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.
Hi everyone
ın fact at the beginning ı want to select three points every curve Idid it But now I shold find which points ont the same time AND THEN I shold save its How can do it
My code is here
#include <iostream>
#include <iostream>
#include "opencv2/highgui/highgui.hpp"
#include...
Hello to everyone.
I would like to ask a question: I'm analyzing the properties of CO2 for a project and I have noticed that for temperatures between 220 -300 K the Clasius Clapeyron curve is linear on 1/T, that is in the Antoine equation of the form logP = A-B/(T+C), C is almost 0. In my...
I hope this is in the right forum.
I'm trying to see how to calculate rotation curves for a spiral galaxy.
The data I have is brightness profile (mag/arcsec^2) and observed rotation curve - to see how it fits.
from what I found I need the total mass, and for that I need the surface mass density...
Hi all,
I was wondering if anyone could help me with calculating the angle of lean on a motorbike on a banked curve (with friction). If possible proofs would be very useful.
Thanks
Lee
Let's say I want to calculate some instantaneous speeds during a trip from city a to city b.
I start down the highway from an initial speed of zero then increased to 55. After 15 minutes I increase to 70. Down to 60 for 2 minutes. Then up to 80 for 15. Back down to 70. Then a slow decrease to...
Dear users,
I deal with following problem.
I've got a data of Intenzity of chemiluminezcence as a function of temperature. The data should be fitted by following equation:
I found that in origin one can fit also data with integral function, with fittin parameters P, A, i, E.
The beta and...
Homework Statement
Sketch a picture of the cone x = sqrt(y^2+z^2) , and elliptic paraboloid x = 2−y^2−z^2 on the same grid.
Although the picture does not have to be perfect, indicate clearly the orientation of both figures relative to coordinate axes. Identify the curve at the intersection of...
Homework Statement
Find the domain of the function f(x,y) = 1/sqrt(x^2-y). Sketch the level curves f=1, f=1/2, f=1/3.
Homework EquationsThe Attempt at a Solution
Domain is rather simple to find: x^2>y AND x^2-y cannot equal 0.
Level curves are also simple to find. They are simply parabolas...
Homework Statement
Use, using the result that for a simple closed curve C in the plane the area enclosed is:
A = (1/2)∫(x dy - y dx) to find the area inside the curve x^(2/3) + y^(2/3) = 4
Homework Equations
Green's Theorem:
∫P dx + Q dy = ∫∫ dQ/dx - dP/dy
The Attempt at a Solution
I...
How can we determine which curve has larger slope just by looking at curves.
http://www.meritnation.com/img/shared/userimages/mn_images/image/a_2_12_2.png
Which curve has larger slope?
I went through this article:
http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/ondas_optics/caustica1.pdf
But I think that when we do F=0, we are assuming the centre of the curve to be the origin and assuming the curve to be a circle, because only then will we be able to say that OP=ON...
Homework Statement
The question is- What is the minimum coefficient of static friction that would keep the car from sliding off the curve? The Cars mass is 13500KG and it is traveling at 50.0hm/hr(13.9m/s) and the curve has a radius of 2.00 x102 m. I know the centripetal acceleration of the car...
I have a calculus question and I am not sure where to get started. The question states:"Find the value of c>0 such that the line y=x+1 is the tangent line to the curve c√x (i.e. intersects the curve at one point and shares the same slope at that point)."
Thanks
Homework Statement
f(x,y) = x^2 + xy + y^2 and z=2x+y intersect, find a parameterization of the curve where they intersect.
Homework EquationsThe Attempt at a Solution
I am lost. I know that z is the partial derivative of the original function, if that's of any use. I can visualize it but not...
a) Find the roots of the equation x^{2}+5x-6
b) Sketch the graph of the function x^{2}+5x-6 labeling the points at which the graph crosses the axes and the co-ordinates of the maximum and minimum of the curve
c) Find the equation of the tangent at the point where x=2 on the curve of...
Homework Statement
I have data of an experiment to find absorbtion coefficent of a sample. one curve shows intensity of original beam, the other one is intensity with sample (with wavelenght). Here is the data :
Homework Equations
Where: I = the intensity of photons transmitted across...
I need to perform geometry matching of curves (see http://www.tiikoni.com/tis/view/?id=c54d9b8 ). As it can be seen, the big problem is that curves might be rotated, though they have similar shape.
Do I need to make curve fitting and look at the parameters of analytical models? But, I guess...
There is a 2D array of x and y positions of a vehicle measured at an interval of one second, e.g.:
x y
1, 5
1, 6
1.5, 6.8
...
I need to somehow quantify how a given driver approaches curves (i.e. driving style - does he/she drives aggressively or not). For this, I divided the problem into two...
A problem states " For a banked curve, ignoring friction, prove that tan θ = ν^2/rg". I tried to prove but I thought that as the normal force is at right angle to track then how could be the component of normal force provide the centripetal force as my book is saying. Please someone help. I...
I'd appreciate advice on the correct statistical method to analyse a dataset -
Dataset is basically a titration curve consisting of [0.5, 1, 2, 3, 4, 5, 6] pg of starting material and 8 replicates in each 'pg bin'. In 'stage 1' of the process each bin is labeled separately, in 'stage 2' all...
Hello! Merry Christmas! (Beer) (Party) (Bigsmile)If $E/\mathbb{Q}$ the elliptic curve $y^2=x^3+x^2-25x+29$ and
$$P_1=\left (\frac{61}{4}, \frac{-469}{8}\right ), P_2=\left ( \frac{-335}{81}, \frac{-6868}{729}\right ) , P_3=\left ( 21, 96\right )$$ I have to show that these points are...
Homework Statement
Let C be the curve given parametrically by x = (t^3) - 3t; y = (t^2) - 5t
a) Find an equation for the line tangent to C at the point corresponding to t = 4
b) Determine the values of t where the tangent line is horizontal or vertical.
Homework Equations
dy/dx =...
According to the documents I have read, Plank made two changes to Rayleigh-Jeans approach in order to produce an equation that matched the black-body radiation, experimental curves:
1) As a mathematical convenience he assumed that the oscillators in the walls of black-body cavity could only have...
Hi Folks,
I have a curve that varies sinusoidally calculated from a numerical program as attached "Trace.png". I would like to fit this amplitude modulation expression to it.
f(t)=A[1+B \cos(\omega_1 t+ \phi)] \cos(\omega_2 t+ \theta)
I managed to adjust the parameters manually to get a very...
Homework Statement
Calculate ##\int_C z^{1/3} dz##, where ##C## is the circle of radius ##1## centered at the origin oriented in the clockwise direction. Use the branch ##0 \le \arg z \le 2 \pi## to define ##z^{1/3}##.
Homework Equations
The Attempt at a Solution
[/B]
I was hoping that...
Hey I am new here and not exactly sure how it works. I am stuck on this problem from my professor and would love any help anyone has!When one thinks of the normal distribution the first thing that comes to mind is the bell curve and grades. While this is one example of a normal curve that is...
Hi All,
Having plotted a cell growth curve (fluorescence vs days), I have found that during the lag phase, the fluorescence values decrease. I haven't come across this previously during lectures and can't seem to find any examples online to help explain it. My only thought was that cell death...
Homework Statement
Attached Image
Homework Equations
this is not a simple plane curve or a close plane curve so I use the formula:
∫ F ⋅ dr/dt dt
The Attempt at a Solution
From the point (0,0) to (2,4)
Direction Vector v(t) = <2-0, 4-0>
Parametric Equation:
r(t) = (2t + 0) i + (4t + 0) j...
Homework Statement
Evaluate the line integral over the curve http://webwork.math.ttu.edu/webwork2_files/tmp/equations/33/92ca0e3b907f876e5a974ad1457d1f1.png from to http://webwork.math.ttu.edu/webwork2_files/tmp/equations/6f/bccf9dd59b9c22450c590a042bb77d1.png .
∫-ydx+3xdy (over the curve C)...
Homework Statement
Consider a mass sliding down a frictionless curve in the shape of a quarter circle of radius
2.00 m as in the diagram. Assume it starts from rest. Use Euler’s method to approximate
both the time it takes to reach the bottom of the curve and its speed at the bottom. Hint...
Homework Statement
##C(t) = t =it^2##, where ##-2 \le t \le 2##. Re-parametrize the curve from ##t## to ##\tau## by the following transformation: ##\tau = \frac{t}{2}##.
Homework EquationsThe Attempt at a Solution
So, the variable ##\tau## is half of every value ##t## can be. Therefore, I...
Relative equations:
F = ma
a = v^2/r
Fcent = mv^2/r
ωf^2 – ωi^2 = 2αs
s = θr
ω = v/r
Problem statement and work so far are all included:
(I am having trouble with 5 and no clue how to start on 7-9)
You are designing a roadway for a local business, to include a circular exit ramp from the...
Hello, I was working on a textbook problem that was referencing a paper regarding the mechanism for the production of Terephthalic acid (TPA). It piqued my interest so I found the original paper and did some reading. At least in 1987 when this paper was published, the intermediate reactions were...
Homework Statement
Let A be a set of critical points of the function f(x).
Let B be a set of roots of the equation f''(x)=0.
Let C be a set of points where f''(x) does not exist.
It follows that B∪C=D is a set of potential inflection points of f(x).
Q 1: Can there exist any inflection points...
Hi all,
I want to know the why the Braking force coefficient is increasing first and then decreasing with respect to wheel slip. I attached the graph for your reference
A point moves along the curve y=2x^2+1 in such a way that the y value is decreasing at the rate of 2 units per second. At what rate is x changing when x=(3/2)?Ok here's what I've done...
Dy/dt=(4x)(dx/dt)
-2 = (4*3/2)(dx/dt)
So dx/dt=-1/3 (ie decreasing by 1/3 units per second)
First, is...
The question
I've been given this question by a teacher and I'm clueless as to what b) and c) are. Could you please help? I would really appreciate it. The prize is a place to see a lecture on quantum simulation at our local University.
Thank you :)
Hy, I wonder which is the good solution for this problem:
Nonlinear least square problem: function: y = x / (a + b.x) linearization: 1/y = a/x + b substitution: v = 1/y , u = 1/x >> model v = b + a.u
What we did in school:
x = [1 2 4 7]; y = [2 1 0.4 0.1];
v=1./y;
u=1./x;
n = length(x)...
I'm confused trying to understand the previous steps that lead to the formula for the angle of an ideally banked curve. In absence of friction, this is the angle where the Centripetal Force is provided by the horizontal component of the Normal Force.
I understand that the Normal Force is...
In john oprea's differential geometry book he discusses the notion of a rectifying curve.
where the difference of the position vector along the curve and some constant vector lies in the rectifying plane defined by the tangent and binormal vector of the Frenet Frame.
I'm having a hard time...
Hi, I'm building a fluid model and using the method of characteristics to solve it. I'll not go into the details as they aren't necessary. Basically I have two points $(-\epsilon,70)$ and $(\epsilon, 0)$ and need to create an exponential curve between them. Could someone please tell me of a way...
At what point on the curve
$$y = e^x$$ is the tangent line parallel to the line
$$y = 2x$$
The derivative of y is
$$\frac{dy}{dx} = e^x$$
But I'm unsure how to proceed from here.
Homework Statement
A highway curve of radius 80 m is banked at 45°. Suppose that an ice storm hits, and the curve
is effectively frictionless. What is the speed with which to take the curve without tending to
slide either up or down the surface of the road?
r = 80m
\theta = 45^o
g = 9.8m/s^2...
NB. first time using Latex so apologies if something came out wrong, I've done my best to double check it.
Consider the curve y = \frac{1}{x} from x=1 to x=\infty. Rotate this curve around the x-axis to create a funnel-like surface of revolution. By slicing up the funnel into disks with...
Homework Statement
A civil engineer is asked to design a curved
section of roadway that meets the following
conditions:
With ice on the road, when the coefficient of
static friction between the road and rubber is 0.12, a car at rest must not slide into the ditch
and a car traveling less than 70...
Hi there,
I am trying to understand the dispersion curve(as shown below) of a 1D lattice with diatomic basic.
Here are my questions
1) Can both optical and acoustic branch of phonon can simultaneously exist in crystal?
2)Why there is a band gap between optical and acoustic phonon...