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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 ). DEFINITIONS In . The curve $\backslash !\backslash ,\backslash gamma$ is said to be simple if it is Injective , i.e. if for all $x$, $y$ in $I$, we have $\backslash ,\backslash !\backslash gamma(x)\; =\; \backslash gamma(y)\; ightarrow\; x\; =\; y$. If $I$ is a closed bounded interval $\backslash ,\backslash !b$, we also allow the possibility $\backslash ,\backslash !\backslash gamma(a)\; =\; \backslash gamma(b)$ (this convention makes it possible to talk about closed simple curve). If $\backslash gamma(x)=\backslash gamma(y)$ for some $x$ e y (other than the extremities of $I$), then $\backslash gamma(x)$ is called a double (or: '''multiple''') '''point''' of the curve. A curve $\backslash !\backslash ,\backslash gamma$ is said to be closed or '''a loop''' if $\backslash ,\backslash !I\; =b$ and if $\backslash !\backslash ,\backslash gamma(a)\; =\; \backslash gamma(b)$. A closed curve is thus a continuous mapping of the circle $S^1$; a '''simple closed curve''' is also called a '''Jordan curve'''. A Plane Curve is a curve for which ''X'' is the Euclidean Plane — these are the examples first encountered — or in some cases the Projective Plane . A '''space curve''' is a curve for which ''X'' is of three dimensions, usually Euclidean Space ; a '''skew curve''' is a space curve which lies in no plane. These definitions also apply to Algebraic Curve s (see below). However, in the case of algebraic curves it is very common not to restrict the curve to having points only defined over the real numbers. This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, although it also includes figures that can be hardly called curves in common usage. For example, the image of a curve can cover a Square in the plane ( Peano Curve ). The image of simple plane curve can have Hausdorff Dimension bigger than one (see Koch Snowflake ) and even Positive Lebesgue Measure (the last example can be obtained by small variation of the Peano curve construction). The Dragon Curve is yet another weird example. CONVENTIONS AND TERMINOLOGY 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 common in Vector Calculus and Differential Geometry . LENGTHS OF CURVES If $X$ is a Metric Space with metric $d$, then we can define the ''length'' of a curve $\backslash !\backslash ,\backslash gamma\; :bightarrow\; X$ by : A rectifiable curve is a curve with length. A Parametrization of $\backslash !\backslash ,\backslash gamma$ is called natural (or '''unit speed''' or '''parametrised by arc length''') if for any $t\_1$, $t\_2$ in $b$, we have

  :<math>\mbox{Length}(\gamma) \int_a^b \left \, {d\gamma \over dt} \, ight \, dt </math>