# Module `path`

The `path` module defines several important classes which are documented in the present section.

## Class `path` — PostScript-like paths

class `path.``path`(*pathitems)

  This class represents a PostScript like path consisting of the path elements pathitems.

  All possible path items are described in Sect. Path elements. Note that there are restrictions on the first path element and likewise on each path element after a `closepath` directive. In both cases, no current point is defined and the path element has to be an instance of one of the following classes: `moveto`, `arc`, and `arcn`.

Instances of the class `path` provide the following methods (in alphabetic order):

`path.``append`(pathitem)

  Appends a pathitem to the end of the path.

`path.``arclen`()

  Returns the total arc length of the path. [1]

`path.``arclentoparam`(lengths)

  Returns the parameter value(s) corresponding to the arc length(s) lengths. [1]

`path.``at`(params)

  Returns the coordinates (as 2-tuple) of the path point(s) corresponding to the parameter value(s) params. [1] [2]

`path.``atbegin`()

  Returns the coordinates (as 2-tuple) of the first point of the path. [1]

`path.``atend`()

  Returns the coordinates (as 2-tuple) of the end point of the path. [1]

`path.``bbox`()

  Returns the bounding box of the path.

`path.``begin`()

  Returns the parameter value (a `normpathparam` instance) of the first point in the path.

`path.``curveradius`(params)

  Returns the curvature radius/radii (or None if infinite) at parameter value(s) params. [2] This is the inverse of the curvature at this parameter. Note that this radius can be negative or positive, depending on the sign of the curvature. [1]

`path.``end`()

  Returns the parameter value (a `normpathparam` instance) of the last point in the path.

`path.``extend`(pathitems)

  Appends the list pathitems to the end of the path.

`path.``intersect`(opath)

  Returns a tuple consisting of two lists of parameter values corresponding to the intersection points of the path with the other path opath, respectively. [1] For intersection points which are not farther apart then epsilon (defaulting to \(10^{-5}\) PostScript points), only one is returned.

`path.``joined`(opath)

  Appends opath to the end of the path, thereby merging the last subpath (which must not be closed) of the path with the first sub path of opath and returns the resulting new path. [1] Instead of using the `joined()` method, you can also join two paths together with help of the `<<` operator, for instance `p = p1 << p2`.

`path.``normpath`(epsilon=None)

  Returns the equivalent `normpath`. For the conversion and for later calculations with this `normpath` an accuracy of epsilon is used. If epsilon is None, the global epsilon of the `path` module is used.

`path.``paramtoarclen`(params)

  Returns the arc length(s) corresponding to the parameter value(s) params. [2] [1]

`path.``range`()

  Returns the maximal parameter value param that is allowed in the path methods.

`path.``reversed`()

  Returns the reversed path. [1]

`path.``rotation`(params)

  Returns a transformation or a list of transformations, which rotate the x-direction to the tangent vector and the y-direction to the normal vector at the parameter value(s) params. [2] [1]

`path.``split`(params)

  Splits the path at the parameter values params, which have to be sorted in ascending order, and returns a corresponding list of `normpath` instances. [1]

`path.``tangent`(params, length=1)

  Return a `line` instance or a list of `line` instances, corresponding to the tangent vectors at the parameter value(s) params. [2] The tangent vector will be scaled to the length length. [1]

`path.``trafo`(params)

  Returns a transformation or a list of tranformations, which translate the origin to a point on the path corresponding to parameter value(s) params and rotate the x-direction to the tangent vector and the y-direction to the normal vector. [1]

`path.``transformed`(trafo)

  Returns the path transformed according to the linear transformation trafo. Here, `trafo` must be an instance of the `trafo.trafo` class. [1]

[1] | (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16) This method requires a prior conversion of the path into a `normpath` instance. This is done automatically (using the precision epsilon set globally using `path.set()`). If you need a different epsilon for a normpath, you also can perform the conversion manually.

[2]

(1, 2, 3, 4, 5) In these methods, params may either be a single value or a list. In the latter case, the result of the method will be a list consisting of the results for each parameter. The parameter itself may either be a length (or a number which is then interpreted as a user length) or an instance of the class `normpathparam`. In the former case, the length refers to the arc length along the path.

## Path elements

The class `pathitem` is the superclass of all PostScript path construction primitives. It is never used directly, but only by instantiating its subclasses, which correspond one by one to the PostScript primitives.

Except for the path elements ending in `_pt`, all coordinates passed to the path elements can be given as number (in which case they are interpreted as user units with the currently set default type) or in PyX lengths.

The following operation move the current point and open a new subpath:

class `path.``moveto`(x, y)

  Path element which sets the current point to the absolute coordinates (x, y). This operation opens a new subpath.

class `path.``rmoveto`(dx, dy)

  Path element which moves the current point by (dx, dy). This operation opens a new subpath.

Drawing a straight line can be accomplished using:

class `path.``lineto`(x, y)

  Path element which appends a straight line from the current point to the point with absolute coordinates (x, y), which becomes the new current point.

class `path.``rlineto`(dx, dy)

  Path element which appends a straight line from the current point to the point with relative coordinates (dx, dy), which becomes the new current point.

For the construction of arc segments, the following three operations are available:

class `path.``arc`(x, y, r, angle1, angle2)

  Path element which appends an arc segment in counterclockwise direction with absolute coordinates (x, y) of the center and radius r from angle1 to angle2 (in degrees). If before the operation, the current point is defined, a straight line from the current point to the beginning of the arc segment is prepended. Otherwise, a subpath, which thus is the first one in the path, is opened. After the operation, the current point is at the end of the arc segment.

class `path.``arcn`(x, y, r, angle1, angle2)

  Same as `arc` but in clockwise direction.

class `path.``arct`(x1, y1, x2, y2, r)

  Path element consisting of a line followed by an arc of radius r. The arc is part of the circle inscribed to the angle at x1, y1 given by lines in the directions to the current point and to x2, y2. The initial line connects the current point to the point where the circle touches the line through the current point and x1, y1. The arc then continues to the point where the circle touches the line through x1, y1 and x2, y2.

Bézier curves can be constructed using:

class `path.``curveto`(x1, y1, x2, y2, x3, y3)

  Path element which appends a Bézier curve with the current point as first control point and the other control points (x1, y1), (x2, y2), and (x3, y3).

class `path.``rcurveto`(dx1, dy1, dx2, dy2, dx3, dy3)

  Path element which appends a Bézier curve with the current point as first control point and the other control points defined relative to the current point by the coordinates (dx1, dy1), (dx2, dy2), and (dx3, dy3).

Note that when calculating the bounding box (see Sect. `bbox`) of Bézier curves, PyX uses for performance reasons the so-called control box, i.e., the smallest rectangle enclosing the four control points of the Bézier curve. In general, this is not the smallest rectangle enclosing the Bézier curve.

Finally, an open subpath can be closed using:

class `path.``closepath`

  Path element which closes the current subpath.

For performance reasons, two non-PostScript path elements are defined, which perform multiple identical operations:

class `path.``multilineto_pt`(points_pt)

  Path element which appends straight line segments starting from the current point and going through the list of points given in the points_pt argument. All coordinates have to be given in PostScript points.

class `path.``multicurveto_pt`(points_pt)

  Path element which appends Bézier curve segments starting from the current point. points_pt is a sequence of 6-tuples containing the coordinates of the two control points and the end point of a multicurveto segment.

## Class `normpath`

The `normpath` class is used internally for all non-trivial path operations, cf. footnote [1] in Sect. Class path — PostScript-like paths. It represents a path as a list of subpaths, which are instances of the class `normsubpath`. These `normsubpath`s themselves consist of a list of `normsubpathitems` which are either straight lines (`normline`) or Bézier curves (`normcurve`).

A given path `p` can easily be converted to the corresponding `normpath` `np` by:

    np = p.normpath()

Additionally, the accuracy is used in all `normpath` calculations can be specified by means of the argument epsilon, which defaults to to \(10^{-5}\) where units of PostScript points are understood. This default value can be changed using the module function `path.set()`.

To construct a `normpath` from a list of `normsubpath` instances, they are passed to the `normpath` constructor:

class `path.``normpath`(normsubpaths=[])

  Construct a `normpath` consisting of subnormpaths, which is a list of `subnormpath` instances.

Instances of `normpath` offer all methods of regular `path` instances, which also have the same semantics. An exception are the methods `append()` and `extend()`. While they allow for adding of instances of `subnormpath` to the `normpath` instance, they also keep the functionality of a regular path and allow for regular path elements to be appended. The latter are converted to the proper normpath representation during addition.

In addition to the `path` methods, a `normpath` instance also offers the following methods, which operate on the instance itself, i.e., modify it in place.

`normpath.``join`(other)

  Join other, which has to be a `path` instance, to the `normpath` instance.

`normpath.``reverse`()

  Reverses the `normpath` instance.

`normpath.``transform`(trafo)

  Transforms the `normpath` instance according to the linear transformation trafo.

Finally, we remark that the sum of a `normpath` and a `path` always yields a `normpath`.

## Class `normsubpath`

class `path.``normsubpath`(normsubpathitems=[], closed=0, epsilon=1e-5)

  Construct a `normsubpath` consisting of normsubpathitems, which is a list of `normsubpathitem` instances. If closed is set, the `normsubpath` will be closed, thereby appending a straight line segment from the first to the last point, if it is not already present. All calculations with the `normsubpath` are performed with an accuracy of epsilon (in units of PostScript points).

Most `normsubpath` methods behave like the ones of a `path`.

Exceptions are:

`normsubpath.``append`(anormsubpathitem)

  Append the normsubpathitem to the end of the `normsubpath` instance. This is only possible if the `normsubpath` is not closed, otherwise an `NormpathException` is raised.

`normsubpath.``extend`(normsubpathitems)

  Extend the `normsubpath` instances by normsubpathitems, which has to be a list of `normsubpathitem` instances. This is only possible if the `normsubpath` is not closed, otherwise an `NormpathException` is raised.

`normsubpath.``close`()

  Close the `normsubpath` instance by appending a straight line segment from the first to the last point, if not already present.

## Predefined paths

For convenience, some often used paths are already predefined. All of them are subclasses of the `path` class.

class `path.``line`(x0, y0, x1, y1)

  A straight line from the point (x0, y0) to the point (x1, y1).

class `path.``curve`(x0, y0, x1, y1, x2, y2, x3, y3)

  A Bézier curve with control points (x0, y0), \(\dots\), (x3, y3).

class `path.``rect`(x, y, w, h)

  A closed rectangle with lower left point (x, y), width w, and height h.

class `path.``circle`(x, y, r)

  A closed circle with center (x, y) and radius r.