# How Smooth Line and Smooth Polygon work

Smoothing is a generalization operation that removes sharp angles in a line or outline. Smoothing is often used to improve aesthetic appearance in cartographic output. The Smooth Line, Smooth Polygon and Smooth Shared Edges geoprocessing tools offer two different smoothing algorithms.

## Smoothing algorithms

### Polynomial Approximation with Exponential Kernel (PAEK)

The Polynomial Approximation with Exponential Kernel (PAEK) option calculates smoothed lines using a parametric continuous averaging technique. The current point coordinates are calculated by the weighted averaging of the coordinates of all points of the source line. The weights of each point decrease with the distance along the line to the current point. In addition to averaging, approximation with polynomials of the second degree is used. The smoothed line doesn't necessarily contain all or any vertices of the source line except the end points. The result depends on one parameter. The method is stable—a minor change to the parameter causes a minor change in the result. In general, this algorithm gives better results than the Bezier Interpolation option in terms of the smoothed shapes. This option is based on the algorithm defined by Bodansky, et al, (2002).

The Smoothing Tolerance parameter is used by the (PAEK) algorithm only. This tolerance specifies the length of a moving path along an input line used to calculate the smoothed coordinates by the (PAEK) algorithm. The longer the path, the more smoothed the resulting lines. Each new location is calculated using the information within the specified length of the path that is centered at the location. In this way, the tolerance defines the region within which all coordinates are taken into account.

### Bezier Interpolation

The Bezier Interpolation option fits Bezier curves through every line segment along an input line. The Bessel Tangent is used to connect the curves smoothly at vertices (Farin, 1997). The resulting lines pass through input vertices. This option is based on the algorithm defined by Farin, (1997).

## References

Bodansky, Eugene; Gribov, Alexander; and Pilouk, Morakot (2002) "Smoothing and Compression of Lines Obtained by Raster-to-Vector Conversion", LNCS 2390, Springer, p. 256-265.

Farin, Gerald (1997) Curves and Surfaces for CAGD, a Practical Guide, 4th Edition, Academic Press, USA.