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

Note on Python 3
----------------
While I am hopeful that this package entirely works with Python 3, it was born from a larger project coded in Python 2 and has not been thoroughly tested in 
Python 3.  Please let me know if you find any incompatibilities.

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.

Also, a big thanks to the author(s) of `A Primer on Bézier Curves <http://pomax.github.io/bezierinfo/>`_, an outstanding resource for learning about Bézier curves and Bézier curve-related algorithms.

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:: python

    from __future__ import division, print_function

.. code:: python

    # 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:: python

    # 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:: python

    # 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:: python

    # 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:: python

    # 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


Tangent vectors and 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.

.. code:: python

    # 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)


To illustrate the awesomeness 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 polynomials)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: python

    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:: python

    # 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:: python

    # 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:: python

    # 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:: python

    # 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:: python

    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.