# What is Curve: Definition + 1000 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. ### Curve for a line integral - direction confusion

When I take ##x = 2\cos(t)## and ##y = 2\sin(t)##, the integral becomes ##\int_{t=\frac{\pi}{2}}^0 4(2\cos(t))^2 \cdot 2 dt = -8\pi##. The final answer is ##8\pi##. Why is my method wrong? I played around with desmos and the parameterisation seems correct...
2. ### A Parallel Transport and Geodesics

So i am confused as to what can be parallel transported , can an arbitrary tensor be transported along any curve that we wish , or do we define a curve and then solve the equation of parallel transport (which is a linear first order differential equation ) and then the solutions we get from...
3. ### Calculate the area of this pond with functions given for the perimeter

So the solution is obviously given here, I'm just trying to understand it. I thought that integrating f(x) from -5 to 5 would give the area under the curve (including the areas below the "pond" at the edges of the image but above y=0. I don't really understand why we are subtracting the integral...
4. ### I Seifert -Solutions of Einstein's equations give flat galactic rotation curves

The Importance of Being Symmetric: Flat Rotation Curves from Exact Axisymmetric Static Vacuum Spacetimes ... Analyzing the low-velocity limitcorresponding to the Newtonian approximation of the Schwarzschild metric, we find an effective logarithmic potential. Thisyields flat rotation curves for...
5. ### At what ##x## value is the tangent equally inclined to the given curve?

I had to look this up; will need to read on it. from my research, https://byjus.com/question-answer/the-equation-of-straight-line-equally-inclined-to-the-axes-and-equidistant-from-the-points-1-2-and-3-4-is-ax-by-c-0-where/ ... I have noted that at equally inclined; the slope value is ##1##...
6. ### B Question about units for "area under curve"

Hello everyone, I am curious, suppose you have a function ##f(x)=x^3## and you to find the area under the curve from 0 to x, the area would be ##\frac {x^4}{4}## but this is units of ##L^4## if x is length, but area is units of ##L^2## so what is going on here? The reason I'm curious is I...
7. ### Measure the transmission curve and wavelengths of colored mineral glass

hi , I want to measure the transmission curve and wavelengths of colored mineral glass , which is the simplest way or which instrument can I use ? thanks
8. ### I Degrees of Freedom in Lagrangian Mechanics for a Fractal Path

Degree of freedom along a parabola, or any such tame curve, is one from lagrangian mechanics point of view. It makes sense. However how does degree of freedom accompany a space filling curve. Intuitively degree of freedom is not two, since not all motions are possible along the curve. How would...
9. ### Calculus Help locating a Math Equation to fit this curve

The concentration is the Y axis, and the values are the x axis. What is the equation for finding a target value on a curve between set of points?
10. ### Engineering Produce a resonance curve for this circuit between 6 kHz and 10 kHz

this is what i created so far, but not quite sure how to get the resonance curve, i tried to enter it in settings and do an AC sweep
11. ### Possible to change a thermistor resistance curve?

Currently when set to 1C, the fridge cools to and maintains 4-5C(As a conversion when set to 34f it cools and maintains approx. 40f. ), all functions like fan/defrost are working correctly. Potentially the main board has an issue but all other zones are working correctly so I would prefer to not...
12. ### Analyzing Hysteresis Curve Fitting: Overfitting or Not?

We recently did an experiment to generate the hysteresis curve of a certain material. The experiment involved switching the current in the wire looped around a ring of the material, and recording the first throw of the ballistic galvanometer. I am not going into the details of calculations...
13. ### B Why use this premise behind the Maxwell-Boltzmann curve?

I'm trying to understand the Maxwell-Boltzmann gas molecule speed distribution. Suppose we have a container of gas such that all the molecules are identical. At first I was under the mistaken impression that one starts with the premise that the distribution of their translational kinetic...
14. ### The relation between the normal and the slope of a cylindrical curve

As you can see in this picture: This explanation "relation between the normal and the slope of a curve" is formulated here: $$\frac{1}{\rho} \frac{d\rho }{d\psi }=\tan\left(\frac{\theta+\psi}{2}\right)$$ I got confused because I don't have the curve equation(regarding the slope of the curve...
15. ### I Worldline Curve vs Trajectory: What’s the Difference?

Is a worldline a curve or a trajectory? Why?
16. ### Silvaco Atlas Syntax question - area from curve

I have a several questions on the following block of codes taken from ganfetex01_aux.in: solve save outf="ganfetex01_$'index'.str" extract init inf="ganfetex01_$'index'.str" extract name="2DEG" 1e-4 * area from curve (depth, impurity="Electron Conc" material="All" mat.occno=1 x.val=0.5) \...
17. ### Delta-Scuti phase folded light curve equation

data I'm trying to fit
18. ### I Why Is the Integral Result 175/3 Instead of 45?

i integrated y to get (1/3x^3 + 2x) with upper limit 5 / lower limit 2 but got 45 not 175 / 3
19. ### I Measuring the rotation curve of galaxies

What should I learn to make astrophysical measurements from open data? Suppose I want to measure the rotation speed of galaxies to generate galactic rotation curves like these: https://en.wikipedia.org/wiki/Galaxy_rotation_curve What should I do and what should I learn? I think I should get...
20. ### Find the Cartesian equation of the curve

Find ms solution; My approach; ##xt=t^2+2## and ##yt=t^2-2## ##xt-2=t^2## and ##yt+2=t^2## ##⇒xt-2=yt+2## ##xt-yt=4## ##t(x-y)=4## ##t=\dfrac{4}{x-y}## We know that; ##x+y=2t## ##x+y=2⋅\dfrac{4}{x-y}##...
21. ### Solve the given problem that involves a space curve

Relatively new area to me; will solve one -at- time as i enjoy the weekend with coffee. 1. Unit tangent ##r=xi+yj+zk## ##r=(t-\dfrac{t^3}{3})i+t^2j+(t+\dfrac{t^3}{3})k## ##T=\dfrac{dr}{dt} ⋅\dfrac{dt}{ds}## ##\dfrac{dr}{dt}=(1-t^2)i+2tj+(1+t^2)k##...
22. ### B Verifying AI Chatbot's Answer on Brachistochrone Curve Formula

I was interviewing an AI chatbot and was wondering how good it is at physics… can anyone confirm whether the answer it gave is true or false? This was through a chatbot called ChatGPT. My prompt: What’s the formula for the Brachistochrone curve? The AI answer: The Brachistochrone curve is a...
23. ### POTW Show That the Curve is Straight

Fix points ##p,q\in \mathbb{R}^n##, and let ##\gamma : [a,b] \to \mathbb{R}^n## be a continuously differentiable curve from ##p## to ##q## whose arclength equals the Euclidean distance between the points, ##|q - p|##. Prove that ##\gamma## lies on the straight line passing through ##p## and ##q##.
24. ### Find the coordinates of intersection between tangents and given curve

ooops...this was a bit tricky but anyway my approach; ... ##\dfrac{dy}{dx}=-2x## therefore; ##\dfrac{y-7}{x+1}=-2x## and given that, ##y=4-x^2## then; ##4-x^2-7=-2x^2-2x## ##x^2+2x-3=0## it follows that, ##(x_1,y_1)=(-3,-5)## and ##(x_2,y_2)=(1,3)##. There may be another approach...
25. ### I How to evaluate the enclosed area of this implicit curve?

The implicit curve in question is ##y=\operatorname{arccoth}\left(\sec\left(x\right)+xy\right)##; a portion of the equations graph can be seen below: In particular, I'm interested in the area bound by the curve, the ##x##-axis and the ##y##-axis. As such, we can restrict the domain to ##[0...
26. ### SO2 Conversion Equilibrium Curve

Hi... i want to draw Equilibrium curve for SO2 oxidation to SO3, i found following relations but don't know to use them... kindly tell me how to draw Equilibrium curve using these equations or any source on web from where i can get directly SO2 oxidation data Vs Temp? Equations are following...
27. ### Engineering An issue with these two curved beam homework problems

Hi, good evening/morning/night! I have a problem with this 2 homework problems about curved beams. In both I must calculate the stress at points A and B. My procedure is correct, but for some reason in: First problem - the effort in A is very close to that of the teacher, but the effort in B...
28. ### Find the length of the curve C

This question is from a Further Maths paper; Part (a) is pretty straight forward...No issue here...one has to use chain rule... Let ##U=\dfrac{e^x+1}{e^x-1}## to realize ##\dfrac{du}{dx}=\dfrac{-2e^x}{(e^x-1)^2}## and let ##y=\ln u## on taking derivatives, we shall have...
29. ### Find the two points on the curve that share a tangent line

IMPORTANT: NO CALCULATORS I assumed two points, (a, f(a)) and (b, f(b)) where b is greater than a. Since the tangent line is shared, I did f'(a) = f'(b): 1) 4a^3 - 4a - 1 = 4b^3 - 4b - 1 2) 4a^3 - 4a = 4b^3 - 4b 3) 4(a^3 - a) = 4(b^3 - b) 4) a^3 - a = b^3 - b 5) a^3 - b^3 = a - b 6) (a...
30. ### Find the equation of the curve given: ##\frac{dy}{dx}=2(kx-1)^5##

This is the question...hmmmm it stressed me a little bit.:cool: Find the textbook solution here; no. 6 Now my approach to this was as follows; On integration, ##y=\dfrac{(kx-1)^6}{3k} +c## on using the point ##(0,1)## and ##(1,8)##, we end up with ##1=\dfrac{1}{3k} +c##...
31. ### Precise definition of tangent line to a curve

How do we define tangent line to curve accurately ? I cannot say it is a straight line who intersect the curve in one point because if we draw y = x^2 & make any vertical line, it will intersect the curve and still not the tangent we know. Moreover, tangent line may intersect the curve at other...
32. ### MATLAB Curve Fitting in MATLAB: Error Diagnosis

I have two 1D matrices X(1,j) and Y(1,j) of equal length. To fit Y to a model asin(bx) I tried: fit = fittype(@(a,b,X), a*sin(b*X)); [fitted, gof] = fit(X, Y, fit) coefficients = coeffvalues(fitted) this gives the message: Error using fittype>iDeduceCoefficients The independent variable x does...
33. ### I Do All Galaxies with Dark Matter Halos Have Flat Rotation Curves?

Do all galaxies with dark matter halo have flat rotation curve?
34. ### I have a question about the mathematical model of a reflective curve

Hello : Have a question regarding the mathematical model of reflective curve where could i find information on it ? (pdf , webpages , ebooks ,...etc ) Other than Wikipedia Best Regards HB
35. ### How to determine the car's deceleration at a point along a curve?

So I get the exercise and all and have just solved it. But .. I kind of very very intuitively determined ##\theta## to also be the angle for the circular sector. The problem here is that my geometry bag is very weak, I didn't have any geometry in HS, will fix that sooner or later but anyway...

37. ### MHB Proving $\displaystyle\int_C \vec{F}\cdot d\vec{r} = 0$ with Closed Curve

If C is the simple closed curve in the xy plane not enclosing the origin, how to prove that $\displaystyle\int_C \vec{F}\cdot d\vec{r} =0$ where $$F= \frac{yi +xj}{x^2+y^2}$$ How to answer this question? Any math help will be accepted. I am working on this question. If any member of Math...
38. ### Find the equation of given curve in the form ##e^{3y}=f(x)##

For part (a); $$\int e^{3y} \,dy=\int 3x^2\ln x \,dx$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}+k$$$$\frac{e^{3}}{3}=e^3-\frac{e^3}{3}+k$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}-\frac{e^3}{3}$$$$e^{3y}=3x^3 \ln x-x^3-e^3$$ You may check my working...i do not have the solution.

48. ### I A curve that does not meet rational points

This is just to recall a nice fact: Any two points ##A,B\in\mathbb{R}^n\backslash\mathbb{Q}^n,\quad n>1## can be connected with a ##C^\infty##-smooth curve that does not intersect ##\mathbb{Q}^n##. The proof is surprisingly simple: see the attachment
49. ### I Calculating an increasing angle in Spherical Coordinates for a curve

I'm making a program that generates lines in 3D space. One feature that I need is to have an incrementally increasing angle on a line (a bending line / curve). The problem is simple if the line exists in the xy-plane, then it would be a case of stepping say 1m, increase the azimuthal angle φ...
50. ### B How to Find the Intersection of a Logarithmic Curve and a Tangent Line?

I have a formula y=log(x)/log(0.9) which has this graph: I want to find the intersection of this curve and a tangent line illustrated in this rough approximation: The axes have very different scales, so the line isn't actually a slope of -1, it's just looks that way. How can I figure out: 1)...