Reproduce “The Vector Space of Convex Curves: How to Mix Shapes” on OpenResearch
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About this paper
Dongsung Huh
The paper introduces a representation for convex plane curves based on how their local radius of curvature varies with direction. It takes the logarithm of that radius and adds the resulting profiles pointwise, making addition a direction-dependent rescaling. Illustrations combine an ellipse and a spiral into an elliptic spiral, but the paper reports no perceptual or quantitative evaluation.
Key insights
- Convex curves can be endowed with the structure of a vector space and a Hilbert space through the defined operations on their log-radius profiles, allowing for rigorous mathematical analysis.
- Elementary shapes derived from sinusoidal log-radius profiles preserve their characteristic features with varying amplitudes, a significant improvement over angle profile representations.
- The proposed log-radius profile representation offers a 'scaling' perspective on shape, complementing the 'bending' perspective of angle profiles.
Summary by alphaXiv.
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