Euclidean Geometry

• Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• It is one of the oldest mathematical sciences.
• Geometry arises independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes.

Euclidean Geometry

• Euclidean geometry studies geometrical shapes (plane and solid) based on axioms and theorems.
• It is named after Euclid, a Greek mathematician who compiled and organized geometrical knowledge in "The Elements".
• Key concepts include points, lines, planes, angles, and distances.
• Fundamental shapes include triangles, quadrilaterals, circles, prisms, pyramids, spheres, and cylinders.
• It is characterized by axioms (self-evident truths) and postulates (assumptions). Other geometric propositions (theorems) are derived from these through logical deduction.
• The parallel postulate is a defining feature: given a line and a point not on the line, there is exactly one line through the point parallel to the given line.
• Euclidean geometry provides a framework for understanding spatial relationships used extensively in architecture, engineering, and navigation.
• Congruence (same size and shape) and similarity (same shape, different sizes) are central to Euclidean geometry.
• The Pythagorean theorem (a² + b² = c²) relates the sides of a right triangle and is a fundamental result.
• Coordinate geometry introduces a coordinate system, allowing geometric figures to be represented and analyzed using algebraic equations.
• Transformations like translations, rotations, reflections, and dilations are studied .

Non-Euclidean Geometries

• Non-Euclidean geometries deviate from Euclidean geometry by denying the parallel postulate.

Hyperbolic Geometry

• Hyperbolic geometry is a type of non-Euclidean geometry.
• Given a line and a point not on the line, there are infinitely many lines through the point parallel to the given line, in hyperbolic geometry.
• It is modeled by surfaces with constant negative curvature, like the pseudosphere.

Elliptic Geometry

• Elliptic geometry is a type of non-Euclidean geometry.
• There are no parallel lines in elliptic geometry; any two lines intersect.
• It can be visualized as the geometry on the surface of a sphere.
• Non-Euclidean geometries have applications in cosmology and general relativity.

Coordinate Geometry

• Coordinate geometry uses a coordinate system to represent and analyze geometric shapes using algebraic equations.
• It is also known as analytic geometry.
• The Cartesian coordinate system, with x and y axes, is commonly used to represent points in a plane as ordered pairs (x, y).
• Equations of lines, curves, and other geometric figures can be expressed using algebraic equations
• The distance formula allows calculating the distance between two points in the coordinate plane.
• The slope of a line measures its steepness and direction.
• Coordinate geometry provides a tool for solving geometric problems using algebraic methods.

Solid Geometry

• Solid geometry extends the concepts of plane geometry (two-dimensional geometry) to three-dimensional space.
• It deals with the properties and measurements of three-dimensional shapes, called solids.
• Key solid shapes include cubes, rectangular prisms, spheres, cylinders, cones, pyramids, and polyhedra.
• Surface area is the total area of the surfaces of a solid.
• Volume is the amount of space a solid occupies.
• Solid geometry is used in architecture, engineering, and computer graphics.

Transformations

• Geometric transformations are operations that change the position, size, or shape of a geometric figure.
• Common transformations include translations, rotations, reflections, and dilations.
• A translation shifts a figure without changing its size or shape.
• A rotation turns a figure around a fixed point.
• A reflection flips a figure across a line, creating a mirror image.
• A dilation changes the size of a figure by a scale factor.
• Transformations are used in computer graphics, animation, and geometric modeling.

Trigonometry

• Trigonometry studies the relationships between the angles and sides of triangles.
• Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a right triangle to the ratios of its sides.
• Trigonometry is used in surveying, navigation, engineering, and physics.
• The unit circle provides a geometric representation of trigonometric functions.

Differential Geometry

• Differential geometry uses calculus to study the geometry of curves and surfaces.
• It deals with concepts such as curvature, torsion, and geodesics.
• Differential geometry has applications in physics, engineering, and computer graphics.

Topology

• Topology is a branch of geometry that studies the properties of shapes that are preserved under continuous deformations (stretching, bending, twisting) without tearing or gluing,.
• It is often described as "rubber sheet geometry".
• Topologists study properties such as connectedness, compactness, and continuity.
• Examples of topological spaces include surfaces, knots, and manifolds.

Fractals

• Fractals are geometric shapes that exhibit self-similarity at different scales.
• A fractal looks similar to itself, no matter how closely you zoom in.
• They are generated by repeating a simple process over and over in an ongoing feedback loop.
• Fractals are used to model natural phenomena such as coastlines, mountains, and trees.

Key Concepts

• Point: A location in space, having no dimension.
• Line: A straight, one-dimensional figure extending infinitely in both directions.
• Plane: A flat, two-dimensional surface extending infinitely in all directions.
• Angle: The measure of the rotation between two lines or planes that meet at a point.
• Distance: The length of the shortest path between two points.
• Shape: The external form or outline of an object.
• Area: The amount of surface covered by a two-dimensional shape.
• Volume: The amount of space occupied by a three-dimensional object.
• Congruence: The property of two figures having the same size and shape.
• Similarity: The property of two figures having the same shape, but possibly different sizes.
• Symmetry: The property of a figure remaining unchanged under certain transformations, such as reflection or rotation.

Euclidean geometry explores shapes using axioms and theorems, named after Euclid's "The Elements". It covers points, lines, planes, angles, and distances, with shapes like triangles and circles. Theorems are derived through logical deduction from basic axioms.