From Tutte to Floater and Gotsman: On the Resolution of Planar Straight-line Drawings and Morphs

TL;DR

This work analyzes the readability of planar graph drawings produced by Tutte’s barycentric method, Floater’s F-drawings, and Floater–Gotsman FG-morphs on maximal plane graphs. It derives a tight-looking exponential lower bound on the resolution of F-drawings in terms of the outer-triangle resolution and the smallest positive coefficient in the coefficient matrix, and proves that the corresponding FG-morphs inherit similar hard-to-preserve resolution properties, including the possibility of exponentially small resolution along the morph. The authors also show upper bounds via explicit graph constructions and provide a discretization approach that reduces FG-morphs to piecewise-linear morphs with a polynomially large (in $n$) number of steps. Together, these results reveal inherent resolution limitations of these popular Planar Straight-Line Drawing methods, even when starting from moderately-resolved inputs, and raise questions about polynomial-resolution guarantees for morphs in general. The work also introduces a rigorous algebraic and geometric framework for analyzing how coefficient matrices influence drawing quality and morph behavior, with implications for surface parameterization and mesh morphing applications.

Abstract

The algorithm of Tutte for constructing convex planar straight-line drawings and the algorithm of Floater and Gotsman for constructing planar straight-line morphs are among the most popular graph drawing algorithms. In this paper, focusing on maximal plane graphs, we prove upper and lower bounds on the resolution of the planar straight-line drawings produced by Floater's algorithm, which is a broad generalization of Tutte's algorithm. Further, we use such results in order to prove a lower bound on the resolution of the drawings of maximal plane graphs produced by Floater and Gotsman's morphing algorithm. Finally, we show that such a morphing algorithm might produce drawings with exponentially-small resolution, even when transforming drawings with polynomial resolution.

Figures (10)

• Figure 1: Illustration for the proof of Lemma \ref{['le:resolution-xy-triangle']}. The side $s$ of $\Delta$ is represented by a fat line segment. In (a) two vertices of $\Delta$ belong to the boundary of $\mathcal{R}$, and in (b) three vertices of $\Delta$ belong to the boundary of $\mathcal{R}$.
• Figure 2: Illustration for the proof of Lemma \ref{['le:first-vertex-is-internal']}. (a) If the altitude of $T$ through $v$ does not lie inside $T$, then there is a height of $T$ smaller than $\delta$. (b) If $v$ is an external vertex of $G$, then the distance between one of the end-vertices of $e$ and one of the edges incident to the outer face of $G$ is smaller than $\delta$. The gray angles are equal.
• Figure 3: Illustration for the proof that $y(u)\geq 0$, where $(v,u)$ is the edge that follows $(v,u_e)$ in clockwise order around $v$. (a) The case in which $\widehat{vuu_e}>\widehat{vu_eu}$. (b) The case in which $\widehat{vu_eu}>\widehat{vuu_e}$.
• Figure 4: (a) The subgraph of $G$ composed of the edges incident to $v$. (b) The subgraph $G^v$ of $G$ induced by the neighbors of $v$. (c) The subgraph $G_1$ of $G$ inside $\mathcal{C}_1$.
• Figure 5: Illustration for the proof of Lemma \ref{['le:neighbors-above-below']}. The fat lines are $\ell(u_1)$, $\ell(u_2)$, and $\ell(u_3)$.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• proof
• proof
• proof
• Lemma 2
• proof