Geometric Properties Of Conics On A Hyperbolic Plane

In this thesis we define conics on a hyperbolic plane and investigate some of theirgeometric properties.Explicitly, we first define ellipse and hyperbola on a hyperbolic plane in the sameway as in the Euclidean case, and then derive their standard equations in suitablyestablished rectangular coordinate systems.Furthermore, we prove that, similar to the Euclidean case, ellipse and hyperbolaon a hyperbolic plane have the focus-directrix property.Thus we define parabola on a hyperbolic plane using the equidistant focus-directrixproperty and derive their standard equations in a suitably established rectangularcoordinate system. We also show that the limit curves obtained from varying ellipsesin a similar way as in the Euclidean case are not parabola.Finally, we study a special class of ellipses, which we call the Thales ellipses, havingthe so-called Thales property, that is, the two lines connecting a point on the ellipse tothe two vertices on its major axis always make a right angle. We also find the relationthat the major and minor radii of a Thales ellipse satisfy.Our main results in this thesis are Theorems2.3–2.4,3.1–3.2,4.1–4.2, and5.1.