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Evolutes and involutes of frontals in the euclidean plane

Fukunaga T., Takahashi M.

Demonstratio Mathematica

2015

Vol. 48, nr 2

147--166

We have already defined the evolutes and the involutes of fronts without inflection points. For regular curves or fronts, we can not define the evolutes at inflection points. On the other hand, the involutes can be defined at inflection points. In this case, the involute is not a front but a frontal at inflection points. We define evolutes of frontals under conditions. The definition is a generalisation of both evolutes of regular curves and of fronts. By using relationship between evolutes and involutes of frontals, we give an existence condition of the evolute with inflection points. We also give properties of evolutes and involutes of frontals.

On homotopy classification of phrases and its application

Fukunaga T.

Demonstratio Mathematica

2010

Vol. 43, nr 2

419-432

This paper is a survey on the recent study of homotopy theory of generalized phrases in Turaev's theory of words and phrases. In this paper, we introduce the homotopy classification of generalized phrases with some conditions on numbers of letters. The theory of topology of words and phrases are closely related to the theory of surface-curves. We also introduce applications of topology of words to the topology of surface-curves.