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A given blue circle and a given red circle intersect in two points. In order to obtain bipolar coordinates, a method is required to specify which point is the right one.. An isoptic arc is the locus of points X that sees points C, D under a given oriented angle of vectors i.e. = { | (,) = +}.
The 31 great circles can be seen as the union of edges in 3 polyhedra. The lower right image shows 15 red circles, 10 green circles, and 6 blue circles. In geometry, the 31 great circles of the spherical icosahedron is an arrangement of 31 great circles in icosahedral symmetry. [1]
The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer.Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection.
Book 3 of Euclid's Elements deals with the properties of circles. Euclid's definition of a circle is: A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.
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A straight line through a circle's center is orthogonal to it, and if straight lines are also considered as a kind of generalized circles, for instance in inversive geometry, then an orthogonal pair of lines or line and circle are orthogonal generalized circles.
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While a few of these were limited to chronological reading lists and discussion topics, a majority provided homework problems and exams (often with solutions) and lecture notes. Some courses also included interactive web demonstrations in Java, complete textbooks written by MIT professors, and streaming video lectures. As of May 2018, 100 ...