"if the range"

Not sure what a colorbar has to do with this sentence.

Also the "good" in parenthesis is probably a stand in for some precise term and otherwise not true for categorical data and questionable on discrete data.

This sentence kind of runs on.

I still feel this is more complicated than it needs to be, and not even particularly true if you want to be mathematically rigorous; colormaps are finite discrete ordered points in R^3, not a smooth curve or path, so I find taking about it like this distracting. The first step is normalizing the data between vmin and vmax to the range 0-1, perhaps using a non-linear function, and secondly, using a linear lookup table to find the RGBA color closest to that value between 0 and 1.

I always like code examples to make the abstract concrete:

A color mapping from floating point data values to colors takes place in two steps.  
First the data is normalized between 0 and 1.  This requires a `vmin` and `vmax` the define 
which values are mapped to 0 and 1, respectively, and it requires a function that defines the 
mapping between these two values.    The simplest case is a linear map ::
    
		import matplotlib.colors as mcolors
		norm = mcolors.Normalize(vmin=100, vmax=300) 
		norm([100, 200, 300])  # returns 0.0, 0.5, 1.0

but a logarithmic norm is also possible ::

    		norm = mcolors.LogNorm(vmin=10, vmax=1000)
		norm([10, 100, 1000])  # returns 0.0, 0.5, 1.0

Once the data has been normalized between 0 and 1, it can be passed to a colormap 
to return RGB(A) values ::
		
		cmap = cm.viridis
		cmap([0.0, 0.5, 1.0])  
	      #  returns RGBA array with the first three colors of viridis colormap.
	      # array([[0.267004, 0.004874, 0.329415, 1.      ],
             # [0.127568, 0.566949, 0.550556, 1.      ],
             # [0.993248, 0.906157, 0.143936, 1.      ]])

We use discrete colormaps as an implementation detail (both for performance and because we are targeting 8bit RGB which is inherently discrete), but if we had infinite precision floats and an analytic function for the colormap the whole system would still hold together.

I think the correct solution is to include both versions :)

There is a TOC aspect to this file, and then you've added an explanation. a) not 100% sure the explanation belongs here, versus an intermediate/advanced tutorial. Most long-form explanations are tutorials currently (i.e. transform stack). I'm not adamantly advocating for that, but it would be more consistent with current practice. It would also make your examples easier to write.

Regardless, this large narrative in the middle of the TOC makes the TOC very hard to parse. At the very least, this should go at the end?

The outline here is a little non-linear, and seems to take a while before it settles into an order.

• mapping
• choosing colormap
• mapping
• norm
• norm
• colormapping

I think this could all be three paragraphs, or maybe 5 if you want to be expansive:

1. overview
2. norm step
3. colormap step

If you plan to talk about the "Curve through space" I wonder if a diagram would make this much clearer. Expressing this with math formalisms without diagrams is going to lose your audience.