Interpolation of Contrast

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14/06/14 15:29Maciej Ratajski Interpolation of contrast

Pgina 1 de 7http://www.maciejratajski.com/theory/interpolation-of-contrast/#problem

Interpolation of contrast

A simple formula and an algorithm for interpolating horizontal to

vertical stem width ratio. The aim is for contrast to grow in linear

relationship to weight. The algorithm results in anisotropic

interpolation, hence its usability can be limited.

1. The problem

2. An example

3. The formula

4. The algorithm

5. The implementations

6. The limitations

7. The application

The problem

The width of vertical and horizontal stems is hardly ever equal in

type design. What's more, within one type family the bold weights

often need a bigger difference in stroke weight than the light

weights. At the same time, designing the former with highercontrast leads to problems with interpolation.

The standard isotropic Multiple Master interpolation between a very

light master with almost no stroke contrast and a very bold one with

high contrast produces too thin horizontals in most of the

interpolated weights.

fig 1: Standard isotropic interpolation.

1

2

3


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fig 2: Interpolation of contrast (anisotropic).

That is the case, because even if the stem weights grow linearly

within the family, the ratio between them doesn't. On the contrary,

we observe a negative exponential relationship between vertical

stem width and horizontal to vertical ratio, which leads to uneven

distribution of contrast.

fig 3: Horizontal to vertical stem width ratio (as a percentage) in relation to vertical

stem width for isotropic interpolation (left) and interpolation of contrast (right).

An example

Let's assume we have two masters: an extra light one with no

stroke contrast and an extra bold one with horizontals that are 80%

of verticals. If we interpolate the light, medium and bold weights, we

get the linear interpolation of stem widths:

ExtraLight 2020 = 100%

Light 5565 85%

Medium 90110

82%Bold 125155 80%


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ExtraBold 160200 = 80%

What we want instead is probably closer to the linear interpolation

of contrast:

ExtraLight 2020 = 100%

Light 6265 95%Medium 99110 = 90%

Bold 132155 85%

ExtraBold 160200 = 80%

The formula

The standard formula for isotropic, linear interpolation is:

y= ( x x ) ( x x ) ( y y ) + y

The proposed formula for interpolating yto xratio is:

y= { [ 1 ( x x ) ( x x ) ] ( y x y x ) + y

x } x

fig 4: A comparison between standard isotropic interpolation (red) and interpolation

0 1 0 1 0 0

0 1 0 0 0 1 1

1 1


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of contrast (blue).

The algorithm

The algorithm interpolates ratio between horizontals and verticals. It

needs to be provided with vertical and horizontal stem widths of the

extreme masters (LightWeightX, LightWeightYand BoldWeightX,

BoldWeightY).

fig 5: A good way of providing the algorithm with xand yweight of the extreme

masters is by using stem widths.

For contrast to grow in linear relationship to weight, horizontals

have to grow at a different rate than verticals. This leads to

anisotropic interpolation.

Interpolation of the xcoordinates

Coordinates on the xaxis are interpolated traditionally. We start

with calculating the interpolation point for the desired vertical stem

weight (InterpolationWeightX) of the interpolated instance. It


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designates the position on the interpolation axis (on the 01 scale).

InterpolationPointX= ( InterpolationWeightX LightWeightX

) ( BoldWeightX LightWeightX)

Then we apply it to x coordinates of nodes:

x= InterpolationPointX( x x ) + x

Interpolation of the ycoordinates

We use our formula to calculate the horizontal stem weight for the

interpolated instance:

InterpolationWeightY= [ ( 1 InterpolationPointX) (

LightWeightY LightWeightX BoldWeightY BoldWeightX

) + BoldWeightY BoldWeightX] InterpolationWeightX

Now we can calculate the interpolation point for the ycoordinates:

InterpolationPointY= ( InterpolationWeightY LightWeightY

) ( BoldWeightY LightWeightY)

and apply it:

y= InterpolationPointY( y y ) + y

The implementations

This can be tested in all apps that allow for independent x and y

interpolation.

Glyphs app

Using it in Glyphs app requires adding InterpolationWeightY custom

parameter to the instances. To calculate it, apply the

InterpolationPointYvalue to the xinterpolation axis.

GlyphsInterpolationWeightY= InterpolationPointY(

BoldWeightX LightWeightX) + LightWeightX

Creating instances with this custom parameter is automated by a

great plug-indeveloped by Mekkablue.

1 0 0

1 0 0


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fig 6: Glyphs script developed by Mekkablue.

The limitations

As with any anisotropic interpolation, there are some obvious

limitations to using this method in current type design process. It

doesn't take diagonal contrast into account and, depending on the

positions of nodes and handles, the angles and widths of diagonals

and the curvature might shift. These changes, though small, need

to be corrected manually.

The application

This method can be used as a starting point for the design, since it

automates the process of adjusting horizontal stroke weights. In

some cases it can also be tricked into producing the desired

diagonal angles and widths right away.

Probably the best solution to the problem of contrast is still adding

an intermediate master, so this method can be used to simplify the

process of creating one.

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Provided with stem widths of extremes, the formula calculates

adequate horizontal stem width for a chosen vertical stem width.

The result doesn't need to be applied with the aforementioned

interpolation method. It could be implemented manually or

automatically by a parametric design tool, aware of curves,

diagonals and the axis.

Maciej Ratajski

August 29th, 2013

Thanks for contribution, feedback and critique to: Rainer Erich

Scheichelbauer, MichaJarociski and Tim Ahrens.

1. There are multiple reasons for it, one being the fact that horizontals tend to

look thicker than verticals of the same width. Furthermore, contrast is often

derived from pen-formed writing and is a strong part of the typographical

tradition. It is also a design feature.

2. This is also due to multiple reasons. In most cases there just isn't enough

space for growth for horizontal stems. Bold letters can be wider than the light

ones but they usually cannot be taller.

3. Adequate contrast in medium and light instances is often achieved by

compromising the design of the bold master.

4. One way is to put nodes between which you interpolate at the same height

(and interpolate traditionally). The other is to turn a flaw into a feature by

setting vertical distances between interpolated nodes so that the angle (and

the diagonal stroke) end up right.

Interpolation of Contrast

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