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In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons , the cells of the arrangement, line segments and rays , the edges of the arrangement, and points where two or more lines cross, the vertices of the arrangement.
Thus in differential geometry, a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries, a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a ...
The principal vanishing point is the vanishing point of all horizontal lines perpendicular to the picture plane. The vanishing points of all horizontal lines lie on the horizon line. If, as is often the case, the picture plane is vertical, all vertical lines are drawn vertically, and have no finite vanishing point on the picture plane.
Vertical and horizontal subspaces for the Möbius strip. The Möbius strip is a line bundle over the circle, and the circle can be pictured as the middle ring of the strip. . At each point on the strip, the projection map projects it towards the middle ring, and the fiber is perpendicular to the middle ri
Sheaf-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry.
Through any point P in the plane, there is one and only one vertical line within the plane and one and only one horizontal line within the plane. This symmetry breaks down as one moves to the three-dimensional case. A vertical line is any line parallel to the vertical direction. A horizontal line is any line normal to a vertical line.
Vertical – spanning the height of a body. Longitudinal – spanning the length of a body. Lateral – spanning the width of a body. The distinction between width and length may be unclear out of context. Adjacent – next to; Lineal – following along a given path. The shape of the path is not necessarily straight (compare to linear).
In topology, a curve is defined by a function from an interval of the real numbers to another space. [49] In differential geometry, the same definition is used, but the defining function is required to be differentiable. [53] Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one. [54]