svgpathtools
============

svgpathtools is a collection of tools for manipulating and analyzing SVG
Path objects and Bézier curves.

Features
--------

svgpathtools contains functions designed to **easily read, write and
display SVG files** as well as *a large selection of
geometrically-oriented tools* to **transform and analyze path
elements**.

Additionally, the submodule *bezier.py* contains tools for for working
with general **nth order Bezier curves stored as n-tuples**.

Some included tools:

-  **read**, **write**, and **display** SVG files containing Path (and
   other) SVG elements
-  convert Bézier path segments to **numpy.poly1d** (polynomial) objects
-  convert polynomials (in standard form) to their Bézier form
-  compute **tangent vectors** and (right-hand rule) **normal vectors**
-  compute **curvature**
-  break discontinuous paths into their **continuous subpaths**.
-  efficiently compute **intersections** between paths and/or segments
-  find a **bounding box** for a path or segment
-  **reverse** segment/path orientation
-  **crop** and **split** paths and segments
-  **smooth** paths (i.e. smooth away kinks to make paths
   differentiable)
-  **transition maps** from path domain to segment domain and back (T2t
   and t2T)
-  compute **area** enclosed by a closed path
-  compute **arc length**
-  compute **inverse arc length**
-  convert RGB color tuples to hexadecimal color strings and back

Prerequisites
-------------

-  **numpy**
-  **svgwrite**

Setup
-----

If not already installed, you can **install the prerequisites** using
pip.

.. code:: bash

    $ pip install numpy

.. code:: bash

    $ pip install svgwrite

Then **install svgpathtools**:

.. code:: bash

    $ pip install svgpathtools

Alternative Setup
~~~~~~~~~~~~~~~~~

You can download the source from Github and install by using the command
(from inside the folder containing setup.py):

.. code:: bash

    $ python setup.py install

Credit where credit's due
-------------------------

Much of the core of this module was taken from `the svg.path (v2.0)
module <https://github.com/regebro/svg.path>`__. Interested svg.path
users should see the compatibility notes at bottom of this readme.

Basic Usage
-----------

Classes
~~~~~~~

The svgpathtools module is primarily structured around four path segment
classes: ``Line``, ``QuadraticBezier``, ``CubicBezier``, and ``Arc``.
There is also a fifth class, ``Path``, whose objects are sequences of
(connected or disconnected\ `1 <#f1>`__\ ) path segment objects.

-  ``Line(start, end)``

-  ``Arc(start, radius, rotation, large_arc, sweep, end)`` Note: See
   docstring for a detailed explanation of these parameters

-  ``QuadraticBezier(start, control, end)``

-  ``CubicBezier(start, control1, control2, end)``

-  ``Path(*segments)``

See the relevant docstrings in *path.py* or the `official SVG
specifications <http://www.w3.org/TR/SVG/paths.html>`__ for more
information on what each parameter means.

1 Warning: Some of the functionality in this library has not been tested
on discontinuous Path objects. A simple workaround is provided, however,
by the ``Path.continuous_subpaths()`` method. `↩ <#a1>`__

.. code:: ipython2

    from __future__ import division, print_function

.. code:: ipython2

    # Coordinates are given as points in the complex plane
    from svgpathtools import Path, Line, QuadraticBezier, CubicBezier, Arc
    seg1 = CubicBezier(300+100j, 100+100j, 200+200j, 200+300j)  # A cubic beginning at (300, 100) and ending at (200, 300)
    seg2 = Line(200+300j, 250+350j)  # A line beginning at (200, 300) and ending at (250, 350)
    path = Path(seg1, seg2)  # A path traversing the cubic and then the line
    
    # We could alternatively created this Path object using a d-string
    from svgpathtools import parse_path
    path_alt = parse_path('M 300 100 C 100 100 200 200 200 300 L 250 350')
    
    # Let's check that these two methods are equivalent
    print(path)
    print(path_alt)
    print(path == path_alt)
    
    # On a related note, the Path.d() method returns a Path object's d-string
    print(path.d())
    print(parse_path(path.d()) == path)


.. parsed-literal::

    Path(CubicBezier(start=(300+100j), control1=(100+100j), control2=(200+200j), end=(200+300j)),
         Line(start=(200+300j), end=(250+350j)))
    Path(CubicBezier(start=(300+100j), control1=(100+100j), control2=(200+200j), end=(200+300j)),
         Line(start=(200+300j), end=(250+350j)))
    True
    M 300.0,100.0 C 100.0,100.0 200.0,200.0 200.0,300.0 L 250.0,350.0
    True


The ``Path`` class is a mutable sequence, so it behaves much like a
list. So segments can **append**\ ed, **insert**\ ed, set by index,
**del**\ eted, **enumerate**\ d, **slice**\ d out, etc.

.. code:: ipython2

    # Let's append another to the end of it
    path.append(CubicBezier(250+350j, 275+350j, 250+225j, 200+100j))
    print(path)
    
    # Let's replace the first segment with a Line object
    path[0] = Line(200+100j, 200+300j)
    print(path)
    
    # You may have noticed that this path is connected and now is also closed (i.e. path.start == path.end)
    print("path is continuous? ", path.iscontinuous())
    print("path is closed? ", path.isclosed())
    
    # The curve the path follows is not, however, smooth (differentiable)
    from svgpathtools import kinks, smoothed_path
    print("path contains non-differentiable points? ", len(kinks(path)) > 0)
    
    # If we want, we can smooth these out (Experimental and only for line/cubic paths)
    # Note:  smoothing will always works (except on 180 degree turns), but you may want 
    # to play with the maxjointsize and tightness parameters to get pleasing results
    # Note also: smoothing will increase the number of segments in a path
    spath = smoothed_path(path)
    print("spath contains non-differentiable points? ", len(kinks(spath)) > 0)
    print(spath)
    
    # Let's take a quick look at the path and its smoothed relative
    # The following commands will open two browser windows to display path and spaths
    from svgpathtools import disvg
    from time import sleep
    disvg(path) 
    sleep(1)  # needed when not giving the SVGs unique names (or not using timestamp)
    disvg(spath)
    print("Notice that path contains {} segments and spath contains {} segments."
          "".format(len(path), len(spath)))


.. parsed-literal::

    Path(CubicBezier(start=(300+100j), control1=(100+100j), control2=(200+200j), end=(200+300j)),
         Line(start=(200+300j), end=(250+350j)),
         CubicBezier(start=(250+350j), control1=(275+350j), control2=(250+225j), end=(200+100j)))
    Path(Line(start=(200+100j), end=(200+300j)),
         Line(start=(200+300j), end=(250+350j)),
         CubicBezier(start=(250+350j), control1=(275+350j), control2=(250+225j), end=(200+100j)))
    path is continuous?  True
    path is closed?  True
    path contains non-differentiable points?  True
    spath contains non-differentiable points?  False
    Path(Line(start=(200+101.5j), end=(200+298.5j)),
         CubicBezier(start=(200+298.5j), control1=(200+298.505j), control2=(201.057124638+301.057124638j), end=(201.060660172+301.060660172j)),
         Line(start=(201.060660172+301.060660172j), end=(248.939339828+348.939339828j)),
         CubicBezier(start=(248.939339828+348.939339828j), control1=(249.649982143+349.649982143j), control2=(248.995+350j), end=(250+350j)),
         CubicBezier(start=(250+350j), control1=(275+350j), control2=(250+225j), end=(200+100j)),
         CubicBezier(start=(200+100j), control1=(199.62675237+99.0668809257j), control2=(200+100.495j), end=(200+101.5j)))
    Notice that path contains 3 segments and spath contains 6 segments.


Reading SVGSs
~~~~~~~~~~~~~

| The **svg2paths()** function converts an svgfile to a list of Path
  objects and a separate list of dictionaries containing the attributes
  of each said path.
| Note: Line, Polyline, Polygon, and Path SVG elements can all be
  converted to Path objects using this function.

.. code:: ipython2

    # Read SVG into a list of path objects and list of dictionaries of attributes 
    from svgpathtools import svg2paths, wsvg
    paths, attributes = svg2paths('test.svg')
    
    # Update: You can now also extract the svg-attributes by setting
    # return_svg_attributes=True, or with the convenience function svg2paths2
    from svgpathtools import svg2paths2
    paths, attributes, svg_attributes = svg2paths2('test.svg')
    
    # Let's print out the first path object and the color it was in the SVG
    # We'll see it is composed of two CubicBezier objects and, in the SVG file it 
    # came from, it was red
    redpath = paths[0]
    redpath_attribs = attributes[0]
    print(redpath)
    print(redpath_attribs['stroke'])


.. parsed-literal::

    Path(CubicBezier(start=(10.5+80j), control1=(40+10j), control2=(65+10j), end=(95+80j)),
         CubicBezier(start=(95+80j), control1=(125+150j), control2=(150+150j), end=(180+80j)))
    red


Writing SVGSs (and some geometric functions and methods)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The **wsvg()** function creates an SVG file from a list of path. This
function can do many things (see docstring in *paths2svg.py* for more
information) and is meant to be quick and easy to use. Note: Use the
convenience function **disvg()** (or set 'openinbrowser=True') to
automatically attempt to open the created svg file in your default SVG
viewer.

.. code:: ipython2

    # Let's make a new SVG that's identical to the first
    wsvg(paths, attributes=attributes, svg_attributes=svg_attributes, filename='output1.svg')

.. figure:: https://cdn.rawgit.com/mathandy/svgpathtools/master/output1.svg
   :alt: output1.svg

   output1.svg

There will be many more examples of writing and displaying path data
below.

The .point() method and transitioning between path and path segment parameterizations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

SVG Path elements and their segments have official parameterizations.
These parameterizations can be accessed using the ``Path.point()``,
``Line.point()``, ``QuadraticBezier.point()``, ``CubicBezier.point()``,
and ``Arc.point()`` methods. All these parameterizations are defined
over the domain 0 <= t <= 1.

| **Note:** In this document and in inline documentation and doctrings,
  I use a capital ``T`` when referring to the parameterization of a Path
  object and a lower case ``t`` when referring speaking about path
  segment objects (i.e. Line, QaudraticBezier, CubicBezier, and Arc
  objects).
| Given a ``T`` value, the ``Path.T2t()`` method can be used to find the
  corresponding segment index, ``k``, and segment parameter, ``t``, such
  that ``path.point(T)=path[k].point(t)``.
| There is also a ``Path.t2T()`` method to solve the inverse problem.

.. code:: ipython2

    # Example:
    
    # Let's check that the first segment of redpath starts 
    # at the same point as redpath
    firstseg = redpath[0] 
    print(redpath.point(0) == firstseg.point(0) == redpath.start == firstseg.start)
    
    # Let's check that the last segment of redpath ends on the same point as redpath
    lastseg = redpath[-1] 
    print(redpath.point(1) == lastseg.point(1) == redpath.end == lastseg.end)
    
    # This next boolean should return False as redpath is composed multiple segments
    print(redpath.point(0.5) == firstseg.point(0.5))
    
    # If we want to figure out which segment of redpoint the 
    # point redpath.point(0.5) lands on, we can use the path.T2t() method
    k, t = redpath.T2t(0.5)
    print(redpath[k].point(t) == redpath.point(0.5))


.. parsed-literal::

    True
    True
    False
    True


Bezier curves as NumPy polynomial objects
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

| Another great way to work with the parameterizations for ``Line``,
  ``QuadraticBezier``, and ``CubicBezier`` objects is to convert them to
  ``numpy.poly1d`` objects. This is done easily using the
  ``Line.poly()``, ``QuadraticBezier.poly()`` and ``CubicBezier.poly()``
  methods.
| There's also a ``polynomial2bezier()`` function in the pathtools.py
  submodule to convert polynomials back to Bezier curves.

**Note:** cubic Bezier curves are parameterized as

.. math:: \mathcal{B}(t) = P_0(1-t)^3 + 3P_1(1-t)^2t + 3P_2(1-t)t^2 + P_3t^3

where :math:`P_0`, :math:`P_1`, :math:`P_2`, and :math:`P_3` are the
control points ``start``, ``control1``, ``control2``, and ``end``,
respectively, that svgpathtools uses to define a CubicBezier object. The
``CubicBezier.poly()`` method expands this polynomial to its standard
form

.. math:: \mathcal{B}(t) = c_0t^3 + c_1t^2 +c_2t+c3

 where

.. math::

   \begin{bmatrix}c_0\\c_1\\c_2\\c_3\end{bmatrix} = 
   \begin{bmatrix}
   -1 & 3 & -3 & 1\\
   3 & -6 & -3 & 0\\
   -3 & 3 & 0 & 0\\
   1 & 0 & 0 & 0\\
   \end{bmatrix}
   \begin{bmatrix}P_0\\P_1\\P_2\\P_3\end{bmatrix}

``QuadraticBezier.poly()`` and ``Line.poly()`` are `defined
similarly <https://en.wikipedia.org/wiki/B%C3%A9zier_curve#General_definition>`__.

.. code:: ipython2

    # Example:
    b = CubicBezier(300+100j, 100+100j, 200+200j, 200+300j)
    p = b.poly()
    
    # p(t) == b.point(t)
    print(p(0.235) == b.point(0.235))
    
    # What is p(t)?  It's just the cubic b written in standard form.  
    bpretty = "{}*(1-t)^3 + 3*{}*(1-t)^2*t + 3*{}*(1-t)*t^2 + {}*t^3".format(*b.bpoints())
    print("The CubicBezier, b.point(x) = \n\n" + 
          bpretty + "\n\n" + 
          "can be rewritten in standard form as \n\n" +
          str(p).replace('x','t'))


.. parsed-literal::

    True
    The CubicBezier, b.point(x) = 
    
    (300+100j)*(1-t)^3 + 3*(100+100j)*(1-t)^2*t + 3*(200+200j)*(1-t)*t^2 + (200+300j)*t^3
    
    can be rewritten in standard form as 
    
                    3                2
    (-400 + -100j) t + (900 + 300j) t - 600 t + (300 + 100j)


The ability to convert between Bezier objects to NumPy polynomial
objects is very useful. For starters, we can take turn a list of Bézier
segments into a NumPy array

Numpy Array operations on Bézier path segments
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`Example available
here <https://github.com/mathandy/svgpathtools/blob/master/examples/compute-many-points-quickly-using-numpy-arrays.py>`__

To further illustrate the power of being able to convert our Bezier
curve objects to numpy.poly1d objects and back, lets compute the unit
tangent vector of the above CubicBezier object, b, at t=0.5 in four
different ways.

Tangent vectors (and more on NumPy polynomials)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython2

    t = 0.5
    ### Method 1: the easy way
    u1 = b.unit_tangent(t)
    
    ### Method 2: another easy way 
    # Note: This way will fail if it encounters a removable singularity.
    u2 = b.derivative(t)/abs(b.derivative(t))
    
    ### Method 2: a third easy way 
    # Note: This way will also fail if it encounters a removable singularity.
    dp = p.deriv() 
    u3 = dp(t)/abs(dp(t))
    
    ### Method 4: the removable-singularity-proof numpy.poly1d way  
    # Note: This is roughly how Method 1 works
    from svgpathtools import real, imag, rational_limit
    dx, dy = real(dp), imag(dp)  # dp == dx + 1j*dy 
    p_mag2 = dx**2 + dy**2  # p_mag2(t) = |p(t)|**2
    # Note: abs(dp) isn't a polynomial, but abs(dp)**2 is, and,
    #  the limit_{t->t0}[f(t) / abs(f(t))] == 
    # sqrt(limit_{t->t0}[f(t)**2 / abs(f(t))**2])
    from cmath import sqrt
    u4 = sqrt(rational_limit(dp**2, p_mag2, t))
    
    print("unit tangent check:", u1 == u2 == u3 == u4)
    
    # Let's do a visual check
    mag = b.length()/4  # so it's not hard to see the tangent line
    tangent_line = Line(b.point(t), b.point(t) + mag*u1)
    disvg([b, tangent_line], 'bg', nodes=[b.point(t)])


.. parsed-literal::

    unit tangent check: True


Translations (shifts), reversing orientation, and normal vectors
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython2

    # Speaking of tangents, let's add a normal vector to the picture
    n = b.normal(t)
    normal_line = Line(b.point(t), b.point(t) + mag*n)
    disvg([b, tangent_line, normal_line], 'bgp', nodes=[b.point(t)])
    
    # and let's reverse the orientation of b! 
    # the tangent and normal lines should be sent to their opposites
    br = b.reversed()
    
    # Let's also shift b_r over a bit to the right so we can view it next to b
    # The simplest way to do this is br = br.translated(3*mag),  but let's use 
    # the .bpoints() instead, which returns a Bezier's control points
    br.start, br.control1, br.control2, br.end = [3*mag + bpt for bpt in br.bpoints()]  # 
    
    tangent_line_r = Line(br.point(t), br.point(t) + mag*br.unit_tangent(t))
    normal_line_r = Line(br.point(t), br.point(t) + mag*br.normal(t))
    wsvg([b, tangent_line, normal_line, br, tangent_line_r, normal_line_r], 
         'bgpkgp', nodes=[b.point(t), br.point(t)], filename='vectorframes.svg', 
         text=["b's tangent", "br's tangent"], text_path=[tangent_line, tangent_line_r])

.. figure:: https://cdn.rawgit.com/mathandy/svgpathtools/master/vectorframes.svg
   :alt: vectorframes.svg

   vectorframes.svg

Rotations and Translations
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython2

    # Let's take a Line and an Arc and make some pictures
    top_half = Arc(start=-1, radius=1+2j, rotation=0, large_arc=1, sweep=1, end=1)
    midline = Line(-1.5, 1.5)
    
    # First let's make our ellipse whole
    bottom_half = top_half.rotated(180)
    decorated_ellipse = Path(top_half, bottom_half)
    
    # Now let's add the decorations
    for k in range(12):
        decorated_ellipse.append(midline.rotated(30*k))
        
    # Let's move it over so we can see the original Line and Arc object next
    # to the final product
    decorated_ellipse = decorated_ellipse.translated(4+0j)
    wsvg([top_half, midline, decorated_ellipse], filename='decorated_ellipse.svg')

.. figure:: https://cdn.rawgit.com/mathandy/svgpathtools/master/decorated_ellipse.svg
   :alt: decorated\_ellipse.svg

   decorated\_ellipse.svg

arc length and inverse arc length
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Here we'll create an SVG that shows off the parametric and geometric
midpoints of the paths from ``test.svg``. We'll need to compute use the
``Path.length()``, ``Line.length()``, ``QuadraticBezier.length()``,
``CubicBezier.length()``, and ``Arc.length()`` methods, as well as the
related inverse arc length methods ``.ilength()`` function to do this.

.. code:: ipython2

    # First we'll load the path data from the file test.svg
    paths, attributes = svg2paths('test.svg')
    
    # Let's mark the parametric midpoint of each segment
    # I say "parametric" midpoint because Bezier curves aren't 
    # parameterized by arclength 
    # If they're also the geometric midpoint, let's mark them
    # purple and otherwise we'll mark the geometric midpoint green
    min_depth = 5
    error = 1e-4
    dots = []
    ncols = []
    nradii = []
    for path in paths:
        for seg in path:
            parametric_mid = seg.point(0.5)
            seg_length = seg.length()
            if seg.length(0.5)/seg.length() == 1/2:
                dots += [parametric_mid]
                ncols += ['purple']
                nradii += [5]
            else:
                t_mid = seg.ilength(seg_length/2)
                geo_mid = seg.point(t_mid)
                dots += [parametric_mid, geo_mid]
                ncols += ['red', 'green']
                nradii += [5] * 2
    
    # In 'output2.svg' the paths will retain their original attributes
    wsvg(paths, nodes=dots, node_colors=ncols, node_radii=nradii, 
         attributes=attributes, filename='output2.svg')

.. figure:: https://cdn.rawgit.com/mathandy/svgpathtools/master/output2.svg
   :alt: output2.svg

   output2.svg

Intersections between Bezier curves
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython2

    # Let's find all intersections between redpath and the other 
    redpath = paths[0]
    redpath_attribs = attributes[0]
    intersections = []
    for path in paths[1:]:
        for (T1, seg1, t1), (T2, seg2, t2) in redpath.intersect(path):
            intersections.append(redpath.point(T1))
            
    disvg(paths, filename='output_intersections.svg', attributes=attributes,
          nodes = intersections, node_radii = [5]*len(intersections))

.. figure:: https://cdn.rawgit.com/mathandy/svgpathtools/master/output_intersections.svg
   :alt: output\_intersections.svg

   output\_intersections.svg

An Advanced Application: Offsetting Paths
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Here we'll find the `offset
curve <https://en.wikipedia.org/wiki/Parallel_curve>`__ for a few paths.

.. code:: ipython2

    from svgpathtools import parse_path, Line, Path, wsvg
    def offset_curve(path, offset_distance, steps=1000):
        """Takes in a Path object, `path`, and a distance,
        `offset_distance`, and outputs an piecewise-linear approximation 
        of the 'parallel' offset curve."""
        nls = []
        for seg in path:
            ct = 1
            for k in range(steps):
                t = k / steps
                offset_vector = offset_distance * seg.normal(t)
                nl = Line(seg.point(t), seg.point(t) + offset_vector)
                nls.append(nl)
        connect_the_dots = [Line(nls[k].end, nls[k+1].end) for k in range(len(nls)-1)]
        if path.isclosed():
            connect_the_dots.append(Line(nls[-1].end, nls[0].end))
        offset_path = Path(*connect_the_dots)
        return offset_path
    
    # Examples:
    path1 = parse_path("m 288,600 c -52,-28 -42,-61 0,-97 ")
    path2 = parse_path("M 151,395 C 407,485 726.17662,160 634,339").translated(300)
    path3 = parse_path("m 117,695 c 237,-7 -103,-146 457,0").translated(500+400j)
    paths = [path1, path2, path3]
    
    offset_distances = [10*k for k in range(1,51)]
    offset_paths = []
    for path in paths:
        for distances in offset_distances:
            offset_paths.append(offset_curve(path, distances))
    
    # Note: This will take a few moments
    wsvg(paths + offset_paths, 'g'*len(paths) + 'r'*len(offset_paths), filename='offset_curves.svg')

.. figure:: https://cdn.rawgit.com/mathandy/svgpathtools/master/offset_curves.svg
   :alt: offset\_curves.svg

   offset\_curves.svg

Compatibility Notes for users of svg.path (v2.0)
------------------------------------------------

-  renamed Arc.arc attribute as Arc.large\_arc

-  Path.d() : For behavior similar\ `2 <#f2>`__\  to svg.path (v2.0),
   set both useSandT and use\_closed\_attrib to be True.

2 The behavior would be identical, but the string formatting used in
this method has been changed to use default format (instead of the
General format, {:G}), for inceased precision. `↩ <#a2>`__

Licence
-------

This module is under a MIT License.