Explore the fundamental concepts of hyperbolic geometry in this 45-minute lecture from the Universal Hyperbolic Geometry series. Delve into the algebraic definitions of points and lines using proportions of three numbers, highlighting the duality between these objects. Learn how to plot points and lines, derive formulas for joining points and finding intersections of lines, and understand the J function for computations. Discover applications to Cartesian geometry and gain insights into the computational nature of hyperbolic geometry. Visualize concepts through three-dimensional representations and grasp the connection between lines and planes in space and their projections on the viewing plane.