By Gunnar E. Carlsson, Ralph L. Cohen, Wu-Chung Hsiang, John D.S. Jones

ISBN-10: 0387940987

ISBN-13: 9780387940984

In 1989-90 the Mathematical Sciences learn Institute performed a software on Algebraic Topology and its purposes. the most components of focus have been homotopy concept, K-theory, and functions to geometric topology, gauge conception, and moduli areas. Workshops have been performed in those 3 components. This quantity includes invited, expository articles at the issues studied in this application. They describe fresh advances and aspect to attainable new instructions. they need to turn out to be beneficial references for researchers in Algebraic Topology and comparable fields, in addition to to graduate scholars.

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**Sample text**

Each cycloidal sector swept by the normal segment from the point of contact of the disk to the cycloid has area three times that of the overlapping circular segment cut from the rolling disk. This surprising result is extended to epicycloids (and hypocycloids), obtained by rolling a disk of radius r externally (or internally) around a ﬁxed circle of radius R. The factor 3 is replaced by (3 + 2r/R) for the epicycloid, and by (3 − 2r/R) for the hypocycloid. This leads to several interesting consequences.

8: The sector OCP has area ω± times that of the overlapping circular segment cut oﬀ by chord P C. 1. 2. Every epicycloidal or hypocycloidal cap PODT has area κ± times that of wedge P CT of the rolling disk, with κ+ = 1 + 2r/R for the epicycloid, and κ− = 1 − 2r/R for the hypocycloid. 2. We treat the epicycloid ﬁrst. 9a shows an epicycloidal arc OP traced by a point P on a disk of radius r as it rolls along the outer circumference of a ﬁxed circle of radius R. The epicyloid lies inside the annular ring between the ﬁxed circle of radius R and the concentric circle of radius R + 2r.

9 Epicycloidal and Hypocycloidal Caps Revisited..................................... 64 Notes............................................................................................................... 1. INTRODUCTION 33 2 A point on the boundary of a circular disk that rolls once along a straight line traces a cycloid. The cycloid divides its circumscribing rectangle into a cycloidal arch below the curve and a cycloidal cap above it. The area of the arch is three times that of the disk, and the area of the cap is equal to that of the disk.

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