"In geometry, a three-dimensional space (3-D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point."
Introduction to geometric concepts in three dimensions.
Coordinate systems: An introduction to three-dimensional coordinate systems and their representations.
Distance formula: The distance formula in three-dimensional space and its applications.
Planes: The equation of a plane and its characteristics such as normal vector and distance from origin.
Vector operations: Addition, subtraction, dot product, cross product and their applications in three-dimensional geometry.
Equations of lines: The equation of a line in three-dimensional space and its applications such as intersection and projection.
Intersection of planes: Finding the intersection of two or more planes in three-dimensional space.
Parametric equations: Conversion of linear equations into parametric form and applications.
Distance from a point to a line and plane: Finding the shortest distance from a point to a line or plane in three-dimensional space.
Conic sections: Analyzing conic sections such as parabolas, hyperbolas, and ellipses in three-dimensional space.
Quadric surfaces: Analyzing quadric surfaces such as spheres, cylinders, cones, and tori in three-dimensional space.
Curves and surfaces: Analyzing curves and surfaces in three-dimensional space such as level curves and level surfaces.
Tangent planes and normal lines: Defining tangent planes and normal lines to a surface and their applications.
Calculus in three dimensions: Understanding how calculus is applied in the context of three-dimensional geometry.
Transformations: Understanding how objects can be transformed in three-dimensional space through translation, rotation, and scaling.
Applications in engineering: Real-world applications of three-dimensional geometry in engineering such as designing and 3D printing objects.
Point: The most basic three-dimensional object, a point has no length, width, or height and is often represented as a dot.
Line: A one-dimensional object with infinite length and zero width. In three-dimensional space, a line is often represented as a series of points.
Plane: A two-dimensional object with infinite length and width but zero height. It extends infinitely in both directions and forms the basis of many geometric shapes.
Sphere: A three-dimensional object with a curved surface that is equidistant from a central point. A sphere can be thought of as a set of points that are all the same distance from a center point.
Cone: A three-dimensional object with a circular base that narrows to a single point at the top. Cones can be classified as either right (the apex is directly above the center of the base) or oblique (the apex is off-center).
Cylinder: A three-dimensional object with parallel circular bases connected by a curved surface. It can be thought of as a stack of circles that are connected by parallel lines.
Cube: A three-dimensional object with six square faces, each of which is perpendicular to the adjacent faces. All sides and angles of a cube are equal.
Prism: A three-dimensional object with two parallel, congruent bases that are connected by rectangular faces. The cross-section of a prism is always the same shape as its base.
Pyramid: A three-dimensional object with a polygonal base and triangular faces that meet at a single point at the top (the apex).
Torus: A doughnut-shaped object with a circular cross-section that is generated by rotating a circle around an axis that is coplanar with the circle.
"Most commonly, it is the three-dimensional Euclidean space, the Euclidean n-space of dimension n=3 that models physical space."
"More general three-dimensional spaces are called 3-manifolds."
"A tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space."
"The set of these n-tuples is commonly denoted R^n."
"When n = 3, this space is called the three-dimensional Euclidean space (or simply 'Euclidean space' when the context is clear)."
"It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists."
"It is only one example of a large variety of spaces in three dimensions called 3-manifolds."
"When the three values refer to measurements in different directions (coordinates), any three directions can be chosen."
"Provided that vectors in these directions do not all lie in the same 2-space (plane)."
"These three values can be labeled by any combination of three chosen from the terms width/breadth, height/depth, and length."
"While this space remains the most compelling and useful way to model the world as it is experienced..."
"...three values (coordinates) are required to determine the position of a point."
"In geometry, a three-dimensional space... is a mathematical space..."
"More general three-dimensional spaces are called 3-manifolds."
"...and can be identified to the pair formed by an n-dimensional Euclidean space and a Cartesian coordinate system."
"The Euclidean n-space of dimension n=3..."
"It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists."
"...any combination of three chosen from the terms width/breadth, height/depth, and length."
"The set of these n-tuples is commonly denoted R^n."