Approximate convolution with pairs of cubic Bézier LN curves

• Published in: Computer aided geometric design Vol. 28; no. 6; pp. 357 - 367
• Main Authors: Ahn, Young Joon, Hoffmann, Christoph M.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01-08-2011; Elsevier
• Subjects: [formula omitted] Bezier approximation; Approximation; Bezier; Compatibility; Computer aided design; Convolution; Convolution curve; Cubic LN curve; Error estimate; Exact sciences and technology; Linear normal map; Mathematical analysis; Mathematical models; Mathematics; Minkowski sum; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Offsets; Sciences and techniques of general use; Splines; Cubic LN curve; Minkowski sum; Error estimate; Convolution curve; Offsets; Linear normal map; G − 1 Bezier approximation; Offset curve; G ― 1 Bezier approximation; Error estimation; Necessary and sufficient condition; Minkowski metric; Spline approximation; Cubic spline; Numerical approximation; Letter; Convolution; Bézier curve; Cubics; Curve fitting; Computer aided design
• ISSN: 0167-8396; 1879-2332
• DOI: 10.1016/j.cagd.2011.06.006

In this paper we present an approximation method for the convolution of two planar curves using pairs of compatible cubic Bézier curves with linear normals (LN). We characterize the necessary and sufficient conditions for two compatible cubic Bézier LN curves with the same linear normal map to exist. Using this characterization, we obtain the cubic spline approximation of the convolution curve. As illustration, we apply our method to the approximation of a font where the letters are constructed as the Minkowski sum of two planar curves. We also present numerical results using our approximation method for offset curves and compare our method to previous results.
► An algorithm is presented for convolving two planar curves using cubic Bezier LN curves. ► Necessary and sufficient conditions are given for the approximation using compatible LN curves. ► Several examples illustrate our method and comparisons with prior work are given.