End of sub-interval this curve represents.

See start for more information.

root

Curve: The “root” curve that contains the current curve.

This indicates that the current curve is a section of the “root” curve. For example:

>>> _, right = curve.subdivide()
>>> right
<Curve (degree=2, dimension=2, start=0.5, end=1)>
>>> right.root is curve
True
>>> right.evaluate(0.0) == curve.evaluate(0.5)
array([[ True, True]], dtype=bool)
>>>
>>> mid_left, _ = right.subdivide()
>>> mid_left
<Curve (degree=2, dimension=2, start=0.5, end=0.75)>
>>> mid_left.root is curve
True
>>> mid_left.evaluate(1.0) == curve.evaluate(0.75)
array([[ True, True]], dtype=bool)
edge_index

Optional [ int ] : The index of the edge among a group of edges.

>>> curve.edge_index
1
>>> curve.previous_edge
<Curve (degree=1, dimension=2)>
>>> curve.previous_edge.edge_index
0
>>> curve.next_edge
<Curve (degree=1, dimension=2)>
>>> curve.next_edge.edge_index
2

This is intended to be used when a Curve is created as part of a larger structure like a Surface or CurvedPolygon.

next_edge

Optional [ Curve ] : An edge that comes after the current one.

This is intended to be used when a Curve is created as part of a larger structure like a Surface or CurvedPolygon.

previous_edge

Optional [ Curve ] : An edge that comes before the current one.

This is intended to be used when a Curve is created as part of a larger structure like a Surface or CurvedPolygon.

evaluate(s)

Evaluate \(B(s)\) along the curve.

See evaluate_multi() for more details.

../_images/curve_evaluate.png
>>> nodes = np.array([
...     [0.0  , 0.0],
...     [0.625, 0.5],
...     [1.0  , 0.5],
... ])
>>> curve = bezier.Curve(nodes, 2)
>>> curve.evaluate(0.75)
array([[ 0.796875, 0.46875 ]])
Parameters:s (float) – Parameter along the curve.
Returns:The point on the curve (as a two dimensional NumPy array with a single row).
Return type:numpy.ndarray
evaluate_multi(s_vals)

Evaluate \(B(s)\) for multiple points along the curve.

This is done by first evaluating each member of the Bernstein basis at each value in s_vals and then applying those to the control points for the current curve.

This is done instead of using de Casteljau’s algorithm. Implementing de Casteljau is problematic because it requires a choice between one of two methods:

  • vectorize operations of the form \((1 - s)v + s w\), which requires a copy of the curve’s control points for each value in s_vals
  • avoid vectorization and compute each point in serial

Instead, we can use vectorized operations to build up the Bernstein basis values.

>>> nodes = np.array([
...     [0.0, 0.0, 0.0],
...     [1.0, 2.0, 3.0],
... ])
>>> curve = bezier.Curve(nodes, 1)
>>> curve
<Curve (degree=1, dimension=3)>
>>> s_vals = np.linspace(0.0, 1.0, 5)
>>> curve.evaluate_multi(s_vals)
array([[ 0.  , 0.  , 0.  ],
       [ 0.25, 0.5 , 0.75],
       [ 0.5 , 1.  , 1.5 ],
       [ 0.75, 1.5 , 2.25],
       [ 1.  , 2.  , 3.  ]])
Parameters:s_vals (numpy.ndarray) – Parameters along the curve (as a 1D array).
Returns:The points on the curve. As a two dimensional NumPy array, with the rows corresponding to each s value and the columns to the dimension.
Return type:numpy.ndarray
plot(num_pts, ax=None, show=False)

Plot the current curve.

Parameters:
Returns:

The axis containing the plot. This may be a newly created axis.

Return type:

matplotlib.artist.Artist

Raises:

NotImplementedError – If the curve’s dimension is not 2.

subdivide()

Split the curve \(B(s)\) into a left and right half.

Takes the interval \(\left[0, 1\right]\) and splits the curve into \(B_1 = B\left(\left[0, \frac{1}{2}\right]\right)\) and \(B_2 = B\left(\left[\frac{1}{2}, 1\right]\right)\). In order to do this, also reparameterizes the curve, hence the resulting left and right halves have new nodes.

../_images/curve_subdivide.png
>>> nodes = np.array([
...     [0.0 , 0.0],
...     [1.25, 3.0],
...     [2.0 , 1.0],
... ])
>>> curve = bezier.Curve(nodes, 2)
>>> left, right = curve.subdivide()
>>> left
<Curve (degree=2, dimension=2, start=0, end=0.5)>
>>> left.nodes
array([[ 0.   , 0.   ],
       [ 0.625, 1.5  ],
       [ 1.125, 1.75 ]])
>>> right
<Curve (degree=2, dimension=2, start=0.5, end=1)>
>>> right.nodes
array([[ 1.125, 1.75 ],
       [ 1.625, 2.   ],
       [ 2.   , 1.   ]])
Returns:The left and right sub-curves.
Return type:Tuple [ Curve, Curve ]
intersect(other)

Find the points of intersection with another curve.

See Curve-Curve Intersection for more details.

../_images/curve_intersect.png
>>> nodes1 = np.array([
...     [0.0  , 0.0  ],
...     [0.375, 0.75 ],
...     [0.75 , 0.375],
... ])
>>> curve1 = bezier.Curve(nodes1, 2)
>>> nodes2 = np.array([
...     [0.5, 0.0 ],
...     [0.5, 0.75],
... ])
>>> curve2 = bezier.Curve(nodes2, 1)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[ 0.5, 0.5]])
Parameters:

other (Curve) – Other curve to intersect with.

Returns:

Array of intersection points (possibly empty).

Return type:

numpy.ndarray

Raises:
elevate()

Return a degree-elevated version of the current curve.

Does this by converting the current nodes \(v_0, \ldots, v_n\) to new nodes \(w_0, \ldots, w_{n + 1}\) where

\[\begin{split}\begin{align*} w_0 &= v_0 \\ w_j &= \frac{j}{n + 1} v_{j - 1} + \frac{n + 1 - j}{n + 1} v_j \\ w_{n + 1} &= v_n \end{align*}\end{split}\]
Returns:The degree-elevated curve.
Return type:Curve
specialize(start, end)

Specialize the curve to a given sub-interval.

../_images/curve_specialize.png
>>> curve = bezier.Curve.from_nodes(np.array([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ]))
>>> new_curve = curve.specialize(-0.25, 0.75)
>>> new_curve
<Curve (degree=2, dimension=2, start=-0.25, end=0.75)>
>>> new_curve.nodes
array([[-0.25 , -0.625],
       [ 0.25 ,  0.875],
       [ 0.75 ,  0.375]])

This is generalized version of subdivide(), and can even match the output of that method:

>>> left, right = curve.subdivide()
>>> also_left = curve.specialize(0.0, 0.5)
>>> np.all(also_left.nodes == left.nodes)
True
>>> also_right = curve.specialize(0.5, 1.0)
>>> np.all(also_right.nodes == right.nodes)
True
Parameters:
  • start (float) – The start point of the interval we are specializing to.
  • end (float) – The end point of the interval we are specializing to.
Returns:

The newly-specialized curve.

Return type:

Curve

degree

int: The degree of the current shape.

dimension

int: The dimension that the shape lives in.

For example, if the shape lives in \(\mathbf{R}^3\), then the dimension is 3.

locate(point)

Find a point on the current curve.

Solves for \(s\) in \(B(s) = p\).

Note

A unique solution is only guaranteed if the current curve has no self-intersections. This code assumes, but doesn’t check, that this is true.

../_images/curve_locate.png
>>> curve = bezier.Curve.from_nodes(np.array([
...     [0.0, 0.0],
...     [1.0, 2.0],
...     [3.0, 1.0],
...     [4.0, 0.0],
... ]))
>>> point1 = np.array([[3.09375, 0.703125]])
>>> s = curve.locate(point1)
>>> s
0.75
>>> point2 = np.array([[2.0, 0.5]])
>>> curve.locate(point2) is None
True
Parameters:point (numpy.ndarray) – A (1xD) point on the curve, where \(D\) is the dimension of the curve.
Returns:The parameter value (\(s\)) corresponding to point or None if the point is not on the curve.
Return type:Optional [ float ]
Raises:ValueError – If the dimension of the point doesn’t match the dimension of the current curve.
nodes

numpy.ndarray: The nodes that define the current shape.