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Although chord and sagitta, when considered in tandem, may reflect manyunderlying geometric properties of circles on the Euclidean plane, theirimplications on the digital plane are not yet well-understood. In this paper,we explore some of their fundamental properties on the digital plane that havea strong bearing on the unsupervised detection of circles and circular arcs ina digital image. We show that although the chord-and-sagitta properties of areal circle do not readily migrate to the digital plane, they can indeed beused for the analysis in the discrete domain based on certain bounds on theirdeviations, which are derived from the real domain. In particular, we derive anupper bound on the circumferential angular deviation of a point in the contextof chord property, and an upper bound on the relative error in radiusestimation with regard to the sagitta property. Using these two bounds, wedesign a novel algorithm for the detection and parameterization of circles andcircular arcs, which does not require any heuristic initialization or manualtuning. The chord property is deployed for the detection of circular arcs,whereas the sagitta property is used to estimate their centers and radii.Finally, to improve the accuracy of estimation, the notion of restricted Houghtransform is used. Experimental results demonstrate superior efficiency androbustness of the proposed methodology compared to existing techniques.

On Chord and Sagitta in ${\mathbb Z}^2$: An Analysis towards Fast and Robust Circular Arc Detection