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.

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()
```

(Source code, png, hires.png, pdf)

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.

(Source code, png, hires.png, pdf)

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])
```

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.

```
import numpy as np
import matplotlib.pyplot as plt
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()
```

(Source code, png, hires.png, pdf)

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 numpy as np
import matplotlib.pyplot as plt
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()
```

(Source code, png, hires.png, pdf)

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 numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as patches
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()
```

(Source code, png, hires.png, pdf)

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()
```

One use of transformations is to create a new transformation that is offset from another transformation, eg 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 <http://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.

```
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as patches
import matplotlib.transforms as transforms
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()
```

(Source code, png, hires.png, pdf)

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
*api example code: custom_projection_example.py*.