Interpolation Theory

Luc(as) de Groot’s Interpolation Theory is a theory about the optical behaviour of a family of fonts of increasing weight. It has resulted in a formula for calculating intermediate weights that has been influential among type designers the world over.

About interpolation

When producing large type families with multiple stroke weights, a certain degree of automation is not only agreeable (as a way to avoid tedious work), it may even be necessary (for making it feasible at all to generate such a huge number of fonts). Interpolation is a common method for doing this: to generate intermediate weights between two basic values, after which, of course, the automatically produced fonts are fine-tuned by hand.

Type designers are often inclined to establish intermediate weights by drawing, as it were, a straight line between the boldest and lightest versions of a family and then choose the intermediate weights somewhere along that graph.

Interpolation theory

As early as 1987, when developing intermediate versions of an existing typeface during an internship, Luca(as) realized that the optically correct in-between weights are on a hollow curve that produces values which are lower than those on the straight line of the “average values”. In other words: if the verticals of the Regular weight have a value of 40 units and those of the Bold weight 70, then the SemiBold verticals should not be 55 units wide but slightly less, in order to give the optical impression of being exactly “in the middle”.
Luc(as)’s tests resulted in a formula and a graph that precisely define the optimum value for any possible intermediate weight.

Anisotropic Topology-dependent Interpolation Theory

The 1987 Interpolation Theory has its limitations. Says Luc(as): ‘it is one-dimensional. Yet the world has more dimensions; it is the same with type. On Earth the options for moving vertically are rather limited, while one can walk endlessly when moving forward or backward. Type behaves similarly. Therefore, the fine-tuning of the Interpolation Theory led to an anisotropic version, i.e.: a system that works differently in the “x” and in the “y” directions. In this case it means that the stroke weights grow slower on the vertical axis than on the horizontal  one.

Limited growth

In order to explore this phenomenon I made two proof weights: an extremely bold one and a very light one. Now what happens if what interpolates between these extremes? It turns out that at a certain point the horizontals cannot get any thicker, because then the letter will disappear. However, there’s much more leeway in the vertical strokes. Here, too, the optically optimum proportions cannot be calculated by simple interpolation. When the proportion between verticals and horizontals is 1:1 in the light typeface and 3:1 in the bold one, it is not self-evident that in the intermediate weight, 2:1 is the proportion that looks best.  Here, too, linear interpolation won’t do for calculating the in-between values. A slightly lower contrast will work better.

Topology of single letters

Moreover, the possibilities for growth of the horizontals depends on the topology of the single letters: when there is only one horizontal stroke, as in “L”, it has more room to expand than strokes that must share the same vertical space with 1, 2 or 3 other strokes. Each of these four topologies needs its own interpolation curve. Therefore, the new  version of the Interpolation Formula is not only anisotropic, but also topology-dependent.

When a letter has only one horizontal stroke (“L”), that stroke can expand almost up to the glyph’s full height in increasing weights. With two strokes (“F” or “C”), less than half the height is available for each stroke, etc. One of the most difficult glyphs in this respect is the Euro sign, as it is among the few glyphs in which four horizontal strokes must share the vertical space available.