Figure 3
From: The quantitation of buffering action I. A formal & general approach
Graphical representation of two-partitioned systems of functions. The unordered combination of two functions π1(x), π2(x) of a single independent variable x is termed a "two-partitioned system of functions". The two functions may represent the two complementary parts of a whole, e.g. "bound H+ ions" vs. "free H+ ions" in an aqueous solution. The sum of the two functions is termed "sigma function" σ(x) (see main text for detailed explanation).. A, Family of curves. The individual functions π1(x), π2(x), and σ(x) may be plotted individually as a family of curves (this is possible for multi-partitioned systems as well). B & C, Area plots. The individual partitioning functions of partitioned systems can be plotted "on top of each other" such that the value of each function is represented by the vertical distance between consecutive curves. In a partitioned system, their order is not constrained, and thus two equally valid representations exist for a two-partitioned system (B,C). A limitation of area plots is that they do not allow visualization of negative-valued partitioning functions. D, Three-Dimensional Space Curve. The independent variable x and the values of the partitioning functions π1(x), π2(x) of a two-partitioned system may be interpreted as x-, y- and z-coordinates, respectively. This results in a three-dimensional space curve. Such a curve can display both positive and negative values. Again, there are two different, equally valid representations. Projections of that curve on the xy-plane (red) and xz-plane (blue) correspond to the individual partitioning functions π1(x) and π2(x). Projection of the space curve on the yz-plane (gray) corresponds to a plot of the composite relations π1(π2(x)) or π2(π1(x)); these projections are not necessarily single-valued functions. The projection on the yz-plane is suited particularly well to assess the proportion between the individual rates of change of the two functions. Importantly, these proportions provide the clue to the quantitation of "buffering action".