Questions tagged [plane-curves]

Plane curves are continuous (or smooth) functions $\gamma\colon I\to\mathbb R^2$ from a real interval to the plane. Sometimes also the image $\gamma(I)$ is called curve.

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Finding points outside two lanes

I have two set of points. One set defines the left lane and other set the right lane. And I also have another set of points(black and purple below) and I need to find points in third set which are ...
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307 views

Bending Energy of regular planar curves is parameterization invariant

I need to show that the Bending Energy of a Planar Curve $$\int_I {\kappa^{2}||\gamma^{'}||}$$ is invariant under a reparameterization of $\gamma$ I'm not really sure how exactly I go about doing ...
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Elliposid in $\mathbb{R}^3$ is given by $2x^2+2y^2+z^2=338$. Find radius of sphere that touches ellipsoid in points $(x,y,10)$

Elliposid in $\mathbb{R}^3$ is given by $2x^2+2y^2+z^2=338$. Find radius of sphere that touches ellipsoid in points $(x,y,10)$. Sphere has center on $z-axis$ This is what I have so far. Sphere ...
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187 views

How can I find the normalization of a singular hyperelliptic curve?

Given a hyperelliptic plane curve $$ X = \text{Spec}\left( \frac{\mathbb{C}[x,y]}{y^2 - x(x-1)(x-2)^4} \right) $$ how can I find its normalization? I know I am suppose to compute the integral ...
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How to find a plane curve $\gamma$ such that $G\circ\gamma=\Gamma$?

Let $G: \mathbb{R}^2\longrightarrow\mathbb{R}^2$ be a smooth function. Let also $\Gamma:[0,1]\longrightarrow\mathbb{R}^2$ be a smooth curve. I am working on the problem of finding a curve $\gamma$ ...
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181 views

Periodic curves

I am building an Android library with different kinds of progress views represented with curves. As you can see it in the gif below, I have used the following curves: Lemniscate of Bernoulli, ...
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26 views

Rate of increase of the function

How do I find the value of the rate of increase of the function $z=x^2+y^2$ along the direction of the given line $y=3x-1$ and in the $xy$ plane at the point $(x, y, z)=(1,2,5)$?
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Strictly convex curves and lines

Let $\gamma\colon [0,1] \to \mathbb R^2$ (continuous) be a simple closed plane curve and $\,\mathcal C$ its image. Let $x,y$ be some functions such that $\gamma(t)=(x(t),y(t))$. $\gamma$ is said ...
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275 views

Property of strictly convex curves

Let $\gamma\colon [0,1] \to \mathbb R^2$ (continuous) be a simple closed plane curve and $\,\mathcal C$ its image. Let $x,y$ be some functions such that $\gamma(t)=(x(t),y(t))$. $\gamma$ is said ...
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39 views

closed planar curves invariants

http://www.mathcurve.com/courbes2d/generique/generique.shtml In this french website, he presents an invariant of closed planar curves: number of double points. He says that there is no curves with 1 ...
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187 views

Jordan curve theorem (Maehara's proof)

There is one very simple proof for Jordan curve theorem http://www.maths.ed.ac.uk/~aar/jordan/maehara.pdf But Maehara's proof contains one "trivial" step when we fit Jordan curve $J$ in a ...
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33 views

Techniques for finding functions with known values and derivatives at two points

I want to find a function $\gamma(t) = (x(t),y(t))$ such that for two values of $t$, we have $\gamma'(t)$ and $\gamma(t)$ have some value, and at no point does the curvature ever exceed $r$. What ...
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62 views

Equation for sickle shaped plane curve

Is there a parametric equation for a plane curve with the shape of a sickle cell, e.g. half nephroid and half circle? I couldn't find one so far. Thanks! I'm looking for an equation consisting of ...
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123 views

Collinearity of triangle vertices on circular paths

Suppose as a function of time, the vertices of a triangle move with constant speed along circular trajectories: $$ \vec p_i(t) = a_i \left( \begin{array}{c} \cos(b_i t+ c_i)\\ \sin(b_i t+ c_i) \end{...
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181 views

Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
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271 views

Find volume of cube with the help of eqn of plane

The volume of cube whose two faces lie on the plane 6x-3y+2z+1=0 and 6x-3y+2z+4=0?
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Which curve (surface) is this?

We're having trouble fitting our data... well, we don't even know which function we should fit onto. Anybody knows if this function is well defined mathematically?
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935 views

Log power rule problem

According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian plane....
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134 views

Intersection of sphere and plane is a point, find c, such that (0,0,c) is center of sphere

Sphere: $$ x^2+y^2+(z-c)^2=1 $$ Plane: $$ x+2y+3z=0 $$ Find the values of $c$, for which the intersection of the sphere and the plane is a point. Well, I know that the sphere has the center (0,0,c) ...
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92 views

Graph theory: creating surfaces

If we draw a circle in the plane , we separate the plane into 2 regions , inside and outside. On the following surfaces, determine the maximal number of circles that you can draw on the surface ...
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30 views

Finding the radii of an ellipse from the intersection of a plane and a sphere

I'm trying to solve the following problem, regarding Stokes Theorem: $F = z i + xj + yk $; C the curve of intersection of the plane $x + y + z = 0$ and the sphere $x^2 + y^2 + z^2 = 1$ [Hint: ...
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Curve fitting: How to identify the appropriate function for a beat-like phenomena?

I have a time series data which shows some beat like behaviour. The envelope does not look exponentially decreasing, as it is impossible from a physics point of view. The envelope is likely to be a ...
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87 views

use of wolfram in determining area between two curves

I am new to the use of Wolfram (that for the limited cases I have used is very impressive). However I wonder if anyone can tell me what I am doing wrong. I wanted to calculate the area between the two ...
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51 views

On a formulation in Hilberts original paper about the space-filling Hilbert curve

I have a question on the famous paper Über die stetige Abbildung einer Linie auf ein Flächenstück (which translates roughly as On the continuous mapping of a line onto a square) by D. Hilbert. Let the ...
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42 views

The change of the angle of the gradient as moving along the curve

I'm given a curve $g = 0$ in 2D specified by g(x,y) = f(x) - y. The normal to the curve is the gradient of $g$ - $(f', -1)$. Now I want express the change in the angle $\theta$ of the normal as I move ...
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31 views

Reparametarize a curve to move a unit length

I'm interested in the general case when we have a curve $(x,f(x))$ parameterized by $x$ to find a parametrization $x=g(t)$ such that $ds/dt=1$ along the curve. So far what I came up: $$\frac{ds}{dt}^...
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45 views

Plane integral for continuous curves

I'm trying to understand complex path integral $\int_C f(z)dz$ for continuous closed curve $C$. Is it necessary that $C$ is rectifiable and not just generally continuous? Do we get all the ...
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232 views

Describe curves by words

I am trying to describe the general aspect of the following curves My ideas: The curves are continuous, and defined for positive values of x, and giving negative values. Have logarithmic growth (...
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478 views

Angle between tangent vector and point of a cardiod.

Consider the cardioid $\rho=2a\left( 1 - \cos \phi \right) $. Show that the angle between the tangent vector and an arbitrary point (different from the origin) of the curve is half the polar angle. ...
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44 views

advice for curve fitting

I have numerically obtained some curves, corresponding with it I have also obtained some roots. I strongly believed these curves can be fitted with some (elliptic) functions taken the roots as ...
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31 views

When working with trochoids what does θ stand for?

These are the formulas with which you can draw trochoids. $x = aθ - b sin(θ)$ $y = a - b cos(θ)$ I'm trying to make trochoids but I got hung up on this symbol $θ$, what is it and how do I use it, I ...
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500 views

How to find a plane that is tangent to 3 spheres?

So there are spheres with radius of 1 centered at (1,2,0), (4,5,0) and (1,3,2). How can one find a plane that is tangent to all 3 spheres? Visually, it looks like as if the spheres are sitting on a ...
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172 views

Find the maximum volume of the pyramid bounded by the plane and the coordinate planes?

Surface $\sqrt{c}=\sqrt{x}+\sqrt{y}+\sqrt{z}$ , $(c>0)$ I found that at $(x_{0},y_{0},z_{0})$ a tangent plane to the surface is : $\frac{x-x_{0}}{2\sqrt{x_{0}}}+\frac{y-y_{0}}{2\sqrt{y_{0}}}+\...
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104 views

Intersecting convex figures

Take in the real plane a finitely long horizontal line segment and connect the two endpoints by a convex path, above the segment, with the property that the only extreme points of the convex hull of ...
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47 views

Unique solution of a simple functional equation

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in C^{\...
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71 views

Curvature in $\mathbb{R}^2$

Let $f(t) = (x(t),y(t))$, not necessarily parametrized by arclength. We define the unit tangent vector, $T(t) = (1/|f'(t)|)(x',y')$. Also the normal vector, $N(t) = (1/|f'(t)|)(-y',x')$, which is ...
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A 4th grade curve meets a line in one point with multiplicity 4

Suppose a 4th grade curve meets a line in one point with multiplicity 4. Example: the lemniscate $(x^2 + y^2)^2 = y^2 - x^2$ meets the line $x=y$ when the condition $x^4=0$ holds. This shows that ...
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1answer
96 views

Deriving tangent plane equation from scalar equation of plane

There's one step in the derivation that I don't understand. I'll explain some of the derivation and then explain which step I don't understand. Begin with scalar equation of plane: A(x-x0) + B(y-yo)...
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1answer
29 views

Equation for a Vinyl curve

This video seems to show an explicit map from the torus to $\mathbb{R}^2$. Does it factorize through the projection $\mathbb{R}^3 \to \mathbb{R}^2$? What is the equation of the curve?
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200 views

Problem with calculating winding number in sum of curves

Let $$ \begin{align} \gamma= \gamma_1 +\gamma_2+\gamma_3,\\ \gamma_1(t)=e^{it}, t\in[0,2\pi] \\ \gamma_2(t)=-1+2e^{-2it}, t\in [0,2\pi]\\ \gamma_3(t)=1-i+e^{it},t\in [\frac{\pi}{2},\frac{9\pi}{2}] \...
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Image of a parametrised curve and its geometrical meaning

$\gamma_1(t)= t^2 + i\, t^4 , t\in [ -1, 1]$ which is the Image of this curve and / or what is the geometrical description of the set of points $ z : z=\gamma_1(t)$ I am helping a friend study ...
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512 views

Maximum area enclosed by a string attached at fixed points

Two fixed points A and B have a string of length L attached between them. Supposing that the string does not intersect the line segment AB, then the string and AB will form a closed figure. What shape ...
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2k views

Lattice Points in x-y plane

What are Lattice Points? Which points in x-y planes are Lattice Points? Is (m,n) a lattice point where m,n are any integers?
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675 views

convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
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39 views

Intersection of planes

A line perpendicular to the plane $ 3x-5y+4z-11=0 $ passes through the origin. At what point does this normal intersects the plane?
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What is the meaning of “slope of ca”?

I'm reading a paper, when this article refers to the function: $$\beta(v)=\frac{(\frac{v}{I})^k}{1+(\frac{v}{I})^k}$$ It say that "around the $I$, $\beta$ is approximately linear in $I$, and has a ...
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353 views

Curvature of the boundary curve of convex set

I've got a very simple question, but I can't get a rigorous proof. Suppose that $X\subset\mathbb{R}^2$ is a convex set, and suppose further that $\gamma$ is a closed regular curve with image $\partial ...
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116 views

The area of a region around a curve

If we are given a simple closed curve with length $L$ in the plane, and we have a fixed number $r$ such that for each point $x$ on the curve there is a related disc $D(x,r)$ with radius $r$, how can ...
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118 views

Creating a $C^2$ smooth convex curve which joins circular arcs

Suppose that I have many points $p_1$,..., $p_n$, all very close to origin 0. For R very, very large, I have circular arcs of radius R, $\alpha_i$, about $p_i$. I do not speak about their ...
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54 views

Domain and Range problem(plane)

Consider the function $z = \ln{(y + 1)}+\sqrt{x-3}$. Find the domain and range, and sketch the domain in the plane.