A Brief Course In Analytic Geometry [Yefimov]

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N. Yefimov’s A Brief Course in Analytic Geometry is a concise university-level introduction to the theoretical foundations of plane and solid analytic geometry. The book begins with coordinate methods: representing points and geometric objects algebraically, calculating distances and ratios, describing curves through equations and parametric representations, and studying intersections. It then develops the geometry of first- and second-order curves—especially lines, circles, ellipses, parabolas, and hyperbolas—and shows how changing coordinates can simplify their equations and reveal their geometric properties. 

The second half extends these ideas into three-dimensional space. Yefimov introduces vectors and their linear operations, scalar products, vector and triple-scalar products, and uses them to formulate equations of planes, straight lines, curves, and surfaces. A substantial final section is devoted to quadric surfaces, providing the analytic framework for objects such as ellipsoids, paraboloids, and hyperboloids; the book concludes with an elementary treatment of determinants. Overall, the approach emphasizes the translation between geometric ideas and algebraic equations, making analytic geometry a systematic language for studying both plane and spatial figures. 


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