Bézier curves and the Takagi function

TL;DR

The paper connects Bézier subdivision via the de Casteljau algorithm to affine iterated function systems by representing subdivision as two complex-parameter maps with coefficients in Z[t], and identifies a complex domain where the IFS is contractive and has a unique connected attractor. In a regime where t=1/2+iβ and β→0, a suitably scaled attractor converges to the Takagi curve T, with the Takagi function appearing as the leading vector field on the base Bézier curve. The results extend to Bézier curves with more control points, yielding a Takagi component in the corresponding vector field and illustrating a deep link between subdivision fractals and classical fractal functions. Overall, the work bridges subdivision-based geometric modeling with fractal theory and provides explicit IFS constructions for complexified Bézier curves and their fractal limits.

Abstract

We consider Bézier curves with complex parameters, and we determine explicitly the affine iterated function system (IFS) corresponding to the de Casteljau subdivision algorithm, together with the complex parametric domain over which such an IFS has a unique global connected attractor. For a specific family of complex parameters having vanishing imaginary part, we prove that the Takagi fractal curve is the attractor, under suitable scaling.

Figures (7)

• Figure 1: Attractors of the IFS for quadratic Bézier curve with complex parameter $t = 0.4 -0.55 i$ with control points (plotted as orange polyline) from left to right: $(-1 +i, i, 2+i)^\top$, $(-1 +i, -i, 2+i)^\top$, and $(-1 +i, 2-i, 2+i )^\top$. The initial point $X\in\mathbb{C}^{1\times (n+1)}$ is an arbitrary point in dimension $n$ in homogenous coordinates. We show the state of the system after 20 iterations. In order to visualize the higher dimensional attractor, we project it to the complex plane. In all cases the fixed point of $f_0$ is the first row of matrix $\mathbf{P}$ of (\ref{['eq:BezierIFS']}), i.e., $(-1 +i,1,1)$, while the fixed point of $f_1$ is $(2+i,0,1)$ --- the last row of $\mathbf{P}$.
• Figure 2: The domain of $t$, over which the IFS $\{\mathbb{C};f_1,f_2\}$ is hyperbolic, is the intersection of two open discs $|t|<1$ and $|1-t|<1$.
• Figure 3: Attractor of the IFS for the Bézier curve of degree 6 with control points $(-2,-1+i,2 i,1+i,1,2+i,3+i)^\top$ and complex parameters (from top to bottom and left to right): $t = 0.1 + 0.3 i$, $t = 0.4 + 0.4 i$, $t = 0.5 + 0.25 i$, and $t = 0.5 + 0.5 i$. We show the state of the system after 20 iterations. To visualize the higher dimensional attractor, we project this attractor to the complex plane. Note that the fixed point of $f_0$ is the first row of matrix $\mathbf{P}$ of (\ref{['eq:BezierIFS']}), i.e., $(-2,1,0,0,0,0,1)$, while the fixed point of $f_1$ is $(3+i,0,0,0,0,0,1)$ --- the last row of $\mathbf{P}$.
• Figure 4: The blue curve is $\mathcal{A}^\ast(\beta)$, which for $\beta=\frac{1}{2}$ becomes the Lévy C curve. The orange curve is the graph $\mathcal{T}$ of the Takagi function $\mathrm{T}$.
• Figure 5: Left: the scaled attractor $\mathcal{A}^\ast(\beta)$ for $\beta=\frac{1}{4}$ (blue) and the Takagi curve (orange). Right: the same for $\beta=\frac{1}{8}$. For $\beta=\frac{1}{4}$, the attractor features crunodes; these crunodes appear to have turned into cusps for $\beta=\frac{1}{8}$, although this is not the case (see remark after theorem \ref{['thm:Takagi']}).

Theorems & Definitions (23)

• Theorem 2.1
• Definition
• Theorem 2.2: Hutchinson
• Lemma 3.1
• proof
• Remark
• Lemma 3.2
• proof
• Corollary
• Theorem 3.1