Curves Definition and 9 Discussions

In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve. In this article, these curves are sometimes called topological curves to distinguish them from more constrained curves such as differentiable curves. This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves, since they are generally defined by implicit equations.
Nevertheless, the class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn. This is the case of space-filling curves and fractal curves. For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve.
A plane algebraic curve is the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field k, the curve is said to be defined over k. In the common case of a real algebraic curve, where k is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a finite field are widely used in modern cryptography.

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  1. B

    I Questions about algebraic curves and homogeneous polynomial equations

    It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...
  2. cianfa72

    I Maps with the same image are actually different curves?

    Hi, I've a doubt about the definition of curve. A smooth curve in ##\mathbb R^2## is defined by the application ##\gamma : I \rightarrow \mathbb R^2##. Consider two maps ##\gamma## and ##\gamma'## that happen to have the same image (or trace) in ##\mathbb R^2##. At a given point on the...
  3. B

    Find the osculating plane and the curvature

    I know the osculating plane is normal to the binormal vector ##B(t)=(a,b,c)##. And since the point on which I am supposed to find the osculating plane is not given, I'm trying to find the osculating plane at an arbitrary point ##P(x_0,y_0,z_0)##. So, if ##R(x,y,z)## is a point on the plane, the...
  4. EEristavi

    Curve's Length

    Homework Statement I have to find length of the curve: y2 = (x-1)3 from (1,0) to (2,1) Homework Equations s = ∫ √(1 + (f '(x) )2 ) dx where we have integral from a to b The Attempt at a Solution I'm bit confused: I'm thinking of writing function regarding x, f(x)...
  5. F

    Curve and admissible change of variable

    Homework Statement If I have the two curves ##\phi (t) = ( \cos t , \sin t ) ## with ## t \in [0, 2\pi]## ##\psi(s) = ( \sin 2s , \cos 2s ) ## with ## s \in [\frac{\pi}{4} , \frac{5 \pi}{4} ] ## My textbook says that they are equivalent because ##\psi(s) = \phi \circ g^{-1}(s) ## where ##...
  6. T

    Find a piecewise smooth parametric curve to the astroid (a hypocycloid with four cusps)

    Homework Statement Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve? Homework Equations $\phi(\theta) =...
  7. 5

    Parametric curve tangent help

    Homework Statement Consider the parametric curve given by: x=6cos(2t), y=t5/2. Calculate the equation of the tangent to this curve at the point given by t=π/4, in the form y=mx+c. The tangent is given by y= Homework Equations The Attempt at a Solution [/B] the answer that I got was...
  8. C

    B Do curves, circles and spheres really exist?

    Obviously, they exist as mathematical concepts, and those concepts are real, but in physical reality, everything is made up of subatomic particles and, if the theory is ever verified, strings. So if you try to construct a curve, circle or sphere, you are necessarily stacking a bunch of subatomic...
  9. ydonna1990

    How to draw curves (cardioid, lemniscate, devil's curve)?

    Hi everyone, Thanks for visiting my post. I was wondering if you guys know what kind of software I must use to draw complicated curves. I already have the equations. For example, for lemniscate I would use following: 2(x^2+y^2)=25(x^2-y^2) I appreciate the help.
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