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This familiar equation for a plane is called the general form of the equation of the plane or just the plane equation. [6] Thus for example a regression equation of the form y = d + ax + cz (with b = −1) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a ...
The 10 orientable flat 3-manifolds are: [3] Euclidean 3-space, . The 3-torus, made by gluing opposite faces of a cube. The manifold made by gluing opposite faces of a cube with a 1/2 twist on one pair. The manifold made by gluing opposite faces of a cube with a 1/4 twist on one pair.
For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In the three-dimensional case a surface normal, or simply normal, to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P.
For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Euclidean space is a Riemannian manifold with a Riemannian metric coming from the way it sits inside the ambient space.
Plane equation in normal form. In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.
Parallel Coordinates plots are a common method of visualizing high-dimensional datasets to analyze multivariate data having multiple variables, or attributes. To plot, or visualize, a set of points in n-dimensional space, n parallel lines are drawn over the background representing coordinate axes