# The Mathematics behind Font Shapes --- Bézier Curves and More

## Contents

Steve Jobs once said that “technology alone is not enough”. He thinks that truly great products reside on the intersection of technology and liberal arts. He had also told a story about how he dropped Reed college and attended the calligraphy class, which had a great impact on the font he decided to adopt on Macintosh later^{1}.

In my opinion, computer font is a great demonstration of arts and aesthetics combined seamless with technology. In this post, I try to understand the mathematics behind the beauty of font. Below is the Source Han Serif font designed by Google and Adobe.

You may have noticed already that most fonts in use nowadays can be resized to arbitrary size and still looks good and smooth. This kind of fonts are called outline fonts (or vector fonts), as opposed to the already deprecated bitmap fonts. The fonts we now use are mostly opentype fonts, which is an extension of the truetype font format^{2}.

The word “outline” means that the shape of each glyph in a font is made up of straight and curved lines. Various fonts mainly use two major outline types — truetype outline and postscript outline. Truetype outlines are represented using quadratic Bézier curves, while postscript outlines are represented using cubic Bézier curves. Quadratic Bézier curve is a specical case for cubic Bézier curve. As a result, quadratic curves can be converted to cubic curves without any loss of accuracy, while cubic curves can only be approximated by several quadratic curves.

Since opentype font can either use truetype or postscript outlines, we can convert truetype font to opentype format without loss of accuracy, but not the other way around^{3}.

## How does Bézier curves represent the font outline?

Bézier curves are parametric curves whose shapes are controlled by a parameter `t`

and some on and off curve points. In this part, we focus on quadratic Bézier curves. The shape of quadratic curves are determined by two on-curve points (or end points) and one off-curve point. The off-curve point is used to control the shape of the curve. Suppose we have three points： \((A_x, A_y)\) (point A), \((B_x, B_y)\) (point B), and \((C_x, C_y)\) (point C). A and C are the end points and B is the control point. The parametric equation for the underlying curve is:

\[\begin{align} P_x &= (1-t)^2A_x + 2t(1-t)B_x + t^2C_x \\ P_y &= (1-t)^2A_y + 2t(1-t)B_y + t^2C_y \\ \end{align}\]

When parameter \(t\) is changed from 0 to 1 in the above equations, we can get all the points \((P_x, P_y)\) in the underling curve. That is, this curve can be uniquely determined once we know the position of point A, B and C.

See the figure above for an example of quadratic B-curve. In this example, A is \((1.0, 1.0)\), B is \((1.5, 2.0)\) and C is \((3.0, 1.5)\). The blue line is the generated curve when we change \(t\) from 0 to 1.

For a glyph in a font, its outline can be split into multiple small curves until it can be represented by B-curves^{4}, which is in turn represented by three points. Ultimately, if we simplify things a bit, we can represent each glyph in a font with a set of points.

In the below image, I show the outline for character `e`

from a font^{5}:

In the above image, the character `e`

is made up of two outlines. The red dots are the end points of B-curves and the cross symbols are the control points. You can see that for complex characters, the outline consists of a dozen small curves.

## How are the characters filled?

Knowing only the glyph outline is not enough. How do we actually fill the characters, i.e., how do you decide what is inside the character and what is outside? In fact, there is a rule. Each point is given a unique number starting from zero. When you traverse the outline from lower number to higher number, the “inside” of character is always on your right. So the outer outline of a character is numbered clockwise and the inner outline is numbered counter-clockwise. See the below image for an example:

Furthermore, if one on-curve point is exactly in the center of two control points, it can be omitted without loss of information. So you may see there are several consecutive control points with no on-curve points in between. In the above image, for example, point 7, 8, 9 and 10 are both control points. More discussion regarding this issue can be found here.

# Conclusion

In this post, I have roughly covered how font outline works, that is, how the glyph shapes for a font are represented. But how we should display or print the font at various size (usually measured in pt) is another long story. Hopefully, I will try to describe that on another posts soon.

# References

- https://fontforge.github.io/bezier.html
- Truetype outlines
- How outline font is rendered on screen
- Very good documentation about font outline and how to convert hand design to acutal font outline
- Microsoft truetype font documentation