Resume Planimetry includes metric part and trigonometry. In geometries of metric spaces from the end of XIX age their tensor forms are widely used. However the trigonometry is remained only in its scalar form in a plane. The tensor trigonometry is development of the flat scalar trigonometry from Leonard Euler classic forms into general multi-dimensional tensor forms with vector and scalar orthoprojections and with step by step increasing complexity and opportunities. Described in the book are fundamentals of this new mathematical subject with many initial examples of its applications. In theoretic plan, the tensor trigonometry complements naturally Analytic Geometry and Linear Algebra. In practical plan, it gives the clear instrument for solutions of various geometric and physical problems in homogeneous isotropic spaces, such as Euclidean, quasi- and pseudo-Euclidean ones. In these spaces, the tensor trigonometry gives very clear general laws of motions in complete forms and with polar decompositions into principal and secondary motions, their descriptive trigonometric vector models, which are applicable also to n-dimensional non-Euclidean geometries in subspaces of constant radius embedded in enveloping metric spaces, and in the theory of relativity. In STR, these applications were considered till a trigonometric 4D pseudoanalog of the 3D classic theory by Frenet–Serret with absolute differentially-geometric, kinematic and dynamic characteristics in the current points of a world line. New methods of the tensor trigonometry can be also useful in other domains of mathematics and physics. The book is intended for researchers in the fields of multi-dimensional spaces, analytic geometry, linear algebra with theory of matrices, non-Euclidean geometries, theory of relativity and also to all those who is interested in new knowledges and applications, given by exact sciences. It may be useful for educational purposes on this new subject in the university departments of algebra, geometry and physics. This book is an updated author’s English version of the original Russian scientific monograph “Tensor Trigonometry. Theory and Applications.” – Moscow: Publisher MIR, 2004, 336p., ISBN-10: 5-03-003717-9 and ISBN-13: 978-5-03-003717-2. On the Google books there is an original Russian edition of this book (2004): https://books.google.ru/books/about?id=HGgjEAAAQBAJ