Arcfitting

Arcfitting is the process of determining the parameters of a circular arc that best fits a set of two-dimensional points or other observational data. An arc refers to a portion of a circle, and is defined by a center (cx, cy), a radius r, and a start and end angle (or equivalently two end points). The objective is to minimize a chosen distance measure between the data and the arc, typically the perpendicular distance to the circle or the arc itself.

Common approaches start with an initial estimate obtained from algebraic circle fitting, such as the Kasa method, which linearizes the circle equation and solves a linear least squares problem. This provides a fast, if approximate, solution that can serve as a starting point for more accurate methods. Geometric (orthogonal) distance fitting then minimizes the true distance from each point to the circle using nonlinear least squares (for example Gauss-Newton or Levenberg–Marquardt). This yields parameter estimates that better reflect measurement errors.

Robust variants address outliers and noise: Taubin and Pratt circle fitting offer improved stability, while RANSAC-based methods can identify and exclude outliers before fitting. Arc-specific fitting often imposes constraints so the solution represents a finite arc rather than a full circle, by binding the arc to start/end angles or by specifying arc endpoints.

Practical considerations include data preprocessing (outlier removal, normalization), weighting by measurement uncertainty, and distinguishing between fitting a full circle versus a finite arc. Degenerate cases, such as nearly colinear points or extremely large radii, require special handling.

Applications of arcfitting span image analysis, computer vision and edge detection, GIS for curved road or river centerlines, vector graphics and CAD, and the study of circular features in astronomy. Evaluation metrics typically report radial or angular residual errors, and may consider endpoint alignment or arc length accuracy.