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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77 views

Can every possible curve be expressed mathematically [closed]

Can every possible curve/parabola shown on a graph (for example $x^2$ or somthing much more complicated) be expressed in an equation like $y=x^2$. Or are there some lines you can't express?
5
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1answer
96 views

Prove this : $\left(a\cos\alpha\right)^n + \left(b\sin\alpha\right)^n = p^n$

I have this question: If the line $x\cos\alpha + y\sin\alpha = p$ touches the curve $\left(\frac{x}{a}\right)^\frac{n}{n - 1} + \left(\frac{y}{b}\right)^\frac{n}{n - 1} = 1$ then prove that $\left(a\...
0
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32 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 ...
2
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1answer
56 views

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?
3
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0answers
46 views

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 $F(\...
1
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1answer
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 ...
2
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1answer
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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0answers
164 views

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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0answers
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 ...
1
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1answer
33 views

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 edge ...
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2answers
23 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 ...
0
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1answer
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: $$\frac{ds}{dt}^...
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0answers
26 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 ...
0
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0answers
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) $ ...
2
votes
1answer
50 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 ...
1
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1answer
54 views

Path integral over Koch curve

If a curve $C$ is a Koch curve (snowflake fractal) centered at origo, then is the following path integral defined? $$\int_C \frac{1}{z}\space dz.$$ If it exists, the value must be $2\pi i$ for ...
2
votes
2answers
178 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 ...
0
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1answer
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 $\mathbb{...
0
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0answers
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 ...
1
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1answer
40 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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0answers
51 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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0answers
39 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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0answers
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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0answers
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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1answer
18 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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0answers
38 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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0answers
38 views

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 ...
0
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1answer
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 ...
2
votes
1answer
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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1answer
41 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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2answers
48 views

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 ...
0
votes
2answers
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 ...
2
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0answers
47 views

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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0answers
39 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 $f:D^2\to\mathbb R^2$ ...
1
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1answer
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 $A(0;4)$, $B(6;4)$, $C(1;-1)$. ...
1
vote
2answers
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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0answers
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 ...
0
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1answer
63 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 $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
3
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1answer
48 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 ...
0
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0answers
20 views

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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vote
3answers
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 - ...
6
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53 views

Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
2
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2answers
39 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 ...
4
votes
2answers
42 views

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 $(1,0,...
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2answers
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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0answers
70 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 ...
0
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0answers
26 views

Is this the correct way to compute the blow up of a curve

I'm trying to calculate the blowup of the curve $y^5=z^2-3z^3+2z^4$ at $(0,0)$ We have the relation $Ay=Bz$, now I split it into two charts: The first chart$(y,a=A/B)$: $y^5=a^2y^2-3a^3y^3+2a^4y^2-y^...
2
votes
3answers
47 views

Circle of radius of Intersection of Plane and Sphere

The plane $x+2y-z=4$ cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius? I tried putting value of y from plane in sphere but then I get a $zx$ term. How to proceed?
0
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0answers
37 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. ...
2
votes
1answer
45 views

When is $t \mapsto \gamma (t) = \big( (1 + a \cos t) \cos t, (1 + a \cos t) \sin t \big)$ simple and closed?

Show that $\gamma (t) = \big( (1 + a \cos t) \cos t, (1 + a \cos t) \sin t \big), t \in [0, 2\pi]$, where $a$ is a constant, is a simple closed curve if $|a| < 1$ , but that if $|a| > 1$ its ...