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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92 views

Proving regularity and orthogonality of two curves

Let $\alpha(s) = (x(s)),y(s))$ be a regular plane curve that is parameterized by arclength, and let $n(s)$ be the normal vector and $k(s)$ be the curvature of $\alpha$. Consider the family of curves: ...
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1answer
117 views

image of parametric quadratic curve with three components contained in a plane

I am studying Differential geometry I tried to prove this by taking all three components as quadratic with $t$ as a parameter but could not be successful. If all three component functions of a ...
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43 views

Proving that strictly monotonic curvature implies no self intersections (more specifically, using the following inequalities)

Let $a(s)$ be a regular curve that is parametrized by arclength. Prove that, if the curvature $k(s)$ is a strictly monotonic function, then $a(s)$ has no self intersections. Suggestions: a) [will be ...
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31 views

How to approximate a level curve?

Let $G$ be a $C^\infty$ function $G:\mathbb{R}^2\rightarrow\mathbb{R}$, and let $C:=G^{\leftarrow}(c)$, i.e. $C$ is a level set of $G$. I know that $C$ is bounded (which implies that it's a closed ...
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93 views

Interpolating a three point curve at any angle using cubic splines

I'm trying to interpolate a curve using cubic splines and three points in the x-y plane. I have some troubles finding the equation for the middle point such that the normal vectors in point P0 is ...
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166 views

Proof of Schur's Theorem for Convex Plane Curves by Guggenheimer

I'm reading Differential Geometry by Heinrich W. Guggenheimer and I have a doubt about the proof of Schur's Theorem for Convex Plane Curves on page 31. I will put the theorem and the proof here before ...
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102 views

Intersection points of a straight line with a closed convex curve

I tried to solve the following problem from Do Carmo. It is very intuitive, but I am having a hard time trying to formalize everything. Show that if a straight line $L$ meets a closed convex curve $C$...
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87 views

Intersection of two plane curves in the residue field

I want to propose this problem. Suppose that $C:F(X,Y,Z)=0,\;C':G(X,Y,Z)=0$ are two plane curves defined over a number field $K$. Suppose that they have no common component and that all the ...
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56 views

What does $\partial L$ in $\int _{\partial L}\:xy\:dx\:+\:z\:dy\:+\:\left(x+2y^3\right)dz\:$ mean?

Let the solid L be given by, $L=\left\{\left(x,y,z\right)\in R^3∣\:0\le \:z\le \:1-x^2-y^2,\:y\ge \:-x\right\}$ Observe that this solid is limited by the plane $z=0$, the plane $y=-x$ and the ...
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99 views

Constructing reducible polynomial with two irreducible polynomial

I'm currently trying to prove following statement. Let f,g be homogeneous polynomials of degree n,m respectively, in $k[X,Y,Z]$. Here k is algebraically closed field, and n $\le$ m. Also f does not ...
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64 views

How do we call this method?

We want to study a parametric curve $f : \mathbb{R} \to \mathbb{R}^2, \ t\mapsto (x(t),y(t))$. To proceed, we use the cartesian equation of a line $y=ax+b$ with $a\in \mathbb{R},b\in \mathbb{R}^{*}$ ...
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193 views

Complement of Simply Connected Subsets in the Plane

I know that if $X$ is a connected and simply connected subset of the plane and $\mathbb{A}$ is a 2-annulus, that it does not imply that $\mathbb{R}^2 \setminus X \simeq \mathbb{A}$ (where $\simeq$ ...
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28 views

if two consecutive copies of a curve do not intersect, can we add an infinite sequence?

Start with a simple non-self intersecting curve on the plane from point $A$ to $B$. Then add a translated copy of that curve next to it, so they from a new curve, and assume that also the new curve ...
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1answer
118 views

Is there something wrong with the problem statement?

I'm doing the following question relating to parametrising surfaces and finding the tangent plane, Question: I parametrised the surface in term of sphereical coordinates, such that $$x=\sin\theta \...
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1answer
77 views

Radius of curvature for a strictly convex curve

Suppose I have a strictly convex curve $\gamma$ parametrised by arc length $s$ in the plane. I take a point $s_{1}$ on $\gamma$ and issue a straight line from $s_{1}$ to another point $s_{2}$ on $\...
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240 views

Jordan curve theorem for rectifiable curves

I am looking for a proof for the Jordan curve theorem in the particular case of rectifiable curves. I consider this case as "intermediate" between the general case and the case of piecewise ...
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154 views

Introductory text on curve shortening flow

I have been using this pdf as a primary source for the introductory part of a project on CSF I'm currently writing. Since it says 'Chapter 2' above the title, it implies this work is part of some ...
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1answer
40 views

Tangent line at $(0,0)$ of plane curve $y^3=x^5$

Let $f(x,y)=y^3-x^5$. $\nabla f(x,y)=(-5x^4,3y^2)$. $\nabla f(0,0)=(0,0)$ so $(0,0)$ is a singular point of the curve $y^3-x^5=0$. On the other hand, it makes sense to define $g(x)=x^{5/3}$ for $x \...
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40 views

How does this curve parametrization work?

I have come across the following procedure, and I do not really understand what it means. Let $C$ be a closed smooth plane curve, with the origin $O$ lying inside it. Here is what I do not understand:...
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54 views

If a closed plane curve has turning number zero how to prove the continuous argument of the unit tangent field has the same period as the plane curve

Consider a closed plane curve $\gamma:\mathbb{R}\mapsto\mathbb{R^2}$, of period $L$ with unit tangent field $T$. The unit tangent field has continuous argument $\theta$ such that $$T = (cos(\theta(s)...
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64 views

Approximation of Lipschitz curves by $C^{1,1}$ curves

Given a curve $\gamma :[0,1] \to \mathbb{R}^2 $ define the set $F(\gamma)=\cup_{t\in [0,1]} B_1(\gamma(t))$. Given a Lipschitz curve $\gamma$, does it exist a curve $\sigma :[0,1]\to \mathbb{R}^2$ ...
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35 views

Proof $C^1$ image of interval cannot contain a square

I'm looking for a proof verification of a solution of seemingly easy exercise. I ask because I often mess these geometrical proofs up. Exercise. Let $\gamma :[0,1]\to \mathbb R^2$ be a smooth ...
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214 views

A curve with Lebesgue measure non zero

In this Continuously Differentiable Curves in $\mathbb{R}^{d}$ and their Lebesgue Measure the domain of the curve is a compact set. I want know if the same answer holds for curves with non-compact ...
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417 views

A space curve with non-vanishing curvature is planar iff its torsion is 0

Intuitively this is simple and to prove the backwards direction: $\tau = 0 \Rightarrow \mathbf{b'}=0 \Rightarrow \mathbf{b} $ is constant. Then letting $\gamma$ be the parametrisation of the curve ...
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40 views

Normal and tangent vectors to a curve

Assume we have a displacement field $u=u(x,z)=(u_x(x,z), u_z(x,z)))$ which takes the point $R=(x,z)$ to $r=R+u=(x+u_x(x,z), z+u_z(x,z))$. We want to find the tangent and normal to the curve which was $...
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286 views

Can someone explain the detail of the proof (rectifiable curve)

Curve $C$ in the plain, \begin{align} C:= \begin{cases} x=\phi(t) \\ y=\psi(t) \end{cases} \quad, \quad a \le t \le b \end{align} The graph of C is $\{(x, y): x=\phi(t), y=\psi(t), a\le t\le b\}$ $\...
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1answer
80 views

The circle does not intersect the unbounded component of $\mathbb{R}^2 \setminus C$

We get the following problem in our differential geometry class. Let $ C $ be a smooth, non-degenerate simple closed curve traveling counterclockwise. Suppose that the curvature $ \kappa $ of C is ...
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33 views

Manifolds: Showing a curve is given locally by a function $\phi_1$

Image link at bottom I'm not sure how to go about showing that $y=\phi_1(x)$ gives the curve $4y^3-3y-x=0$ locally. I may be able to show that $DF(a) = D\phi_1(a)$, but that doesn't prove it over the ...
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99 views

On the set of points “inside” a closed curve

consider the following: one has a simple closed rectifiable curve $\gamma$ in the plane, and there is a point $a$ such that for all $p\in\gamma$ the segment $\overline{ap}$ intersects $\gamma$ only in ...
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43 views

Plane curves admitting several ways to seal it up by the disc

Let $\gamma:S^1\to\mathbb R^2$ be an immersed closed curve with only isolated transversal self-intersections. Say that $\gamma$ is sealed by the disc if there is an immersion $f:D^2\to\mathbb R^2$ ...
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1answer
36 views

Parametric problem with circumference and tangents

Given the circumference $(x-3)^2+(y-2)^2=13$ find $k$ where $k$ is a coefficient in the parametric equation $(k+1)x+8ky-6k+2=0$ of the lines passing through the points $A(0;4)$, $B(6;4)$, $C(1;-1)$. ...
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268 views

Genus-Degree formula gives the wrong answer: ordinary points?

I'm trying to compute the genus of the normalization of the curve: $y^5=x(x-1)(x-2)$ Now I calculate the ramification points of the projection x: they are $(0,0),(1,0),(2,0)$ and they are of ...
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93 views

Involute of a Curve

A string of length ℓ is attached to the point γ(0) of a unit-speed plane curve γ(s). Show that when the string is wound onto the curve while being kept taught, its endpoint traces out the curve ι(s) = ...
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34 views

Describe a curve by other than a fomula, fitting or interpolation

I have a curve defined by a set of $(x,y)$ given points. After ploting these points I get : My aim is to study the sensitivity of the operation that results these points. In order to do that, the ...
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2answers
693 views

Point normal equation of plane

Im doing an homeassignment in linear algebra! This is the question I'm having problems with! The plane M contains two lines; l1 : (x, y, z) = t(2, −1, 0), t ∈ R, l2 : (x, y, z) = t(0, 1, ...
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25 views

move curve normal to itself

I have a plane curve given by $y = f(x)$. At every point on this curve, I construct a normal direction to this curve and move the point a fixed distance $s$ along the normal to a new point. What is ...
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338 views

Green's theorem and 'simple regions'

I'm looking through at my notes and couldn't understand why, in the notes below, there is a need to compute the curve C2 and C4. It's hard to see why isn't computing C1 and C3 a sufficient condition.
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159 views

Reciprocal relations in Roulette /glissette rollings

If a catenary rolls on a straight line its focus traces out a parabola and vice versa. Is it true? Are there more such examples and how are they co-related? In case of a circle rolling on a fixed ...
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57 views

The curvature blows up

A curve evolves according to the evolution equation $\displaystyle\frac{\partial X}{\partial t}= k \times N$ where $k$ is the curvature of the curve and $N$ is the inward unit normal vector. Then ...
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122 views

Plane and Ellipse Intersection

Short Version: If some can solve the easier to read form as follows, I would be thankful. \begin{equation} B = \frac{1 - d^{T}Bd}{ K_{1} } A \end{equation} \begin{equation} B^{T}d = \frac{1 - d^{...
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2answers
328 views

Solving non-linear second order differential equation: radius of curvature $= k \theta$

I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates radius of curvature $= k \theta$ $$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + ...
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79 views

Reparametrization with non-vanishing lateral derivatives

Let $\mathbf{r}:I\subseteq\mathbb{R}\to\mathbb{R}^2$ be a $C^{1}(I)$ function, non-constant on any subinterval, such that $\forall\ t_0\in I$, the following limits exist: $\lim\limits_{t\nearrow t_0}\...
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2answers
74 views

If $0 < \theta < \frac{\pi}{2}$, then $\gamma$ is a logarithmic spiral

Let $\gamma$ be a plane curve parametrized by the arc length, having the property that its tangent vector $T(t)$ forms a fixed angle $\theta$ with $\gamma(t)$. Before explaining where I am, let ...
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108 views

Direction of outward normal differential $\hat n\ ds$

In vector calculus, why is the outward normal differential $\hat n\ ds$ around a closed curve $dy\ \hat i - dx\ \hat j$? Why isn't it $-dy\ \hat i + dx\ \hat j$ or something else?
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97 views

Embedding a $deg \geq 4$ curve in $\mathbb{P}^2$

Reading the paper by Kollar "The structure of algebraic threefolds", among some examples he does talking about intrinsic and extrinsic geometry over $\mathbb{C}$, he mentions that "an irreducible ...
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1answer
141 views

critical point for the curvature does not correspond to a local maximum/minimum.

Draw an example where a critical point for the curvature does not correspond to a local maximum/minimum Does the curve for infinity sign satisfy this? I am having trouble seeing why it's true, if of ...
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75 views

Gradient Vectors as Normal Vectors: How to Find D in Form Ax + By + Cz = D?

I've got a question that looks like this: Use the normal gradient vector to determine the equation of the line/plane tangent to the given curve/surface at point P. $x^4 + xy + y^2 = 19$, P(2,-3) I ...
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453 views

Total curvature of a piecewise smooth planar curve

I am trying to show that the total curvature of a piecewise smooth planar curve $\alpha$ has the property that $\displaystyle \int_{C}\kappa $ $\text{d}s + \displaystyle \sum_{j=1}^{l} \epsilon_{j} =...
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303 views

Tangent vector for a curve defined by a discrete set of points

I have a curve defined by a discrete set of points (x,y). How can I approximate the tangent vector at a point for such a curve?
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1answer
33 views

Question on a function defined on some plane curve.

Let $C$ be the plane curve defined by $y^d =f(x)$, where $f(x) \in \mathbb{C}[x]$ is a polynomial. Let $g(x,y)$ and $h(x,y)$ are polynomials relatively prime with each other. We consider $r(x,y) =g(x,...