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Application of a Fourier-Type Series Approach based on Triangles of Constant Width to Letterforms

Micha Wasem, Florence Yerly

Abstract

In this work, we present a novel approach to type design by using Fourier-type series to generate letterforms. We construct a Fourier-type series for functions in $L^2(S^1,\mathbb C)$ based on triangles of constant width instead of circles to model the curves and shapes that define individual characters. In order to compute the coefficients of the series, we construct an isomorphism $\mathcal R:L^2(S^1,\mathbb C)\to L^2(S^1,\mathbb C)$ and study its application to letterforms, thus presenting an alternative to the common use of Bézier curves. The proposed method demonstrates potential for creative experimentation in modern type design.

Application of a Fourier-Type Series Approach based on Triangles of Constant Width to Letterforms

Abstract

In this work, we present a novel approach to type design by using Fourier-type series to generate letterforms. We construct a Fourier-type series for functions in based on triangles of constant width instead of circles to model the curves and shapes that define individual characters. In order to compute the coefficients of the series, we construct an isomorphism and study its application to letterforms, thus presenting an alternative to the common use of Bézier curves. The proposed method demonstrates potential for creative experimentation in modern type design.
Paper Structure (10 sections, 5 theorems, 34 equations, 9 figures)

This paper contains 10 sections, 5 theorems, 34 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Parametrizations of Triangles of Constant Width
  3. Support Functions
  4. Triangles of Constant Width
  5. An Isomorphism of $L^2(S^1,\mathbb C)$ based on Triangles of Constant Width
  6. Application to Letterforms
  7. Approximation with $a=\frac{1}{24}$
  8. Approximation with $a=\frac{1}{8}$
  9. Approximation with $a=\frac{1}{5}$
  10. Conclusion

Key Result

Lemma 3.1

If $a>0$, the sequence $(j_n)_{n\in\mathbb N}$ with $j_0=1, j_1 = 2a$ verifying jacobrecurrence admits an explicit representation

Figures (9)

  • Figure 1: Truncated S for the choices $m=4,10,25$ and $m=100$.
  • Figure 2: The images of $\gamma$ for the choices $a=\frac{1}{24}$ and $a=\frac{1}{8}$ as subsets of $\mathbb R^2\cong\mathbb C$
  • Figure 3: The graphs of $\mathcal{R}(\sin)$ and $\mathcal{R}(\cos)$ as real-valued functions if $a=\frac{1}{24}$.
  • Figure 4: The graphs of $\mathcal{R}(\sin)$ and $\mathcal{R}(\cos)$ as real-valued functions if $a=\frac{1}{8}$.
  • Figure 5: The graphs of $\mathcal{R}(\sin)$ and $\mathcal{R}(\cos)$ as real-valued functions if $a=\frac{1}{5}$.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Remark
  • Proposition 3.5
  • Remark