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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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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Is there a plane filling function calculator online?

I recently read about the "Hilbert Curve" and found it very interesting. Does anyone know of a place online where I could extrapolate different shapes and explore this field of mathematics?
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Boundary of faces of plane graph

Theorem. Let $G$ be a plane graph with at least 3 edges drawn on $\mathbb{R}^2$. Then every face of $G$ is bounded by at least 3 edges. We define vertices to be points in $\mathbb{R}^2$ and an ...
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42 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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Polar forms of algebraic curves & surfaces

A paper I'm reading says the following ... With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}) = 0$ be the equation of a surface of degree $n$. The first polar form of ...
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Smoothness of solutions of the curve shortening flow given bounded curvature

I've been looking at the Lemma 1.5 of The Heat Equation Shrinks Embedded Plane Curves to Round Points (here), where Matthew Grayson proved that Suppose that $\kappa$ is bounded for $t\in[0,t_0)$. ...
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40 views

Example of a periodic plane curve whose image is a triangle

I got the following problem as homework for my differential geometry class. Find a $ C^\infty $ function $ \gamma: \mathbb{R} \to \mathbb{R}^2 $ satisfying (i) $ \gamma $ is periodic with period $ 3 ...
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49 views

Every point of Ext(C) lies on some tangent line to C

In our differential geometry class our professor gave the following problem as a homework problem: Let $ C $ be a smooth non-singular simple closed curve in plane. Prove that every point of the ...
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25 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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35 views

Finding the Euler parametrization of a curve

I have the following question as a homework problem for my differential geometry class: find the curvature and the explicit Euler parametrization of the ellipse $ \gamma(t) = (a \cos t, b \sin t) $ ...
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21 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: ...
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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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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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1answer
38 views

Converting all arcs to polygonal arcs in a plane graph

I am trying to understand a proof on the conversion of arcs to polygonal arcs in plane graphs, in the book "Graphs on Surfaces" by Mohar and Thomassen. In the book, an arc joining two points $x,y \in ...
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69 views

Maximum of the length of a curve and its curvature

I'm trying to solve this problem but I can't find any way to do it. Some hints or helps will be very useful and I'll be very thankful. The problem says: Let $\alpha : (a,b) \rightarrow \mathbb{R}^2$ ...
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26 views

Curve connecting two points in $\mathbb{R}^n$ passing through a hyperplane

Let $\pi$ and $\lambda$ be two distinct permutations of $1, 2, . . . , n$, and consider the points $p := (\pi(1),\pi(2), ... , \pi(n))$ and $r:= (\lambda(1), \lambda(2), ... , \lambda(n))$ in ...
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155 views

Punctured plane is not simply connected

Adapt the following definition of "simply connected space" (taken from Wikipedia): A space $X$ is simply connected if it's path connected and for any continuous map $f:S^1\rightarrow X$ can be ...
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21 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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40 views

Integers characterizing singularities of algebraic curves

The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...
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46 views

Unit normal vector at inflection point for any curve: Defined or Undefined?

Consider an arbitrary parametric planar (for simplicity) curve: $ \vec{r}(t) = f(t) \,\hat{i} \, + \, g(t) \, \hat{j}$ Differentiable twice over its domain. $ \vec{r'}(t) = f'(t) \,\hat{i} \, + \, ...
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35 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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400 views

Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
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11 views

Are involutes always regular?

Suppose the curve $\alpha:I \rightarrow \mathbb{R}^2$ is an involute of a regular curve. Does $\alpha'(t)\neq 0$ hold for all $t\in I$?
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17 views

Is the evolute always a regular curve? [closed]

Is the evolute always a regular curve? (that is, the tangent vector of this curve is nonzero at any time)
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37 views

Differentiable, Parametrized Curve for Trace $y = |x|$

my professor gave practice problems for an upcoming midterm, and one is to find a differentiable parametrized curve with trace $y = |x|$ with $-1 \leq x \leq 1$. My thinking is that I should find a ...
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Are all curves with equation of the form $(\xi x +n) \cdot x = \text{const}$ circles?

Let $x(t)=(x_1(t),x_2(t))$ with $t\in [a,b]$ be a smooth curve in $\mathbb{R}^2$ and $\xi \in \mathbb{R}$ such that $$(\xi x +n) \cdot x = \text{const}$$ Here $n$ is the unit normal to the curve. Is ...
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117 views

Examples of functions whose arc-length from the origin is given by their derivative

I'm looking for functions $y:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\int_{0}^{a} \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx = \frac{dy}{dx}\Bigg|_{a}$$ (this kind of feels like a ...
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5k views

Equation of a plane from 2 lines

I have two lines with the following equation $$D1 : (x, y, z) = (2,0,0) + k(0,3,0)$$ $$D2 : (x, y, z) = (2,0,2) + k(0,0,1)$$ and I must find out the equation of the plane that they make. I ...
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47 views

Are all rationally parametrized plane curves algebraic? How does one find their degree?

Suppose a plane curve is given parametrically by $x=p(t),y=q(t)$, where $p,q$ are rational functions. I originally assumed that this means that the parametrized curve is algebraic, i.e. that it is the ...
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22 views

Prove that for any piecewise smooth curve it is possible to find the parametrisation

Prove that for any piecewise smooth curve it is possible to find the parametrisation $\phi$ that is consistent with its length, ie. length of a curve segment between $\phi(a)$ and $\phi(b)$ is equal ...
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500 views

Equation of the locus of centre of the ellipse?

An ellipse slides between two perpendicular lines. To which family does the locus of the centre of the ellipse belong to?
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37 views

Geodesics on surface of revolution of regular curve

I was recently presented with this in differential geometry stating the following: Let us define the regular curve on the XZ plane as: $ \gamma (t) = (sin(t)+2,0,t) $ on XZ plane for $ t \in R $, ...
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40 views

Can a curve cross its asymptote infinitely many times?

Can a curve cross infinitely many times its asymptote? If so, is there a special name for this behaviour?
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Can a surface of revolution be built from a self-intersected curve?

I'm reading "Differential Geometry of Curves And Surfaces" of Manfredo Do Carmo. There's a point in his book about Surfaces of Revolution which confuses me a lot. Here is the part: The part ...
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44 views

Problem about curves. A particle is running along circumference $x^2+y^2=25$

I'm considering a problem about curves. A particle is running along circumference $$x^2+y^2=25$$ with a costant modulus speed compliting a turn in 2 second. I need to determinate the acceleration in ...
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What's so special about involute curves??

An involute curve (specifically, an involute of a circle) is very commonly used to define the shape of the teeth on a gear. Apparently this idea goes back to Euler. Why is this? What special ...
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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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32 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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35 views

Calculating double integral $\iint_{D}xy\,{\rm d}x\,{\rm d}y$ where $D$ is the plane limited by lines $y+x=1$, $y=0$, $x=0$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $a$ and graded ...
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20 views

Intersection Multiplicity of Rational Plane Curves

Suppose I have two rational curves in the complex projective plane. I know their parametrizations, $<x_1(t),y_1(t)>$ and $<x_2(t),y_2(t)>$ I know I can use Grobner bases to find an ...
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1answer
53 views

How do you find the angle of intersection between two given polar curves?

How does one find the angle of intersection between two given polar curves? For example, between $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
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Intersection point of the normal lines at α(t), α(t+h) converges as h→∞ given α is parametrized by arc lenght and it´s curvature is non zero

Let α(t):I→R2 be a curve parametrized by arc lenght and k(t) (curvaure) be non zero. Need to show the intersection point of the normal lines at α(t), α(t+h) converges to a point in the trace of the ...
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51 views

Does every graph have an algebraic form?

Let's say I take a pencil and start drawing a curve on $xy$ plane. The curve is continuous and for each value of $x$ there is only one corresponding value of $y$. So question that interests me is - ...
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Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
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176 views

Measure of curve smoothness

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness? I was thinking that since curvature measures how fast a curve changes, ...
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38 views

Vector analysis: understanding formulas for normal and tangent

I'm studying a chapter on vector analysis and I don't really understand why the following equations are switched for the two situations. (My course is not in English, so I apologize if I translated ...
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41 views

Set-theoretic equality in double dual graph

Someone in the math fandom on tumblr was explaining the concept of a dual graph, and he was asked a tough question: Is the dual of the dual [equal to] the original graph always, or just ...
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Why is this the equation of the tangent plane?

I want to find the equation of the tangent plane of the surface patch $\sigma (r, \theta)=(r\cosh \theta , r\sinh \theta , r^2)$ at the point $(1,0,1)$. I have done the following: The point ...
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36 views

Problem with $arg(\gamma (t))$

I am see my notes about curves on complex spaces and I do not understand why it is so... I need help. I do not understand what way take to do it, I need someone explain slowly please, thanks
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Given two tangents $\varepsilon_{1},\varepsilon_{2}$ of the curve $c_{1}$, on two specific points $x_1, x_2$, find the tangents

Let $\varepsilon_1,\varepsilon_2$ be the equations of the tangents to the curve $c: y = 1+x^2$, on the points $x_1 = 2$ and $x_2 = -2$, respectively. Prove that the equations for the two tangents ...