Plot Option Precedence while combining Plots with Show[]

Question

I like to build sophisticated plots by combining simpler ones with Show[]. Typically this involves setting non-default Plot-Options with the different Plot-Commands, like

Show[ ListPlot[ ,Op1], Plot[ ,Op2], Op3]

Unfortunately the Show[] command is not commutative, as

Show[ Plot[ ,Op2], ListPlot[ ,Op1], Op3]

can produce different results. My expectation was that putting settings in Op3 should overwrite the ones in Op1 and Op2 however this does not work with options like PlotMarkers which are only available within ListPlot[].

The description of the Show[g_1, g_2, g_3, ... ,g_i]-command gives only two hints:

Options explicitly specified in Show override those included in the graphics expression.

and

The lists of non-default options in the g_i are concatenated.

I’m not sure what this precisely means. Is

Show[ ListPlot[ ,Op1], Plot[ ,Op2], Op3]

equivalent to

Show[ ListPlot[ ,Union[Op1,Op2]], Plot[ ,Union[Op1,Op2]], Op3]?

while Op3 overwrites whatever is in Union[Op1,Op2]?

And there is one more question: In

Show[ g_1, g_2, g_3, ..., g_i ]

the Plot in g_1 seems to be treated specially as it defines the PlotRange for the final image generated.

I would like to know the full set of rules how the Plot-Options are combined and to which Plot or Plots they are applied.

Answer 1

First a little background:

All of Mathematica's plotting functions produce a Graphics expression (or Graphics3D, but let's talk about Graphics now). The Graphics expression is simply a representation of what you see in the graphic. You can look at it by converting the output cell to InputForm (Ctrl-Shift-I). For example, Plot will produce Graphics with Lines in it.

Some of the options to plotting functions are passed on directly to Graphics an affect its appearance (how its contents get rendered). An example is Axes. Some others control what the plotting function will put into the graphics. Examples are PlotStyle or PlotMarkers. These are specific to (and different for) each plotting function.

How Show works:

It combines several Graphics expressions into one. The returned Graphics expression will inherit its options from the first one passed to Show. In Show you can override some Graphics options directly, but of course this will only override options for Graphics, and not the plotting functions that produced the graphics (as those have already finished running by the time Show sees their output).

So

Show[ListPlot[... , Op1], Plot[... , Op2], Op3] 

is equivalent to

Show[ListPlot[... , Op3, Op1], Plot[...]]

or to

Show[ListPlot[...], Plot[...], Op3, Op1]

but this is only valid for ListPlot options that are also Graphics options. It is not valid for PlotMarkers.

Also note that if the same option is specified several times in the same Graphics, the first one takes precedence. (Thanks J. M.!)

Answer 2

Just to drive Szabolcs's points home, here are a few examples to demonstrate how Show[] interacts with graphics objects and options given to it.

Let's start with

Show[Plot[Sin[x], {x, -1, 1}], Plot[Exp[x], {x, -2, 2}]]

Compare this image with the output from a version with options tweaked in places.

Show[Plot[Sin[x], {x, -1, 1}, PlotRange -> All], Plot[Exp[x], {x, -2, 2}]]

The same output is obtained from Show[Plot[Sin[x], {x, -1, 1}], Plot[Exp[x], {x, -2, 2}], PlotRange -> All]. On the other hand, Show[Plot[Sin[x], {x, -1, 1}], Plot[Exp[x], {x, -2, 2}, PlotRange -> All]] gives the same output as the first version.

However,

Show[Plot[Sin[x], {x, -1, 1}], Plot[Exp[x], {x, -2, 2}],
     PlotRange -> {{-3, 3}, All}]

and an explicit option setting in Show[], as expected, overrides the PlotRange settings of the component graphics.

---

You don't really have to worry about

The lists of non-default options in the g_i are concatenated.

See, in functions like Plot[], the options, as with usual rules, are parsed left to right, and the first rule encountered that applies is the one taken. Thus, compare

Plot[Sin[x], {x, -2, 2}, MaxRecursion -> 1, PlotPoints -> 5, PlotPoints -> 20]

and

Plot[Sin[x], {x, -2, 2}, MaxRecursion -> 1, PlotPoints -> 20, PlotPoints -> 5]