In the definition of surface plots in my book, (An introduction to data mining, chapter 3) it is said that:

Surface plots are often used to describe mathematical functions or physical surfaces that vary in a relatively smooth manner.

I don't understand what exactly is meant by the above sentence. What special property do mathematical functions and physical surfaces have that is appropriate for the case of surface plot?


Smooth in terms of mathematical functions typically means some function that is differentiable. Surface plots of naturally occurring phenomenon aren't necessarily that smooth (imagine a Fjord or a very steep cliff face), but the more jagged or irregular the data the more difficult it is to visualize the nature of the data without certain elements being occluded in the plot. Here is an example of a discontinuous function from Brunsdon, 2011

discontinuous surface

Imagine rotating the plot on the lower left so the lower surface is entirely obstructed from the view.

A good point to bring up here is also what type of data can be visualized with a surface plot. Data is typically over a regular grid (x,y) and has one unique value, z, associated with each x,y pair. Geographers sometimes call this 2.5d data. If you have truely 3 dimensional data (that is an x,y pair can have potentially multiple z values), and the data is not regular over a grid, a surface plot may not be possible nor appropriate.

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