Plane geometry can be established in a general way using four of the Euclid´s postulates. In this way, we get the affine geometry, which lacks metric and orthogonality but underlies the other plane geometries.
Including the fifth postulate or any equivalent statement, we arrive to the Euclidean geometry. The circle is ists main feature.
Alternative postulates allowing for a metric and orthogonality relations can be introduced to get other types of plane geometries.
A very special case is the geometry of spacetime or Minkowski geometry. Hyperbolic orthogonality defines the metric, and equilateral hyperbolas substitute circles as the main geometric feature. A new property arises which is completely absent in its euclidean counterpart: The existence of null directions. They expand the centre of a circle to two diagonal axeswhich correspond physically to the light cones.
To compare some euclidean properties with their corresponding spacetime counterparts you can visit these sites:
Euclidean properties and Modern Physics Geometry
Conformal mappings of these geometries can be also found in this site (in Galician language): Proxeccion Conforme Espazotempo
Both geometries are strictly related, and this can have pedagogical advantages, as shown by the the Transformation geometry,