DOC: Explain parameters linthresh and linscale of symlog scale

Closes matplotlib#29335 via addressing matplotlib#29335 (comment).

galleries/examples/scales/symlog_demo.py

@@ -1,7 +1,12 @@
 """
-===========
-Symlog Demo
-===========
+============
+Symlog scale
+============
+
+The symmetric logarithmic scale is an extension of the logarithmic scale that
+also covers negative values. As with the logarithmic scale, it is particularly
+useful for numerical data that spans a broad range of values, especially when there
+are significant differences between the magnitudes of the numbers involved.
 
 Example use of symlog (symmetric log) axis scaling.
 """
@@ -34,12 +39,85 @@
 plt.show()
 
 # %%
-# It should be noted that the coordinate transform used by ``symlog``
-# has a discontinuous gradient at the transition between its linear
-# and logarithmic regions. The ``asinh`` axis scale is an alternative
-# technique that may avoid visual artifacts caused by these discontinuities.
+# Linear threshold
+# ----------------
+# Since each decade on a logarithmic scale covers the same amount of visual space
+# and there are infinitely many decades between a given number and zero, the symlog
+# scale must deviate from logarithmic mapping in a small range (-x0, x0), so that
+# that range is mapped to a finite visual space.
+#
+# The symlog scale achieves this by defining a parameter *linthresh* and switching
+# to a linear mapping in the region *(-linthresh, linthresh)*.
+
+
+def format_axes(ax, title=None):
+    """A helper function to better visualize properties of the symlog scale."""
+    ax.xaxis.get_minor_locator().set_params(subs=[2, 3, 4, 5, 6, 7, 8, 9])
+    ax.grid()
+    ax.xaxis.grid(which='minor')  # minor grid on too
+    linthresh = ax.xaxis.get_transform().linthresh
+    linscale = ax.xaxis.get_transform().linscale
+    ax.axvspan(-linthresh, linthresh, color='0.9')
+    if title:
+        ax.set_title(title.format(linthresh=linthresh, linscale=linscale))
+
+
+x = np.linspace(-60, 60, 201)
+y = np.linspace(0, 100.0, 201)
+
+fig, (ax1, ax2) = plt.subplots(nrows=2, layout="constrained")
+
+ax1.plot(x, y)
+ax1.set_xscale('symlog', linthresh=1)
+format_axes(ax1, title='Linear region: linthresh={linthresh}')
+
+ax2.plot(x, y)
+ax2.set_xscale('symlog', linthresh=5)
+format_axes(ax2, title='Linear region: linthresh={linthresh}')
 
 # %%
+# Generally, *linthresh* should be chosen so that no or only a few
+# data points are in the linear region. As a rule of thumb,
+# :math:`linthresh \approx \mathrm{min} |x|`.
+#
+#
+# Linear scale
+# ------------
+# Additionally, the *linscale* parameter determines how much visual space should be
+# used for the linear range. More precisely, it defines the ratio of visual space
+# of the region (0, linthresh) relative to one decade.
+
+fig, (ax1, ax2) = plt.subplots(nrows=2, layout="constrained")
+
+ax1.plot(x, y)
+ax1.set_xscale('symlog', linthresh=1)
+format_axes(ax1, title='Linear region: linthresh={linthresh}, linscale={linscale}')
+
+ax2.plot(x, y)
+ax2.set_xscale('symlog', linthresh=1, linscale=0.1)
+format_axes(ax2, title='Linear region: linthresh={linthresh}, linscale={linscale}')
+
+# %%
+# The suitable value for linscale depends on the dynamic range of data. As most data
+# will be outside the linear region, you typically the linear region only to cover
+# a small fraction of the visual area.
+#
+# Limitations and alternatives
+# ----------------------------
+# The coordinate transform used by ``symlog`` has a discontinuous gradient at the
+# transition between its linear and logarithmic regions. Depending on data and
+# scaling, this will be more or less obvious in the plot.
+
+fig, ax = plt.subplots()
+ax.plot(x, y)
+ax.set_xscale('symlog', linscale=0.05)
+format_axes(ax, title="Discontinuous gradient at linear/log transition")
+
+# %%
+# The ``asinh`` axis scale is an alternative transformation that supports a wide
+# dynamic range with a smooth gradient and thus may avoid such visual artifacts.
+# See :doc:`/gallery/scales/asinh_demo`.
+#
 #
 # .. admonition:: References
 #