Classical geometry

Classical geometry refers to the study of geometric shapes, sizes, properties, and positions based on the principles established in ancient times, particularly by Greek mathematicians such as Euclid, Archimedes, and Pythagoras. This field encompasses various fundamental concepts, including points, lines, angles, surfaces, and solids.

Affine geometry is a branch of geometry that studies the properties of figures that remain invariant under affine transformations. These transformations include operations such as translation, scaling, rotation, and shearing, which can alter the size and orientation of shapes but do not change their basic structure or ratios of distances. Here are some key concepts in affine geometry: 1. **Affine Transformations**: An affine transformation is a function between affine spaces that preserves points, straight lines, and planes.

Hyperbolic geometry is a non-Euclidean geometry that arises from altering Euclid's fifth postulate, the parallel postulate. In hyperbolic geometry, the essential distinction is that, given a line and a point not on that line, there are infinitely many lines through that point that do not intersect the original line. This contrasts with Euclidean geometry, where there is exactly one parallel line that can be drawn through a point not on a line.

Interactive Geometry Software (IGS) refers to computer programs that allow users to create, manipulate, and analyze geometric shapes and constructions in a dynamic and visual manner. This type of software enables users to explore mathematical concepts related to geometry through direct interaction, often using a graphical interface. Key features of interactive geometry software typically include: 1. **Dynamic Construction**: Users can create geometric figures (like points, lines, circles, polygons, etc.) and manipulate them in real time.

Non-Euclidean geometry refers to any form of geometry that is based on axioms or postulates that differ from those of Euclidean geometry, which is the geometry of flat surfaces as described by the ancient Greek mathematician Euclid. The most notable feature of Non-Euclidean geometry is its treatment of parallel lines and the nature of space.

Absolute geometry is a type of geometry that studies the properties and relations of points, lines, and planes without assuming the parallel postulate of Euclidean geometry. Instead, it can be considered a framework that encompasses both Euclidean and non-Euclidean geometries by focusing on the common properties shared by them.

Elliptic geometry is a type of non-Euclidean geometry characterized by its unique properties and the nature of its parallel lines. In contrast to Euclidean geometry, where the parallel postulate states that through a point not on a given line, there is exactly one line parallel to the given line, in elliptic geometry, there are no parallel lines at all. Every pair of lines eventually intersects.

Spherical geometry is a branch of mathematics that deals with geometric shapes and figures on the surface of a sphere, as opposed to the flat surfaces typically studied in Euclidean geometry. It is a non-Euclidean geometry, meaning that it does not abide by some of the postulates of Euclidean geometry, particularly the parallel postulate.