Geometry is a branch of mathematics concerned with the properties, measurement, and relationships of points, lines, angles, surfaces, and solids. Originating in ancient civilizations for land measurement and construction, geometry has evolved into a formal deductive system, with Euclidean geometry as its classical foundation. Modern geometry extends into non-Euclidean and abstract spaces, playing a fundamental role in fields such as physics, engineering, computer graphics, and topology.
1.1 Basic Primitives
1.1.1 Points and Lines
A point is a fundamental object in geometry, representing a specific location in space with no dimension (length, width, or height). It is typically denoted by a dot and labeled with a capital letter. A line is a straight, one-dimensional set of points extending infinitely in both directions. It has no thickness and is the shortest path between any two points. In Euclidean geometry, a line is defined by two distinct points.
1.1.2 Planes and Space
A plane is a two-dimensional flat surface that extends infinitely in all directions. It contains an infinite number of points and lines. Space, in geometry, refers to three-dimensional (or higher-dimensional) environments where points, lines, and planes interact. In Euclidean space, three dimensions are defined by length, width, and height.
1.2 Angles and Measurement
1.2.1 Types of Angles
An angle is formed by two rays sharing a common endpoint called the vertex. Angles are classified by their measure in degrees or radians: acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), straight (180°), reflex (between 180° and 360°), and full rotation (360°).
1.2.2 Angle Relationships
Angles can be related through their positions and measures. Complementary angles sum to 90°, supplementary angles sum to 180°, adjacent angles share a vertex and a side, and vertical angles are formed by intersecting lines and are equal. Corresponding, alternate interior, and alternate exterior angles arise when a transversal crosses parallel lines.
1.3 Axioms and Postulates
Axioms (or postulates) are fundamental statements assumed true without proof, serving as the foundation for geometric reasoning. Euclid’s five postulates, such as “a straight line can be drawn from any point to any other point,” form the basis of Euclidean geometry. Modern geometry also relies on systems of axioms (e.g., Hilbert’s axioms) to ensure logical consistency.
2.1 Plane Geometry
2.1.1 Triangles
A triangle is a polygon with three sides and three interior angles. The sum of its interior angles is 180°. Triangles are classified by side lengths (scalene, isosceles, equilateral) and by angles (acute, right, obtuse). Key properties include the Pythagorean theorem for right triangles and the triangle inequality.
2.1.1.1 Congruence and Similarity
Two triangles are congruent if their corresponding sides and angles are equal. Congruence criteria include Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS). Triangles are similar if their corresponding angles are equal and sides are proportional. Similarity is established by Angle-Angle (AA), Side-Side-Side (SSS), or Side-Angle-Side (SAS) similarity.
2.1.2 Quadrilaterals
A quadrilateral is a four-sided polygon. Common types include squares (equal sides and right angles), rectangles (opposite sides equal, right angles), rhombuses (equal sides, opposite angles equal), parallelograms (opposite sides parallel and equal), trapezoids (one pair of parallel sides), and kites (two distinct pairs of adjacent equal sides). The sum of interior angles is 360°.
2.1.3 Circles
A circle is the set of all points in a plane at a fixed distance (radius) from a center point. Important elements include the center, radius, diameter (twice the radius), circumference, and area (πr²). Circles are studied through central angles, inscribed angles, and arc measures.
2.1.3.1 Arcs, Chords, and Tangents
An arc is a portion of the circumference, measured in degrees or length. A chord is a line segment whose endpoints lie on the circle; the longest chord is the diameter. A tangent is a line that touches the circle at exactly one point, perpendicular to the radius at that point. The tangent-secant theorem relates lengths of tangents and secants.
2.2 Solid Geometry
2.2.1 Polyhedra
A polyhedron is a solid bounded by flat polygonal faces. Faces meet at edges, and edges meet at vertices. Euler’s formula (V – E + F = 2) relates the numbers of vertices (V), edges (E), and faces (F) for convex polyhedra. The five Platonic solids—tetrahedron, cube, octahedron, dodecahedron, icosahedron—are regular polyhedra.
2.2.1.1 Prisms and Pyramids
A prism has two congruent, parallel polygonal bases connected by rectangular lateral faces. A pyramid has a polygonal base and triangular faces meeting at a common apex. Volume formulas: for a prism, V = area of base × height; for a pyramid, V = (1/3) × area of base × height.
2.2.2 Cylinders, Cones, and Spheres
A cylinder has two parallel circular bases and a curved lateral surface. A cone has a circular base and a curved lateral surface tapering to an apex. A sphere is the set of points at a constant distance from a center in three-dimensional space. Volume formulas: cylinder V = πr²h; cone V = (1/3)πr²h; sphere V = (4/3)πr³. Surface areas are similarly defined.
2.3 Geometric Constructions
Classical geometric constructions use only an unmarked straightedge and compass. Common constructions include bisecting a segment or angle, constructing perpendicular lines, drawing a circle through three points, and creating regular polygons. These constructions demonstrate the logical foundations of Euclidean geometry.
3.1 Coordinate Systems
3.1.1 Cartesian Coordinates
The Cartesian coordinate system uses perpendicular axes (x and y in two dimensions, plus z in three) to specify points by ordered pairs (x, y) or triples (x, y, z). Distances between points are given by the Euclidean distance formula: √[(x₂–x₁)² + (y₂–y₁)²].
3.1.2 Polar Coordinates
In the polar coordinate system, a point is represented by a distance r from the origin (radius) and an angle θ from a reference direction. Conversion to Cartesian: x = r cos θ, y = r sin θ. Polar coordinates simplify equations of curves like spirals and circles centered at the origin.
3.2 Equations of Lines and Curves
3.2.1 Linear Equations and Slopes
A line in the plane can be expressed as y = mx + b, where m is the slope (rise over run) and b is the y-intercept. The slope indicates the line’s steepness and direction. Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.
3.2.2 Conic Sections
Conic sections are curves obtained by intersecting a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas, each with a standard quadratic equation. Their properties are used in astronomy (orbits) and engineering (reflective surfaces).
3.2.2.1 Parabolas, Ellipses, and Hyperbolas
A parabola is defined by a focus and directrix, with equation y = ax² + bx + c. An ellipse is the set of points with constant sum of distances to two foci; its standard equation is (x²/a²) + (y²/b²) = 1. A hyperbola is the set of points with constant difference of distances to two foci; its equation is (x²/a²) – (y²/b²) = 1.
3.3 Transformations and Vectors
Geometric transformations include translation (sliding), rotation (turning), reflection (mirroring), and dilation (scaling). Vectors are mathematical objects with magnitude and direction, used to represent translations and forces. Vector operations—addition, subtraction, dot product, cross product—are essential in analytic geometry and physics.
4.1 Spherical Geometry
4.1.1 Great Circles and Spherical Triangles
On a sphere, the analogue of a straight line is a great circle—the intersection of the sphere with a plane through its center. Spherical triangles are formed by arcs of great circles; the sum of their interior angles exceeds 180°. Spherical geometry models Earth’s surface for navigation and astronomy.
4.2 Hyperbolic Geometry
4.2.1 Models of the Hyperbolic Plane
Hyperbolic geometry is a non-Euclidean geometry where the parallel postulate is replaced: through a point not on a line, infinitely many parallel lines exist. Common models include the Poincaré disk (points inside a disk, geodesics as circular arcs) and the upper half-plane model. Hyperbolic geometry has applications in topology and theoretical physics.
5.1 Differential Geometry
5.1.1 Curves and Surfaces
Differential geometry studies curves and surfaces using calculus. Curves are described by curvature and torsion; surfaces by the first and second fundamental forms. Gaussian curvature measures how a surface bends intrinsically (positive for spheres, zero for planes, negative for saddles).
5.1.2 Riemannian Manifolds
A Riemannian manifold is a smooth space that locally resembles Euclidean space, equipped with a metric tensor to define distances and angles. General relativity describes spacetime as a four-dimensional Lorentzian manifold. Riemannian geometry is foundational for modern theoretical physics and geometric topology.
5.2 Topology
5.2.1 Basic Concepts
Topology studies properties of spaces preserved under continuous deformations (stretching, bending, twisting, but not tearing or gluing). Key concepts include open sets, continuity, compactness, connectedness, and boundary. Topological equivalence is called homeomorphism.
5.2.2 Manifolds and Homeomorphism
A manifold is a topological space that locally resembles Euclidean space (e.g., a circle is a 1‑manifold, a sphere is a 2‑manifold). Homeomorphism is a continuous bijection with a continuous inverse; two homeomorphic spaces are considered topologically the same. Classification of manifolds (e.g., the classification of surfaces) is a central problem in topology.