Planar curve offset based on circle approximation

New Approximation Methods for Planar Offset and Convolution Curves

“…This offset approximation method is called CAO (Circle Approximation Offset). The offset approximation error is simply r times that of approximating the unit circular arc by a quadratic curve Q(s), assuming there is no local selfintersection loop in the computed offset approximation (see Reference [26] for more details). In spite of the high degree in the resulting approximation, the size of output data from CAO is smaller than those of current offset approximation methods using the control polygon based approach.…”

“…Since the circle approximation error is the same as the error in CAO, we can estimate the offset approximation error in an a priori fashion (see Reference [26] for more details). That is, we can compute the error bound even without constructing an offset approximation curve.…”

“…When the approximation error between Q 2 (s) and C 2 (s) is , the resulting approximated convolution curve C 1 (t) + Q 2 (s(t)) also has the same approximation error . (In fact, this is true only for non-trimmed (closed or infinite) input curves and for the convolution curves after eliminating self-intersection loops; see Reference [26] for more details.) We call this method QAC (Quadratic Approximation Convolution).…”

A Novel Method of Offset Approximation along the Normal Direction with High Precision

“…Hoschek [9] suggested a least squares solution, Piegl and Tiller [10] employed sample interpolation, and Cheng and Wang [11] used Jacobi series to approximate the offset curve. Lee et al [12] regarded the offset curve as a convolution of a sweeping circle, Ahn et al [13] used conic approximation to approximate the sweeping circle, and Zhao and Wang [14] employed second-order rational polynomials, and achieved high-precision approximation.…”

“…Some algorithms, such as those proposed by Piegl and Tiller [10] and Lee et al [12], employ the point-sample technique for fast and convenient calculation. However, when the input curve changes its parameterization, the algorithms will take it as a new curve, and generate a wholly different approximation result.…”