Titlebook: Circles, Spheres and Spherical Geometry; Hiroshi Maehara,Horst Martini Textbook 2024 The Editor(s) (if applicable) and The Author(s), unde

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https://doi.org/10.1007/978-0-333-97727-9 of figures. Girard’s formula for the area of a spherical triangle is also proved. From the area formula for spherical polygons obtained by applying Girard’s formula, Legendre’s proof of Euler’s polyhedral formula is derived. The theorem on inscribed angles is presented, and the notion of polar set is introduced.

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Spherical Geometry I, of figures. Girard’s formula for the area of a spherical triangle is also proved. From the area formula for spherical polygons obtained by applying Girard’s formula, Legendre’s proof of Euler’s polyhedral formula is derived. The theorem on inscribed angles is presented, and the notion of polar set is introduced.

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Spherical Geometry II, cosine law is applied to prove a triangle comparison theorem for spheres. Euler’s formula for spherical excess is applied to prove a theorem for the area of a spherical triangle with two fixed edges and one variable edge. We also derive an isoperimetric theorem for spherical quadrilaterals.

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,Casey’s Theorem,replaced by common tangent distances between circles. The proof of Casey’s theorem is elaborated and requires a few new techniques. Casey’s theorem can be also extended to a set of circles consisting of more than four circles.

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Jingting Wang,Tianxing Wu,Jiatao Zhangime camera of a regular Android cell phone supported with ultrasonic sensors. The main focus of this work is how to pre-process images on the fly in order to be able to train and to tune the plastic learning module, improving the object’s trajectory prediction.