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# Transformations Tutorial

Transforming visuals in Matplotlib.

Like any graphics packages, matplotlib is built on top of a transformation framework to easily move between coordinate systems, the userland `data` coordinate system, the `axes` coordinate system, the `figure` coordinate system, and the `display` coordinate system. In 95% of your plotting, you won’t need to think about this, as it happens under the hood, but as you push the limits of custom figure generation, it helps to have an understanding of these objects so you can reuse the existing transformations matplotlib makes available to you, or create your own (see `matplotlib.transforms`). The table below summarizes the existing coordinate systems, the transformation object you should use to work in that coordinate system, and the description of that system. In the `Transformation Object` column, `ax` is a `Axes` instance, and `fig` is a `Figure` instance.

Coordinate | Transformation Object | Description
--- | --- | ---
`data` | `ax.transData` | The userland data coordinate system, controlled by the xlim and ylim
`axes` | `ax.transAxes` | The coordinate system of the `Axes`; (0, 0) is bottom left of the axes, and (1, 1) is top right of the axes.
`figure` | `fig.transFigure` | The coordinate system of the `Figure`; (0, 0) is bottom left of the figure, and (1, 1) is top right of the figure.
`display` | `None` | This is the pixel coordinate system of the display; (0, 0) is the bottom left of the display, and (width, height) is the top right of the display in pixels. Alternatively, the identity transform (`matplotlib.transforms.IdentityTransform()`) may be used instead of None.

All of the transformation objects in the table above take inputs in their coordinate system, and transform the input to the `display` coordinate system. That is why the `display` coordinate system has `None` for the `Transformation Object` column – it already is in display coordinates. The transformations also know how to invert themselves, to go from `display` back to the native coordinate system. This is particularly useful when processing events from the user interface, which typically occur in display space, and you want to know where the mouse click or key-press occurred in your data coordinate system.

## Data coordinates

Let’s start with the most commonly used coordinate, the `data` coordinate system. Whenever you add data to the axes, matplotlib updates the datalimits, most commonly updated with the `set_xlim()` and `set_ylim()` methods. For example, in the figure below, the data limits stretch from 0 to 10 on the x-axis, and -1 to 1 on the y-axis.

    import numpy as np
    import matplotlib.pyplot as plt

    x = np.arange(0, 10, 0.005)
    y = np.exp(-x/2.) * np.sin(2*np.pi*x)

    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.plot(x, y)
    ax.set_xlim(0, 10)
    ax.set_ylim(-1, 1)

    plt.show()

You can use the `ax.transData` instance to transform from your `data` to your `display` coordinate system, either a single point or a sequence of points as shown below:

    In [14]: type(ax.transData)
    Out[14]: <class 'matplotlib.transforms.CompositeGenericTransform'>

    In [15]: ax.transData.transform((5, 0))
    Out[15]: array([ 335.175,  247.   ])

    In [16]: ax.transData.transform([(5, 0), (1, 2)])
    Out[16]:
    array([[ 335.175,  247.   ],
           [ 132.435,  642.2  ]])

You can use the `inverted()` method to create a transform which will take you from display to data coordinates:

    In [41]: inv = ax.transData.inverted()

    In [42]: type(inv)
    Out[42]: <class 'matplotlib.transforms.CompositeGenericTransform'>

    In [43]: inv.transform((335.175,  247.))
    Out[43]: array([ 5.,  0.])

If your are typing along with this tutorial, the exact values of the display coordinates may differ if you have a different window size or dpi setting. Likewise, in the figure below, the display labeled points are probably not the same as in the ipython session because the documentation figure size defaults are different.

    x = np.arange(0, 10, 0.005)
    y = np.exp(-x/2.) * np.sin(2*np.pi*x)

    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.plot(x, y)
    ax.set_xlim(0, 10)
    ax.set_ylim(-1, 1)

    xdata, ydata = 5, 0
    xdisplay, ydisplay = ax.transData.transform_point((xdata, ydata))

    bbox = dict(boxstyle="round", fc="0.8")
    arrowprops = dict(
        arrowstyle="->",
        connectionstyle="angle,angleA=0,angleB=90,rad=10")

    offset = 72
    ax.annotate('data = (%.1f, %.1f)' % (xdata, ydata),
                (xdata, ydata), xytext=(-2*offset, offset), textcoords='offset points',
                bbox=bbox, arrowprops=arrowprops)

    disp = ax.annotate('display = (%.1f, %.1f)' % (xdisplay, ydisplay),
                       (xdisplay, ydisplay), xytext=(0.5*offset, -offset),
                       xycoords='figure pixels',
                       textcoords='offset points',
                       bbox=bbox, arrowprops=arrowprops)

    plt.show()

Note

If you run the source code in the example above in a GUI backend, you may also find that the two arrows for the `data` and `display` annotations do not point to exactly the same point. This is because the display point was computed before the figure was displayed, and the GUI backend may slightly resize the figure when it is created. The effect is more pronounced if you resize the figure yourself. This is one good reason why you rarely want to work in display space, but you can connect to the `'on_draw'` `Event` to update figure coordinates on figure draws; see Event handling and picking.

When you change the x or y limits of your axes, the data limits are updated so the transformation yields a new display point. Note that when we just change the ylim, only the y-display coordinate is altered, and when we change the xlim too, both are altered. More on this later when we talk about the `Bbox`.

    In [54]: ax.transData.transform((5, 0))
    Out[54]: array([ 335.175,  247.   ])

    In [55]: ax.set_ylim(-1, 2)
    Out[55]: (-1, 2)

    In [56]: ax.transData.transform((5, 0))
    Out[56]: array([ 335.175     ,  181.13333333])

    In [57]: ax.set_xlim(10, 20)
    Out[57]: (10, 20)

    In [58]: ax.transData.transform((5, 0))
    Out[58]: array([-171.675     ,  181.13333333])

## Axes coordinates

After the `data` coordinate system, `axes` is probably the second most useful coordinate system. Here the point (0, 0) is the bottom left of your axes or subplot, (0.5, 0.5) is the center, and (1.0, 1.0) is the top right. You can also refer to points outside the range, so (-0.1, 1.1) is to the left and above your axes. This coordinate system is extremely useful when placing text in your axes, because you often want a text bubble in a fixed, location, e.g., the upper left of the axes pane, and have that location remain fixed when you pan or zoom. Here is a simple example that creates four panels and labels them ‘A’, ‘B’, ‘C’, ‘D’ as you often see in journals.

    fig = plt.figure()
    for i, label in enumerate(('A', 'B', 'C', 'D')):
        ax = fig.add_subplot(2, 2, i+1)
        ax.text(0.05, 0.95, label, transform=ax.transAxes,
                fontsize=16, fontweight='bold', va='top')

    plt.show()

You can also make lines or patches in the axes coordinate system, but this is less useful in my experience than using `ax.transAxes` for placing text. Nonetheless, here is a silly example which plots some random dots in `data` space, and overlays a semi-transparent `Circle` centered in the middle of the axes with a radius one quarter of the axes – if your axes does not preserve aspect ratio (see `set_aspect()`), this will look like an ellipse. Use the pan/zoom tool to move around, or manually change the data xlim and ylim, and you will see the data move, but the circle will remain fixed because it is not in `data` coordinates and will always remain at the center of the axes.

    import matplotlib.patches as patches

    fig = plt.figure()
    ax = fig.add_subplot(111)
    x, y = 10*np.random.rand(2, 1000)
    ax.plot(x, y, 'go')  # plot some data in data coordinates

    circ = patches.Circle((0.5, 0.5), 0.25, transform=ax.transAxes,
                          facecolor='yellow', alpha=0.5)
    ax.add_patch(circ)
    plt.show()

## Blended transformations

Drawing in `blended` coordinate spaces which mix `axes` with `data` coordinates is extremely useful, for example to create a horizontal span which highlights some region of the y-data but spans across the x-axis regardless of the data limits, pan or zoom level, etc. In fact these blended lines and spans are so useful, we have built in functions to make them easy to plot (see `axhline()`, `axvline()`, `axhspan()`, `axvspan()`) but for didactic purposes we will implement the horizontal span here using a blended transformation. This trick only works for separable transformations, like you see in normal Cartesian coordinate systems, but not on inseparable transformations like the `PolarTransform`.

    import matplotlib.transforms as transforms

    fig = plt.figure()
    ax = fig.add_subplot(111)

    x = np.random.randn(1000)

    ax.hist(x, 30)
    ax.set_title(r'$\sigma=1 \/ \dots \/ \sigma=2$', fontsize=16)

    # the x coords of this transformation are data, and the
    # y coord are axes
    trans = transforms.blended_transform_factory(
        ax.transData, ax.transAxes)

    # highlight the 1..2 stddev region with a span.
    # We want x to be in data coordinates and y to
    # span from 0..1 in axes coords
    rect = patches.Rectangle((1, 0), width=1, height=1,
                             transform=trans, color='yellow',
                             alpha=0.5)

    ax.add_patch(rect)

    plt.show()

Note

The blended transformations where x is in data coords and y in axes coordinates is so useful that we have helper methods to return the versions mpl uses internally for drawing ticks, ticklabels, etc. The methods are `matplotlib.axes.Axes.get_xaxis_transform()` and `matplotlib.axes.Axes.get_yaxis_transform()`. So in the example above, the call to `blended_transform_factory()` can be replaced by `get_xaxis_transform`:

    trans = ax.get_xaxis_transform()

## Using offset transforms to create a shadow effect

One use of transformations is to create a new transformation that is offset from another transformation, e.g., to place one object shifted a bit relative to another object. Typically you want the shift to be in some physical dimension, like points or inches rather than in data coordinates, so that the shift effect is constant at different zoom levels and dpi settings.

One use for an offset is to create a shadow effect, where you draw one object identical to the first just to the right of it, and just below it, adjusting the zorder to make sure the shadow is drawn first and then the object it is shadowing above it. The transforms module has a helper transformation `ScaledTranslation`. It is instantiated with:

    trans = ScaledTranslation(xt, yt, scale_trans)

where `xt` and `yt` are the translation offsets, and `scale_trans` is a transformation which scales `xt` and `yt` at transformation time before applying the offsets. A typical use case is to use the figure `fig.dpi_scale_trans` transformation for the `scale_trans` argument, to first scale `xt` and `yt` specified in points to `display` space before doing the final offset. The dpi and inches offset is a common-enough use case that we have a special helper function to create it in `matplotlib.transforms.offset_copy()`, which returns a new transform with an added offset. But in the example below, we’ll create the offset transform ourselves. Note the use of the plus operator in:

    offset = transforms.ScaledTranslation(dx, dy,
      fig.dpi_scale_trans)
    shadow_transform = ax.transData + offset

showing that can chain transformations using the addition operator. This code says: first apply the data transformation `ax.transData` and then translate the data by `dx` and `dy` points. In typography, a`point <https://en.wikipedia.org/wiki/Point_%28typography%29>`_ is 1/72 inches, and by specifying your offsets in points, your figure will look the same regardless of the dpi resolution it is saved in.

    fig = plt.figure()
    ax = fig.add_subplot(111)

    # make a simple sine wave
    x = np.arange(0., 2., 0.01)
    y = np.sin(2*np.pi*x)
    line, = ax.plot(x, y, lw=3, color='blue')

    # shift the object over 2 points, and down 2 points
    dx, dy = 2/72., -2/72.
    offset = transforms.ScaledTranslation(dx, dy, fig.dpi_scale_trans)
    shadow_transform = ax.transData + offset

    # now plot the same data with our offset transform;
    # use the zorder to make sure we are below the line
    ax.plot(x, y, lw=3, color='gray',
            transform=shadow_transform,
            zorder=0.5*line.get_zorder())

    ax.set_title('creating a shadow effect with an offset transform')
    plt.show()

## The transformation pipeline

The `ax.transData` transform we have been working with in this tutorial is a composite of three different transformations that comprise the transformation pipeline from `data` -> `display` coordinates. Michael Droettboom implemented the transformations framework, taking care to provide a clean API that segregated the nonlinear projections and scales that happen in polar and logarithmic plots, from the linear affine transformations that happen when you pan and zoom. There is an efficiency here, because you can pan and zoom in your axes which affects the affine transformation, but you may not need to compute the potentially expensive nonlinear scales or projections on simple navigation events. It is also possible to multiply affine transformation matrices together, and then apply them to coordinates in one step. This is not true of all possible transformations.

Here is how the `ax.transData` instance is defined in the basic separable axis `Axes` class:

    self.transData = self.transScale + (self.transLimits + self.transAxes)

We’ve been introduced to the `transAxes` instance above in Axes coordinates, which maps the (0, 0), (1, 1) corners of the axes or subplot bounding box to `display` space, so let’s look at these other two pieces.

`self.transLimits` is the transformation that takes you from `data` to `axes` coordinates; i.e., it maps your view xlim and ylim to the unit space of the axes (and `transAxes` then takes that unit space to display space). We can see this in action here

    In [80]: ax = subplot(111)

    In [81]: ax.set_xlim(0, 10)
    Out[81]: (0, 10)

    In [82]: ax.set_ylim(-1, 1)
    Out[82]: (-1, 1)

    In [84]: ax.transLimits.transform((0, -1))
    Out[84]: array([ 0.,  0.])

    In [85]: ax.transLimits.transform((10, -1))
    Out[85]: array([ 1.,  0.])

    In [86]: ax.transLimits.transform((10, 1))
    Out[86]: array([ 1.,  1.])

    In [87]: ax.transLimits.transform((5, 0))
    Out[87]: array([ 0.5,  0.5])

and we can use this same inverted transformation to go from the unit `axes` coordinates back to `data` coordinates.

    In [90]: inv.transform((0.25, 0.25))
    Out[90]: array([ 2.5, -0.5])

The final piece is the `self.transScale` attribute, which is responsible for the optional non-linear scaling of the data, e.g., for logarithmic axes. When an Axes is initially setup, this is just set to the identity transform, since the basic matplotlib axes has linear scale, but when you call a logarithmic scaling function like `semilogx()` or explicitly set the scale to logarithmic with `set_xscale()`, then the `ax.transScale` attribute is set to handle the nonlinear projection. The scales transforms are properties of the respective `xaxis` and `yaxis` `Axis` instances. For example, when you call `ax.set_xscale('log')`, the xaxis updates its scale to a `matplotlib.scale.LogScale` instance.

For non-separable axes the PolarAxes, there is one more piece to consider, the projection transformation. The `transData` `matplotlib.projections.polar.PolarAxes` is similar to that for the typical separable matplotlib Axes, with one additional piece `transProjection`:

    self.transData = self.transScale + self.transProjection + \
        (self.transProjectionAffine + self.transAxes)

`transProjection` handles the projection from the space, e.g., latitude and longitude for map data, or radius and theta for polar data, to a separable Cartesian coordinate system. There are several projection examples in the `matplotlib.projections` package, and the best way to learn more is to open the source for those packages and see how to make your own, since matplotlib supports extensible axes and projections. Michael Droettboom has provided a nice tutorial example of creating a hammer projection axes; see Custom projection.

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