CurveBench: A Benchmark for Exact Topological Reasoning over Nested Jordan Curves

TL;DR

CurveBench is a new benchmark dataset for hierarchical topological reasoning over nested Jordan curves, highlighting the difficulty of exact topological understanding in visual models.

Contribution

The paper introduces CurveBench, a comprehensive dataset and task for structured prediction of containment trees from images of Jordan curves, and evaluates current models' performance.

Findings

Strongest model achieves only 71.1% accuracy on easy and 19.1% on hard configurations.

Fine-tuning vision-language models significantly improves accuracy, surpassing some large language models.

There remains a large gap in exact topological reasoning capabilities of current models.

Abstract

We introduce CurveBench, a benchmark for hierarchical topological reasoning from visual input. CurveBench consists of \textbf{756 images} of pairwise non-intersecting Jordan curves across easy, polygonal, topographic-inspired, maze-like, and dense counting configurations. Each image is annotated with a rooted tree encoding the containment relations between planar regions. We formulate the task as structured prediction: given an image, a model must recover the full rooted containment tree induced by the curves. Despite the visual simplicity of the task, the strongest evaluated model, Gemini 3.1 Pro, achieves only \textbf{71.1\%} tree-generation accuracy on CurveBench-Easy and \textbf{19.1\%} on CurveBench-Hard. We further demonstrate benchmark utility through RLVR-style fine-tuning of open-weight vision-language models. Our trained Qwen3-VL-8B model improves over \texttt{Qwen-3-VL-8B-Thinking} from \textbf{2.8\%} to \textbf{33.3\%} tree-generation accuracy on CurveBench-Easy, exceeding GPT-5.4 and Claude Opus 4.5 under our evaluation protocol. The remaining gap, especially on CurveBench-Hard, shows that exact topology-aware visual reasoning remains far from solved.