Geometry glossary

Clear, one-line definitions of the Geometry terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

angle

A measure of rotation between two rays sharing an endpoint, expressed in degrees or radians.

angular excess

For a spherical triangle, the amount by which its angle sum exceeds π; equal to its area on the unit sphere.

coordinates

An ordered tuple of numbers that identifies a point in a space relative to a chosen reference frame or chart.

See: Cartesian Coordinates and the Plane

cosine

For an angle θ in a right triangle, the ratio (adjacent side) / (hypotenuse); extended via the unit circle.

See: Sine and Cosine on the Unit Circle

degree

An angle measure such that 360° equals one full revolution; one degree equals π/180 radians.

distance

The length of the shortest path between two points; for points in the plane, computed by the Pythagorean formula.

See: Measuring Distance, The Distance Formula

ellipse

The set of points whose distances to two foci sum to a constant; a stretched circle.

See: The Ellipse, Anisotropic ellipsoids and the search ellipse

euclidean norm

For x = (x₁, ..., xₙ) ∈ ℝⁿ, the length ‖x‖ = √(x₁² + ... + xₙ²); induced by the standard dot product.

girard's theorem

On the unit sphere, the area of a spherical triangle equals its angular excess: A = (α + β + γ) − π.

great circles

Circles on a sphere whose center coincides with the sphere's center; the geodesics of spherical geometry.

hyperbola

The set of points whose distances to two foci have a constant difference; two open branches.

See: The Hyperbola

hyperbolic geometry

A non-Euclidean geometry of constant negative curvature where through a point off a line, infinitely many parallels exist.

See: Hyperbolic Geometry

mobius transformation

A map z ↦ (az + b)/(cz + d) with ad − bc ≠ 0; the orientation-preserving isometries of the Poincaré disk are a subset.

parabola

The set of points equidistant from a fixed point (focus) and a fixed line (directrix); the graph of a quadratic.

See: The Parabola

playfair's axiom

Through a point not on a given line there is exactly one line parallel to it; equivalent to Euclid's parallel postulate.

poincare disk

A model of the hyperbolic plane on the open unit disk in which 'lines' are diameters and arcs of circles orthogonal to the boundary.

radian

An angle measure such that 2π radians equals one full revolution (360°).

See: Radians and Arc Length

sine

For an angle θ in a right triangle, the ratio (opposite side) / (hypotenuse); extended via the unit circle.

tangent

For an angle θ, the ratio sin θ / cos θ; geometrically the slope of the terminal ray on the unit circle.

See: The Tangent Function

unit circle

The circle of radius 1 centered at the origin; the standard tool for defining sine and cosine of any angle.

See: Sine and Cosine on the Unit Circle

vector

A quantity with both magnitude and direction; in the plane, an ordered pair (a, b).

See: Vector Calculus Preliminaries, Differential Forms and Vector Fields