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High Quality Content by WIKIPEDIA articles! In Euclidean plane geometry, tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles. A tangent line t to a circle C intersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed. ... Read more

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This volume presents a wide range of results in analytic andprobabilistic number theory. The full spectrum of limit theorems inthe sense of weak convergence of probability measures for the modulesof the Riemann zeta-function and other functions is given by Dirichletseries. Applications to the universality and functional independenceof such functions are also given. Furthermore, similar results arepresented for Dirichlet L-functions and Dirichlet series withmultiplicative coefficients. Audience: This is a self-contained book, useful for researchersand graduate students working in analytic and probabilistic numbertheory and can also be used as a textbook for postgraduate courses. ... Read more

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-A comprehensive and self-contained treatment -Special features include an emphasis on the interplay of ideas from algebra and geometry and their historical origins By virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. This book offers a comprehensive and self-contained treatment of the mathematical theory of PH curves, including algorithms for their construction and examples of their practical applications. Special features include an emphasis on the interplay of ideas from algebra and geometry and their historical origins, detailed algorithm descriptions, and many figures and worked examples. The book will appeal, in whole or in part, to mathematicians, computer scientists, and engineers. ... Read more