# Mathematical Progress in Expressive Image Synthesis IIIAttractive Plane Curves in Differential Geometry

Mathematical Progress in Expressive Image Synthesis III: Attractive Plane Curves in Differential... [The purpose of chapter is to discuss plane curves from differential geometric point of view and applications of plane curves to computer aided designs. Plane curves are determined uniquely by curvatures up to Euclidean motions. Thus geometry of plane curves are formulated by the Euclidean motion group SE(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SE}(2)$$\end{document}. From industrial point of view, other transformation groups are more appropriate for characterizing certain classes of plane curves. For instance, under equiaffine transformation group, conics are characterized as plane curves with constant equiaffine curvatures. Plane curves with monotonous curvature function have been paid much attention in industrial shape design and computer aided geometric design. In this chapter we study plane curves with monotonous curvature function, especially log-aesthetic curves, in terms of similarity transformation group.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Mathematical Progress in Expressive Image Synthesis IIIAttractive Plane Curves in Differential Geometry

Part of the Mathematics for Industry Book Series (volume 24)
Editors: Dobashi, Yoshinori; Ochiai, Hiroyuki
14 pages

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Publisher
Springer Singapore
ISBN
978-981-10-1075-0
Pages
121 –135
DOI
10.1007/978-981-10-1076-7_13
Publisher site
See Chapter on Publisher Site

### Abstract

[The purpose of chapter is to discuss plane curves from differential geometric point of view and applications of plane curves to computer aided designs. Plane curves are determined uniquely by curvatures up to Euclidean motions. Thus geometry of plane curves are formulated by the Euclidean motion group SE(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SE}(2)$$\end{document}. From industrial point of view, other transformation groups are more appropriate for characterizing certain classes of plane curves. For instance, under equiaffine transformation group, conics are characterized as plane curves with constant equiaffine curvatures. Plane curves with monotonous curvature function have been paid much attention in industrial shape design and computer aided geometric design. In this chapter we study plane curves with monotonous curvature function, especially log-aesthetic curves, in terms of similarity transformation group.]

Published: May 22, 2016

Keywords: Transformation group; Log-aesthetic curve; Similarity geometry; Similarity curvature