This paper is dedicated in loving memory to V. V. Alexandrov and A. A. Gol'dberg. Abstract. This is the second of two papers devoted to the topic of conformality at a point and related notions in the plane. We derive representation formulas and estimates for the modules of families of curves that are images of circles, radial segments and arcs of spirals under a µ-homeomorphism. We use them to convert the extremal-length type sufficient and necessary conditions for conformality at a point from

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... art I to analytic sufficient conditions, that depend on directional dilatations and bypass the assumption of K-quaisconformality. Our results extend the Teichmüller-Wittich-Belinskii theorem, results of Reich and Walczak, the author and Jenkins, and Gutlyanskii and Martio.