The answers to both above questions is yes! The above curves are essentially quadratic Bézier curves with three control points. All of them have a relatively easy to pick start and end point. In the examples above we pick (1, 1) and (100, 100). However for any general row count and column count , the end point can — points denoted by convention which is columns first.
Now that we have two points, we can talk about the third control point. When this is the mid-point of the space the curve is a straight diagonal line as all points are collinear. However add an imbalance and we get deviations from the diagonal. For simplicity, these deviations can be parametrized by a single variable , which makes the third point .
where the value sweeps from 0 to 1 in small steps to get the coordinates at each step.
An application of this curve (my use-case) is to model the number of sentences (space) used to describe a movie / TV episode in a plot synopsis (plotted on the y-axis) corresponding to the shots (plotted on the x-axis). Empirical evidence suggests that authors spend more space describing the latter half of a video as it approaches the climax, thus allowing us to use as a model which fits better than the diagonal.