In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve). Curve - Definitions. In ma ...

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• Curve - Definitions
• Curve - Conventions and terminology
• Curve - Lengths of curves
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Main article: differential geometry of curves While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general re ...

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Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History

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In mathematics, a (topological) curve is defined as follows. Let I be an interval of real numbers (i.e. a non-empty connected subset of ). Then a curve is a continuous mapping , where X is a topological space. The curve is said to be simple if it is injective, i.e. if for all x, y in I, we have . If I is a closed ...

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Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History

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The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading. Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more ...

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Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History

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A curve may be a locus, or a path. That is, it may be a graphical representation of some property of points; or it may be traced out, for example by a stick in the sand on a beach. Of course if one says curved in ordinary language, it means bent (not straight), so refers to a locus. This leads to the general idea of curvature. As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachroni ...

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Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History

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Main article: Algebraic curve In the setting of algebraic geometry, a curve is usually defined to be an algebraic curve. These include, for example, elliptic curves, which are studied in number theory and which have important applications to cryptography. Algebraic curves are more akin to surfaces than curves. Non-singular complex projective algebraic curves are in fact compact Riemann surfaces. ...

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Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History

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If X is a metric space with metric d, then we can define the length of a curve by A rectifiable curve is a curve with finite length. A parametrization of is called natural (or unit speed or parametrised by arc length) if for any t1, t2 in [a, ...

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Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History

Read more here: » Curve: Encyclopedia II - Curve - Lengths of curves