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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find the intersection of plane and sphere

If the equation of the sphere is $x^2+y^2+z^2=1$ and the plane is $x+y+z=1$, then how can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the ...
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44 views

Intersection of sphere and ellipsoid

Ellipsoid: $$ x^2+\frac{y^2}{4}+\frac{z^2}{2}=1 $$ Sphere: $$ x^2+(y-1)^2+(z-d)^2=1 $$ For what values of $d$, there is a common tangent plane to both curves? Part of my resolution: Consider ...
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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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Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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654 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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65 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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1answer
43 views

Finding family of curve for given asymptotes

I need to find possible curves, with asymptotes given as $x=0 (x \to -\infty)$ and $y=mx \hspace{0.5cm} m>0$. it is easy to find curves for individual lines, $y= \exp(-\lambda_1 x) + mx$ for $y=mx$ ...
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Closed subset of $\mathbb R^2$ disconnecting $\mathbb R^2$ must contain a curve?

Edit: Per Alex S's answer, my first statement was false, so I've moved it to the bottom. Here is the weaker statement that I am trying to prove (this was included in my original question): Let $C ...
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1answer
14 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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Dual curve of an algebraic curve in affine coordinates

$F$ is an irreductible algebraic curve and we consider the application that sends a non-singular point $(a : b : c)$ in homogenius coordinates, to $\phi(a, b, c) = \bigl(f_x(a, b, c), f_y(a, b, c), ...
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25 views

How to draw cubic plane curve?

In Python, using MatPlotLib, given [vector] parameters $a$ and $b$ and [scalar] parameter $c$, I want to draw a general cubic plane curve in 2-dimensional space (regular plane with $x$ and $y$ axes): ...
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2answers
96 views

What is the Hilbert curve's equation?!

The Hilbert curve has always bugged me because it had no closed equation or function that I could find. What is its equation or function? For example, if I wanted to find the Hilbert's curve point at ...
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1answer
64 views

convex curve as boundary of a convex set

In short, my question is to know if the following statement is true, and how to show it : Theorem Let $\gamma$ be a closed simple $C^2$ convex curve in $\mathbb{R}^2$. We denote $\Gamma$ its image. ...
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1answer
53 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 ...
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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 ...
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12 views

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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10answers
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Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
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71 views

Proof of the formula that computes the genus of smooth projective plane curve

I was searching for a proof of the formula that computes the genus of a smooth projective plane curve of degree $d$: $$g = \frac{(d-1)(d-2)}{2}$$ which do not make use neither of triangulation or ...
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2answers
24 views

Dichotomy in the number of regions on a plane formed by an infinite number of lines

I'm reading Knuth's Concrete Mathematics and we are dealing with recurrence relations. He proves that the number of regions $L_n$ formed by $n$ lines on a plane is $L_n=\frac{n(n+1)}{2}$. I don't ...
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1answer
23 views

What is the rotation index of a figure 8?

Is it 0 since the total turning angle covers one clockwise circle and one counterclockwise circle thus making the total 0 and the rotation index 0?
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1answer
19 views

Existence and uniqueness of a point with horizontal tangent in a convex curve

I had a look on the proof by E. Schmidt of the Schur's Theorem about arcs of convex curves. It states the following: Let $C$ and $C'$ be two arcs of the same length with the endpoints $a,b,a',b'$ ...
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24 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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17 views

3D Denjoy–Riesz theorem

The Denjoy–Riesz theorem states that every totally disconnected subset of $\Bbb R^2$ is the subset of a Jordan arc. Is this true in $\Bbb R^3$? Originally I thought Antoine's necklace would be a ...
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61 views

Curve equidistant to sine and cosine.

If I have the sine and cosine curves plotted, what would be the formula of the curve that is equidistant to both curves? Here's a picture of how it looks like. The original question comes from a ...
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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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40 views

Understand if a curve is parametrized by arc length or not

Show that the curve $$\alpha(t)=(t,1+\frac{1}{t},\frac{1}{t}-t), \quad t\in(0,\infty)$$ is a plane curve. I know $\tau$ must be zero for curve being plane. However, I want to determine the ...
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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 ...
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61 views

Find the highest point on curve defined by intersection of the graph of $g(x,y) = \sqrt{xy}$ and plane $x+y-1=0$

So far this is what I have done: $$F(x,y) = \sqrt{xy} + λ(x+y-1) =0$$ $$F_x = \frac12(xy)^\left(\frac{-1}{2}\right).y + λ=0$$ $$F_y = \frac12(xy)^\left(\frac{-1}{2}\right).x + λ=0$$ $$Fλ = ...
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30 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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73 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?
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31 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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1answer
50 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?
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1answer
31 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 ...
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40 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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148 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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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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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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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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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 ...
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138 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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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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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} \, + \, ...