Plane Euclidean Geometry
Euclidean geometry is the form of geometry defined and studied by Euclid. It is generally distinguished from non-Euclidean geometries by the parallel postulate, which (in Euclid's formulation) states "that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles".
This section covers theorems that relate to Euclidean geometry in two dimensions.
| Name of Topic | Subtopics |
|---|---|
| Lines and Angles | Parallel Lines |
| Polygons | Triangles, Pythagorean Theorem, Quadrilaterals, Polygon Construction |
| Trigonometry | Basic Trigonometry, Sine and Cosine Rules, Trigonometrical Identities |
| Curves | Circles, Conic Sections |
| Coordinate Geometry | Plane Cartesian Coordinates, Plane Polar Coordinates, Coordinate Transformations |
Solid Euclidean Geometry
This section covers theorems that relate to Euclidean geometry in three dimensions. Many proofs in three-dimensional geometry rely on results in plane geometry.
| Name of Topic | Subtopics |
|---|---|
| Lines, Planes and Angles | |
| Polyhedra | Platonic Solids, Archimedean Solids |
| Curved Solids and Surfaces | Spheres, Cylinders, Cones, Quadrics |
Projective Geometry
Elliptic Geometry
Elliptic geometry is a non-Euclidean geometry in which there are no parallel straight lines – any coplanar straight lines will intersect if sufficiently extended. The surface of a sphere, considered as a geometric space in its own right, exhibits this kind of geometry.
Hyperbolic Geometry
Hyperbolic geometry is a non-Euclidean geometry in which every straight line has a continuum of parallel straight lines (in the 'never meeting' sense) through the same point.