Forma, Vol. 32 (No. 1), pp. 29-41, 2017

Original Paper

Generating Geometric Patterns Using Periodic Functions

Katsumi Morita

Department of Fine Arts, Faculty of Music and Fine Arts, Sapporo Otani University, 1-1, Kita 16, Higashi 9, Higashi-ku, Sapporo 065-8567, Japan
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(Received October 31, 2015; Accepted December 25, 2017)

Abstract. Quadratic curves are typical examples of geometric forms. This paper looks at selected quadratic curves and describes their transformation into cubic curves for use as axes for generating geometric patterns. Using periodic functions with t as an intermediary variable, we defined these curves as universal cubic curves expressed as x = f (t), y = g(t) and z = h(t). Using universal cubic curves selected as the axes and selected quadratic curves as motifs, we then generated original geometric patterns by applying affine transformations. This paper describes our investigation of mathematical modeling-based generation of 3D geometric patterns with cubic curves as the axes and quadratic curves as motifs, and of the transformation of the patterns into two dimensions to generate geometric pattern variations. Through pattern generation, this paper aims to provide a basic methodology that can be used in fields such as art and design.

Keywords: Composition Forms, Theory of Plastic Art, Geometric Pattern, Periodic Function, Affine Transformation

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