The TikZ and PGF Packages
Manual for version 3.1.10

This key defines a new axis for the current data visualization called ⟨name⟩. This has two effects:

• 1. A so called scaling mapper is created that will monitor a certain attribute, rescale it, and map it to another attribute. (This will be explained in detail in a moment.)

• 2. The ⟨axis name⟩ is made available as a key that can be used to configure the axis:

  /tikz/data visualization/⟨axis name⟩=⟨options⟩(no default) ¶

  This key becomes available once new axis base=metaaxis name has been called. It will execute the ⟨options⟩ with the path prefix /tikz/data visualization/axis options.

  copy
[new axis base=my axis,
my axis={attribute=some attribute}]


• 3. The ⟨axis name⟩ becomes part of the current set of axes. This set can be accessed through the following key:

  /tikz/data visualization/all axes=⟨options⟩(no default) ¶

  This key passes the ⟨options⟩ to all axes inside the current scope, just as if you had written ⟨some axis name⟩=⟨options⟩ for each ⟨some axis name⟩ in the current scope, including the just-created name ⟨axis name⟩.

There are many ⟨options⟩ that can be passed to a newly created axis. They are explained in the rest of this section.

Specifies that the axis is used to transform the data points according the different values of the key /data point/⟨attribute⟩. For instance, when we create a classical two-dimensional Cartesian coordinate system, then there are two axes called x axis and y axis that monitor the values of the attributes /data point/x and /data point/y, respectively:

In another example, we also create an x axis and a y axis. However, this time, we want to plot the values of the /data point/time attribute on the \(x\)-axis and, say, the value of the height attribute on the \(y\)-axis:

During the data visualization, the ⟨attribute⟩ will be “monitored” during the survey phase. This means that for each data point, the current value of /data point/⟨attribute⟩ is examined and the minimum value of all of these values as well as the maximum value is recorded internally. Note that this works even when very large numbers like 100000000000 are involved.

Here is a real-life example. The scientific axes create two axes, called x axis and y axis, respectively.

This key “fakes” data points for which the attribute’s values are in the comma-separated ⟨list of values⟩. For instance, when you write include value=0, then the attribute range interval is guaranteed to contain 0 – even if the actual data points are all positive or all negative.

This key allows you to simply set the minimum value, regardless of which values are present in the actual data. This key should be used with care: If there are data points for which the attribute’s value is less than ⟨value⟩, they will still be depicted, but typically outside the normal visualization area. Usually, saying include value=⟨value⟩ will achieve the same as saying min value=⟨value⟩, but with less danger of creating ill-formed visualizations.

Works like min value.

The ⟨scaling spec⟩ must have the following form:

⟨\(s_1\)⟩ at ⟨\(t_1\)⟩ and ⟨\(s_2\)⟩ at ⟨\(t_2\)⟩

This means that monitored values in the interval \([s_1,s_2]\) should be mapped to values the “reasonable” interval \([t_1,t_2]\), instead. For instance, we might write

in order to map dates between 1900 and 2000 to the dimension interval \([0\mathrm {cm},5\mathrm {cm}]\).

So much for the basic idea. Let us now have a detailed look at what happens.

Number format and the min and max keywords. The source values \(s_1\) and \(s_2\) are typically just numbers like 3.14 or 10000000000. However, as described in Section 80.2, you can also specify expressions like (pi/2), provided that (currently) you put them in parentheses.

Instead of a number, you may alternatively also use the two key words min and max for \(s_1\) and/or \(s_2\). In this case, min evaluates to the smallest value observed for the attribute in the data, symmetrically max evaluates to the largest values. For instance, in the above example with the year attribute ranging from 1900 to 2000, the keyword min would stand for 1900 and max for 2000. Similarly, for the people attribute min stands for 100 and max for 250. Note that min and max can only be used for \(s_1\) and \(s_2\), not for \(t_1\) and \(t_2\).

A typical use of the min and max keywords is to say

to map the complete range of values into an interval of length of 5cm.

The interval \([s_1,s_2]\) need not contain all values that the ⟨attribute⟩ may attain. It is permissible that values are less than \(s_1\) or more than \(s_2\).

Linear transformation of the attribute. As indicated earlier, the main job of an axis is to map values from a “large” interval \([s_1,s_2]\) to a more reasonable interval \([t_1,t_2]\). Suppose that for the current data point the value of the key /data point/⟨attribute⟩ is the number \(v\). In the simplest case, the following happens: A new value \(v'\) is computed so that \(v' = t_1\) when \(v=s_1\) and \(v'=t_2\) when \(v=s_2\) and \(v'\) is some value in between \(t_1\) and \(t_2\) then \(v\) is some value in between \(s_1\) and \(s_2\). (Formally, in this basic case \(v' = t_1 + (v-s_1)\frac {t_2-t_1}{s_2-s_1}\).)

Once \(v'\) has been computed, it is stored in the key /data point/⟨attribute⟩/scaled. Thus, the “reasonable” value \(v'\) does not replace the value of the attribute, but it is placed in a different key. This means that both the original value and the more “scaled” values are available when the data point is visualized.

As an example, suppose you have written

Now suppose that /data point/x equals 1200 for a data point. Then the key /data point/x/scaled will be set to 22 when the data point is being visualized.

Nonlinear transformations of the attribute. By default, the transformation of \([s_1,s_2]\) to \([t_1,t_2]\) is the linear transformation described above. However, in some case you may be interested in a different kind of transformation: For example, in a logarithmic plot, values of an attribute may range between, say, 1 and 1000 and we want an axis of length 3cm. So, we would write

Indeed, 1 will now be mapped to position 0cm and 1000 will be mapped to position 3cm. Now, the value 10 will be mapped to approximately 0.03cm because it is (almost) at one percent between 1 and 1000. However, in a logarithmic plot we actually want 10 to be mapped to the position 1cm rather than 0.03cm and we want 100 to be mapped to the position 2cm. Such a mapping a nonlinear mapping between the intervals.

In order to achieve such a nonlinear mapping, the function key can be used, whose syntax is described in a moment. The effect of this key is to specify a function \(f \colon \mathbb {R} \to \mathbb {R}\) like, say, the logarithm function. When such a function is specified, the mapping of \(v\) to \(v'\) is computed as follows:

\begin{align*} v' = t_1 + (f(s_2) - f(v))\frac {t_2 - t_1}{f(s_2)-f(s_1)}. \end{align*}

The syntax of the function key is described next, but you typically will not call this key directly. Rather, you will use a key like logarithmic that installs appropriate code for the function key for you.

Default scaling. When no scaling is specified, it may seem natural to use \([0,1]\) both as the source and the target interval. However, this would not work when the logarithm function is used as transformations: In this case the logarithm of zero would be computed, leading to an error. Indeed, for a logarithmic axis it is far more natural to use \([1,10]\) as the source interval and \([0,1]\) as the target interval.

For these reasons, the default value for the scaling that is used when no value is specified explicitly can be set using a special key:

When this key is used with an axis, three things happen:

• 1. The transformation function of the axis is setup to the logarithm.

• 2. The strategy for automatically generating ticks and grid lines is set to the exponential strategy, see Section 82.4.13 for details.

• 3. The default scaling is setup sensibly.

All told, to turn an axis into a logarithmic axis, you just need to add this option to the axis.

Note that this will work with any axis, including, say, the degrees on a polar axis:

Sets scaling to min at 0cm and max at ⟨dimension⟩. The effect is that the range of all values of the axis’s attribute will be mapped to an interval of exact length ⟨dimension⟩.

Sets scaling to 0 at 0cm and 1 at ⟨dimension⟩. In other words, this key allows you to specify how long a single unit should be. This key is particularly useful when you wish to ensure that the same scaling is used across multiple axes or pictures.

The optional per ⟨number⟩ units allows you to apply more drastic scaling. Suppose that you want to plot a graph where one billion corresponds to one centimeter. Then the unit length would be need to be set to a hundredth of a nanometer – much too small for TeX to handle as a dimension. In this case, you can write unit length=1cm per 1000000000 units:

This key is used in conjunction with the logarithmic setting. It cases the scaling to be set to 1 at 0cm and 10 at ⟨dimension⟩. This causes a “power unit”, that is, one power of ten in a logarithmic plot, to get a length of ⟨dimension⟩. Again, this key is useful for ensuring that the same scaling is used across multiple axes or pictures.

This key sets the label of an axis to ⟨text⟩. This text will typically be placed inside a node and the ⟨options⟩ can be used to further configure the way this node is rendered. The ⟨options⟩ will be executed with the path prefix /tikz/data visualization/, so you need to say node style to configure the styling of a node, see Section 82.4.7.

This key creates a new “Cartesian” axis, named ⟨name⟩. For such an axis, the (scaled) values of the axis’s attribute are transformed into a displacement on the page along a straight line. The following key is used to configure in which “direction” the axis points:

This key installs a two-dimensional coordinate system based on the attributes /data point/x and /data point/y.

This axis system is usually a good choice to depict “arbitrary two dimensional data”. Because the axes are automatically scaled, you do not need to worry about how large or small the values will be. The name scientific axes is intended to indicate that this axis system is often used in scientific publications.

You can use the ⟨options⟩ to fine tune the axis system. The ⟨options⟩ will be executed with the following path prefix:

All keys with this prefix can thus be passed as ⟨options⟩.

This axis system will always distort the relative magnitudes of the units on the two axis. If you wish the units on both axes to be equal, consider directly specifying the unit length “by hand”:

The scientific axes have the following properties:

• • The x-values are surveyed and the \(x\)-axis is then scaled and shifted so that it has the length specified by the following key.

  /tikz/data visualization/scientific axes/width=⟨dimension⟩ (no default, initially 5cm) ¶

  The minimum value is at the left end of the axis and at the canvas origin. The maximum value is at the right end of the axis.

• • The y-values are surveyed and the \(y\)-axis is then scaled so that is has the length specified by the following key.

  /tikz/data visualization/scientific axes/height=⟨dimension⟩(no default) ¶

  By default, the height is the golden ratio times the width.

  The minimum value is at the bottom of the axis and at the canvas origin. The maximum value is at the top of the axis.

• • Lines (forming a frame) are depicted at the minimum and maximum values of the axes in 50% black.

The following keys are executed by default as options: outer ticks and standard labels.

You can use the following style to overrule the defaults:

This causes the ticks to be drawn “ on the outside” of the frame so that they interfere as little as possible with the data. It is the default.

This axis system works like scientific axes, only the ticks are on the “inside” of the frame.

This axis system is also common in publications, but the ticks tend to interfere with marks if they are near to the border as can be seen in the following example:

The axes and the ticks are completely removed from the actual data, making this axis system especially useful for scatter plots, but also for most other scientific plots.

The distance of the axes from the actual plot is given by the padding of the axes.

As the name suggests, this is the standard placement strategy. The label of the \(x\)-axis is placed below the center of the \(x\)-axis, the label of the \(y\)-axis is rotated by \(90^\circ \) and placed left of the center of the \(y\)-axis.

Works like scientific axes standard labels, only the label of the \(y\)-axis is not rotated.

Places the labels at the end of the \(x\)- and the \(y\)-axis, similar to the axis labels of a school book axis system.

This axis system is intended to “look like” the coordinate systems often used in school books: The axes are drawn in such a way that they intersect to origin. Furthermore, no automatic scaling is done to ensure that the lengths of units are the same in all directions.

This axis system must be used with care – it is nearly always necessary to specify the desired unit length by hand using the option unit length. If the magnitudes of the units on the two axes differ, different unit lengths typically need to be specified for the different axes.

Finally, if the data is “far removed” from the origin, this axis system will also “look bad”.

The stepping of the ticks is one unit by default. Using keys like ticks=some may help to give better steppings.

The ⟨options⟩ are executed with the key itself as path prefix. Thus, the following subkeys are permissible options:

This axis system creates two axes called x axis and y axis that point right and up, respectively. By default, one unit is mapped to one cm.

This axis system works like xy Cartesian, only it additionally creates an axis called z axis that points left and down. For this axis, one unit corresponds to \(\frac {1}{2}\sin 45^\circ \mathrm {cm}\). This is also known as a cabinet projection.

This axis system works like xy Cartesian, but it introduces two axes called u axis and v axis rather than the x axis and the y axis. The idea is that in addition to a “major” \(xy\)-coordinate system this is also a “smaller” or “minor” coordinate system in use for depicting, say, small vectors with respect to this second coordinate system.

Like xyz Cartesian cabinet, but for the \(uvw\)-system.

This key can be passed to an axis in order to configure which ticks are present for the axis. The possible ⟨options⟩ include, for instance, keys like step, which is used to specify a stepping for the ticks, but also keys like major or minor for specifying the positions of major and minor ticks in detail. The list of possible options is described in the rest of this section.

Note that the ticks option will only configure which ticks should be shown in principle. The actual rendering is done only when the visualize ticks key is used, documented in Section 82.5.4, which is typically done only internally by an axis system.

The ⟨options⟩ will be executed with the path prefix /tikz/data visualization/. When the ticks key is used multiple times for an axis, the ⟨options⟩ accumulate.

This key is similar to ticks, only it is used to configure where grid lines should be shown rather than ticks. In particular, the options that can be passed to the ticks key can also be passed to the grid key. Just like ticks, the ⟨options⟩ only specify which grid lines should be drawn in principle; it is the job of the visualize grid key to actually cause any grid lines to be shown.

If you do not specify any ⟨options⟩, the default text at default ticks is used. This option causes grid lines to be drawn at all positions where ticks are shown by default. Since this usually exactly what you would like to happen, most of the time you just need to all axes=grid to cause a grid to be shown.

This key passes the ⟨options⟩ to both the ticks key and also to the grid key. This is useful when you want to specify some special points explicitly where you wish a tick to be shown and also a grid line.

The value of this key is used to determine the spacing of the major ticks. The key is used by the linear steps and exponential steps strategies, see the explanations in Section 82.4.13 for details. Basically, all ticks are placed at all multiples of ⟨value⟩ that lie in the attribute range interval.

Specifies that between any two major steps (whose positions are specified by the step key), there should be ⟨number⟩ many minor steps. Note that the default of 9 is exactly the right number so that each interval between two minor steps is exactly a tenth of the size of a major step. See also Section 82.4.13 for further details.

See Section 82.4.13 for details on how the phase of steps influences the tick placement.

This key asks the data visualization to place about ⟨number⟩ many ticks on an axis. It is not guaranteed that exactly ⟨number⟩ many ticks will be used, rather the actual number will be the closest number of ticks to ⟨number⟩ so that their stepping is still “good”. For instance, when you say about=10, it may happen that exactly 10, but perhaps even 13 ticks are actually selected, provided that these numbers of ticks lead to good stepping values like 5 or 2.5 rather than numbers like 3.4 or 7. The method that is used to determine which steppings a deemed to be “good” depends on the current tick placement strategy.

Linear steps. Let us start with linear steps: First, the difference between the maximum value \(v_{\max }\) and the minimum value \(v_{\min }\) on the axis is computed; let us call it \(r\) for “range”. Then, \(r\) is divided by ⟨number⟩, yielding a target stepping \(s\). If \(s\) is a number like \(1\) or \(5\) or \(10\), then this number could be used directly as the new value of step. However, \(s\) will typically something strange like \(0.023\,45\) or \(345\,223.76\), so \(s\) must be replaced by a better value like \(0.02\) in the first case and perhaps \(250\,000\) in the second case.

In order to determine which number is to be used, \(s\) is rewritten in the form \(m \cdot 10^k\) with \(1 \le m < 10\) and \(k \in \mathbb Z\). For instance, \(0.023\,45\) would be rewritten as \(2.345 \cdot 10^{-2}\) and \(345\,223.76\) as \(3.452\,2376 \cdot 10^5\). The next step is to replace the still not-so-good number \(m\) like \(2.345\) or \(3.452\,237\) by a “good” value \(m'\). For this, the current value of the about strategy is used:

Once \(m'\) has been determined, the stepping is set to \(s' = m' \cdot 10^k\).

The net effect of all this is that for the default strategy the only valid stepping are the values \(1\), \(2\), \(2.5\) and \(5\) and every value obtainable by multiplying one of these values by a power of ten. The following example shows the effects of, first, setting about=5 (corresponding to the some option) and then having axes where the minimum value is always 0 and where the maximum value ranges from 10 to 100 and, second, setting about to the values from 3 (corresponding to the few option) and to 10 (corresponding to the many option) while having the minimum at 0 and the maximum at 100:

Exponential steps. For exponential steps the strategy for determining a good stepping value is similar to linear steps, but with the following differences:

• • Naturally, since the stepping value refers to the exponent, the whole computation of a good stepping value needs to be done “in the exponent”. Mathematically spoken, instead of considering the difference \(r = v_{\max } - v_{\min }\), we consider the difference \(r = \log v_{\max } - \log v_{\min }\). With this difference, we still compute \(s = r / \meta {number}\) and let \(s = m \cdot 10^k\) with \(1 \le m < 10\).

• • It makes no longer sense to use values like \(2.5\) for \(m'\) since this would yield a fractional exponent. Indeed, the only sensible values for \(m'\) seem to be \(1\), \(3\), \(6\), and \(10\). Because of this, the about strategy is ignored and one of these values or a multiple of one of them by a power of ten is used.

The following example shows the chosen steppings for a maximum varying from \(10^1\) to \(10^5\) and from \(10^{10}\) to \(10^{50}\) as well as for \(10^{100}\) for about=3:

Alternative strategies.

In addition to the standard about strategy, there are some additional strategies that you might wish to use instead:

This is an abbreviation for about=10.

This is an abbreviation for about=5.

This is an abbreviation for about=3.

Switches off the automatic step computation. Unless you use step= explicitly to set a stepping, no ticks will be (automatically) added.

The key can be passed as an option to the ticks key and also to the grid key, which in turn is passed as an option to an axis. The ⟨options⟩ passed to major specify at which positions major ticks/grid lines should be shown (using the at option and also at option) and also any special styling. The different possible options are described later in this section.

Like major, only for minor ticks/grid lines.

Like major, only for subminor ticks/grid lines.

This key allows you to specify ⟨options⟩ that apply to major, minor and subminor alike. It does not make sense to use common to specify positions (since you typically do not want both a major and a minor tick at the same position), but it can be useful to configure, say, the size of all kinds of ticks:

Basically, the ⟨list⟩ must be a list of values that is processed with the \foreach macro (thus, it can contain ellipses to specify ranges of value). Empty values are skipped.

The effect of passing at to a major, minor, or subminor key is that ticks or grid lines on the axis will be placed exactly at the values in ⟨list⟩. Here is an example:

When this option is used, any previously specified tick positions are overwritten by the values in ⟨list⟩. Automatically computed ticks are also overwritten. Thus, this option gives you complete control over where ticks should be placed.

Normally, the individual values inside the ⟨list⟩ are just numbers that are specified in the same way as an attribute value. However, such a value may also contain the keyword as, which allows you so specify the styling of the tick in detail. Section 82.4.6 details how this works.

It is often a bit cumbersome that one has to write things like

A slight simplification is given by the following keys, which can be passed directly to ticks and grid:

This key is similar to at, but it causes ticks or grid lines to be placed at the positions in the ⟨list⟩ in addition to the ticks that have already been specified either directly using at or indirectly using keys like step or some. The effect of multiple calls of this key accumulate. However, when at is used after an also at key, the at key completely resets the positions where ticks or grid lines are shown.

As for at, there are some shorthands available:

This key takes options whose path prefix is /tikz, not /tikz/data visualization. These options will be appended to a current list of such options (thus, multiple calls of this key accumulate). The resulting list of keys is not executed immediately, but it will be executed whenever the data visualization engine calls the TikZ layer to draw something (this placed will be indicated in the following).

Executing this key will cause all “accumulated” TikZ options from previous calls to the key /tikz/data visualization/style to be executed. Thus, you use style to set TikZ options, but you use styling to actually apply these options. Usually, you do not call this option directly since this application is only done deep inside the data visualization engine.

This key works like style, but it has an effect only on nodes that are created during a data visualization. This includes tick labels and axis labels:

Note that in the example the ticks themselves (the little thicker lines) are not red.

Executing this key will cause all “accumulated” node stylings to be executed.

This key is used to specified the layer on which grid lines should be drawn (layers are explained in Section 45). By default, all grid lines are placed on the background layer and thus behind the data visualization. This is a sensible strategy since it avoids obscuring the more important data with the far less important grid lines. However, you can change this style to “get the grid lines to the front”:

When this style is executed, the keys stored in the style will be executed with the prefix /tikz. Normally, you should only set this style to be empty or to on background layer.

This style provides overall configuration options for grid lines. By default, it is set to the following:

This causes grid lines to span all possible values when they are visualized, which is usually the desired behavior (the low and high keys are explained in Section 82.5.4. You can append the style key to this style to configure the overall appearance of grid lines. It should be noted that settings to style inside every grid will take precedence over ones in every major grid and every minor grid. In the following example we cause all grid lines to be dashed (which is not a good idea in general since it creates a distracting background pattern).

This style configures the appearance of major grid lines. It does so by calling the style key to setup appropriate TikZ options for visualizing major grid lines. The default definition of this style is:

In the following example, we use thin major blue grid lines:

As can be seen, this is not exactly visually pleasing. The default settings for the grid lines should work in most situations; you may wish to increase the blackness level, however, when you experience trouble during printing or projecting graphics.

Works like every major grid. The default is

Works like every major grid. The default is

This style allows you to configure the appearance of ticks using the style and node style key. Here is (roughly) the default definition of this style:

The default is

The default is

The default is

Like grid layer, this key specifies on which layer the ticks should be placed.

Like tick layer, but now for the nodes. By default, tick nodes are placed on the main layer and thus on top of the data in case that the tick nodes are inside the data.

This key causes the ⟨options⟩ to be executed for any tick mark(s) at ⟨value⟩ in addition to any options given already for this position:

Shorthand for options at=⟨value⟩ as [no tick text].

The ⟨text⟩ will be put in front of every typeset tick:

Works like tick prefix. This key is especially useful for adding units like “cm” or “\(\mathrm m/\mathrm s\)” to every tick label. For this reason, there is a (near) alias that is easier to memorize:

The key gets called for each number that should be typeset. The argument ⟨value⟩ will be in scientific notation (like 1.0e1 for \(10\)). By default, this key applies \pgfmathprintnumber to its argument. This command is a powerful number printer whose configuration is documented in Section 97.

You are invited to code underlying this key so that a different typesetting mechanism is used. Here is a (not quite finished) example that shows how, say, numbers could be printed in terms of multiples of \(\pi \):

When a tick label is shown at the low position of an even tick, the ⟨distance⟩ is added to the low value, see also Section 82.5.4.

Note that ⟨dimension⟩ should usually be non-positive.

A shorthand for setting tick text odd low padding and tick text odd high padding at the same time.

A shorthand for setting tick text even low padding and tick text even high padding at the same time.

Sets all text paddings to ⟨dimension⟩.

Shorthand for tick text even padding=⟨dimension⟩.

Shorthand for tick text odd padding=⟨dimension⟩. The difference to stack is that the set of value that are “lowered” is exactly exchanged with the set of value “lowered” by stack.

This strategy places ticks at positions that are evenly spaced by the current value of step.

In detail, the following happens: Let \(a\) be the minimum value of the data values along the axis and let \(b\) be the maximum. Let the current stepping be \(s\) (the stepping is set using the step option, see below) and let the current phasing be \(p\) (set using the phase) option. Then ticks are placed all positions \(i\cdot s + p\) that lie in the interval \([a,b]\), where \(i\) ranges over all integers.

The tick positions computed in the way described above are major step positions. In addition to these, if the key minor steps between steps is set to some number \(n\), then \(n\) many minor ticks are introduced between each two major ticks (and also before and after the last major tick, provided the values still lie in the interval \([a,b]\)). Note that is \(n\) is \(1\), then one minor tick will be added in the middle between any two major ticks. Use a value of \(9\) (not \(10\)) to partition the interval between two major ticks into ten equally sized minor intervals.

This strategy produces ticks at positions that are appropriate for logarithmic plots. It is automatically selected when you use the logarithmic option with an axis.

In detail, the following happens: As for linear steps let numbers \(a\), \(b\), \(s\), and \(p\) be given. Then, major ticks are placed at all positions \(10^{i\cdot s+p}\) that lie in the interval \([a,b]\) for \(i \in \mathbb {Z}\).

The minor steps are added in the same way as for linear steps. In particular, they interpolate linearly between major steps.

This key can be used to install a so-called tick placement strategy. Whenever visualize ticks is used to request some ticks to be visualized, it is checked whether some automatic ticks should be created. This is the case when the following key is set:

Provided compute step is set to some nonempty value, upon visualization of ticks the ⟨macro⟩ is executed. Typically, ⟨macro⟩ will first call the ⟨code⟩ stored in the key compute step. Then, it should implement some strategy then uses the value of the computed or desired stepping to create appropriate at commands. To be precise, it should set the keys major, minor, and/or subminor with some appropriate at values.

Inside the call of ⟨macro⟩, the macro \tikzdvaxis will have been set to the name of the axis for which default ticks need to be computed. This allows you to access the minimum and the maximum value stored in the scaling mapper of that axis.