In mathematics, a manifold is a mathematical object that on a small enough scale resembles Euclidean space. For example, seen from far away, the surface of the planet Earth is not flat and Euclidean, but on a smaller scale, one may describe each region via a geographic map, a projection of the surface onto the Euclidean plane. A precise mathematical definition of a manifold is given below. Lines and circles (but not figure eights) are one-dimensional manifolds (1-manifolds).
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Solid
Solid is one of the three classical states of matter. It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a gas does. The atoms in a solid are tightly bound to each other, either in a regular geometric lattice or irregularly.
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Frame fields in general relativity
In general relativity, a frame field (also called a tetrad or vierbein) is a set of four orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by and the three spacelike unit vector fields by . All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field.
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Frenet¿Serret formulas
"Binormal" redirects here. For the category-theoretic meaning of this word, see Normal morphism. In vector calculus, the Frenet¿Serret formulas describe the kinematic properties of a particle which moves along a continuous, differentiable curve in three-dimensional Euclidean space R, or the geometric properties of the curve itself irrespective of any motion.
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Parallel curve
A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of parallel lines. It can also be defined as a curve whose points are at a fixed normal distance of a given curve. It is sometimes called the offset curve but the term "offset" often refers also to translation. The term "offset curve" is used, e.g.
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Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space (M,g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point. The terms are named after German mathematician Bernhard Riemann.
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Tangent
In geometry, the tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point¿that is, coincides with the curve at that point without crossing to the other side of the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f) on the curve and has slope f'(c) where f' is the derivative of f.
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Geodesic
The geodesics are great circle arcs. ]] In mathematics, particularly differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space. In the presence of an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it.
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