Bézier curves off the Diagonal

We all know that we can drag the center point of a diagonal in a plot and then move it around to make beautiful arcs. However, how are they mathematically generated and can they be easily controlled by a single parameter?

Bézier curves for parameter 0, 0.25, 0.5, 0.75 and 1

Bézier curves for \alpha parameter 0, 0.25, 0.5, 0.75 and 1

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 N_R and column count N_C, the end point can (N_C, N_R) — points denoted by (x, y) 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 (0.5 \cdot N_C, 0.5 \cdot N_R) 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 \alpha, which makes the third point (\alpha N_C, (1-\alpha) N_R).

The equations of the curves correspond to rational Bézier curves with three points which have the additional property of also representing conic sections

\mathbf{B}(t) = \frac{\sum_{i=0}^2 {2 \choose i} t^i (1-t)^{2-i}\mathbf{P}_{i} } { \sum_{i=0}^2 {2 \choose i} t^i (1-t)^{2-i} }

where the value t sweeps from 0 to 1 in small steps to get the (x, y) 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 \alpha \sim 0.65 as a model which fits better than the diagonal.


One thought on “Bézier curves off the Diagonal

  1. Martin Bäuml

    Interesting! Of course the follow up question is now, whether \alpha can be learned from training data (?)


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