What is a Curve ?

hi

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that its curvature need not be zero

Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows.

A curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields.

A closed curve is a curve that forms a path whose starting point is also its ending point—that is, a path from any of its points to the same point.

Closely related meanings include the graph of a function (for example, Phillips curve) and a two-dimensional graph.

Interest in curves began long before they were the subject of mathematical study. This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times

Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach

Historically, the term "line" was used in place of the more modern term "curve". Hence the phrases "straight line" and "right line" were used to distinguish what are today called lines from "curved lines". For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length" (Def. 2), while a straight line is defined as "a line that lies evenly with the points on itself" (Def. 4).

The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction.

Curve, In mathematics, an abstract term used to describe the path of a continuously moving point (see continuity). Such a path is usually generated by an equation. The word can also apply to a straight line or to a series of line segments linked end to end.

The conic sections, deeply studied by Apollonius of Perga

A closed curve is a path that repeats itself, and thus encloses one or more regions. Simple examples include circles, ellipses, and polygons. Open curves such as parabolas, hyperbolas, and spirals have infinite length.

The cissoid of Diocles, studied by Diocles and used as a method to double the cube

The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle

The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle

Grading on a curve is a term that describes a variety of different methods that a teacher uses to adjust the scores her students received on a test in some way. Most of the time, grading on a curve boosts the students' grades by moving their actual scores up a few notches, perhaps increasing the letter grade.

The spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius.

Some teachers use curves to adjust the scores received in exams, whereas other teachers prefer to adjust what letter grades are assigned to the actual scores.

A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century. This enabled a curve to be described using an equation rather than an elaborate geometrical construction.

Teachers use the bell curve to analyze their tests, assuming that a bell curve will be visible if the test is a good one of the material she presented.

This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves that can be defined using algebraic equations, algebraic curves, and those that cannot, transcendental curves. Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated

If, for example, a teacher looks at her class scores and sees that the mean (average) grade of her midterm was approximately a C, and slightly fewer students earned Bs and Ds and even fewer students earned As and Fs, then she could conclude that the test was a good design.

Conic sections were applied in astronomy by Kepler. Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid).

If, on the other hand, she plots the test scores and sees that the average grade was a 60 percent, and no one scored above an 80 percent, then she could conclude that the test may have been too difficult.