Curves Definition and 778 Threads

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. cianfa72

    I Notion of congruent curve along a vector field

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  2. S

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  3. K

    I Anomalous contribution to galactic rotation curves due to stochastic s

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  4. S

    B Calculating shaded area: I'm getting discrepancies between methods

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  5. P

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  6. T

    Why are the paths of our cosmic explorations, pretty?

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  7. berlinvic

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  8. F

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  9. S

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  10. chwala

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  11. BloonAinte

    I Characteristic curves for ##u_t + (1-2u)u_x = -1/4, u(x,0) = f(x)##

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  12. Reuben_Leib

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  13. I

    A Finding Minimal Mean Distance Curves on the Unit Sphere

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  14. SaintRodriguez

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  15. V9999

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  16. B

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  17. K

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  18. bakerjay

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  19. LCSphysicist

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  20. cianfa72

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  21. mncyapntsi

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  22. elcaro

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  23. chwala

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  24. S

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  25. Eclair_de_XII

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  26. T

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  27. B

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  28. A

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  29. A

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  30. yucheng

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  31. S

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  32. B

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  33. B

    A Levi-Civita Connection & Length of Curves in GR

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  34. LCSphysicist

    I About groups and continuous curves

    Define $$\phi(A)$$ a transformation which, acting on a vector x, returns $$AxA^{*}$$, in such way that if A belongs to the group $$SL(2,C)$$, $$||\phi(A)x||^2 = ||x||^2$$, so it conserves the metric and so is a Lorentz transformation. $$\phi(AB)x = (AB)x(AB)^{*} = ABxB^{*}A^{*} = A(BxB^{*})A^{*}...
  35. M

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  36. K

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  37. S

    B Solving for integral curves- how to account for changing charts?

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  38. karush

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  39. Astrowolf_13

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    I attempted to solve this problem by finding the angles of an intersection point by equalling both ##r=sin(\theta)## and ##r=\sqrt 3*cos(\theta)##. The angle of the first intersection point is pi/3. The second intersection point is, obviously, at the pole point (if theta=0 for the sine curve and...
  40. bigfooted

    Geometry Is Walker's Textbook the Best Resource for Algebraic Curves?

    I recently became interested in algebraic curves, specifically topics like parametrization and its links to differential equations. I read a number of papers but I'm looking for a good (introduction) textbook on (planar) algebraic curves that gives a solid background, not pure theoretical but...
  41. A

    Differential Equations and Damper Curves

    Good evening, I have been wrestling with the following and thought I would ask for help. I am trying to come up with the equations of motion and energy stored in individual suspension components when a wheel is fired towards the car but, there is a twist! I am assuming a quarter car type...
  42. J

    Exploring Anodic & Cathodic Tafel Curves: Uneven Electrodeposition?

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  43. S

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  44. N

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  45. K

    I Question about Galactic Rotation curves in the Milky Way galaxy

    The graph in Wikipedia, article Milky Way, section Galactic Rotation, shows the actual rotation speeds in blue and the calculated speeds due to observed mass in red. (The graph is to the right of the article.) At about 3 kpc the actual speed is about 205 km/s. To account for the decrease in...
  46. R

    Engineering How to calculate the total central angle of railroad curves

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  47. S

    I Applying the spacetime interval to regular vectors instead of curves

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  48. Z

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  49. S

    I Calculus- Area between two curves (minimize it)

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  50. igorrn

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