One main characteristic of non-Euclidean geometry is hyperbolic geometry. The other is elliptic geometry. Non-Euclidean geometry is still closely related to Euclidean geometry.
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The question refers to the "following". In such circumstances would it be too much to expect that you make sure that there is something that is following?
Euclidean geometry, non euclidean geometry. Plane geometry. Three dimensional geometry to name but a few
There are different kinds of geometry including elementary geometry, Euclidean geometry, and Elliptic Geometry.
Archimedes - Euclidean geometry Pierre Ossian Bonnet - differential geometry Brahmagupta - Euclidean geometry, cyclic quadrilaterals Raoul Bricard - descriptive geometry Henri Brocard - Brocard points.. Giovanni Ceva - Euclidean geometry Shiing-Shen Chern - differential geometry René Descartes - invented the methodology analytic geometry Joseph Diaz Gergonne - projective geometry; Gergonne point Girard Desargues - projective geometry; Desargues' theorem Eratosthenes - Euclidean geometry Euclid - Elements, Euclidean geometry Leonhard Euler - Euler's Law Katyayana - Euclidean geometry Nikolai Ivanovich Lobachevsky - non-Euclidean geometry Omar Khayyam - algebraic geometry, conic sections Blaise Pascal - projective geometry Pappus of Alexandria - Euclidean geometry, projective geometry Pythagoras - Euclidean geometry Bernhard Riemann - non-Euclidean geometry Giovanni Gerolamo Saccheri - non-Euclidean geometry Oswald Veblen - projective geometry, differential geometry