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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Examples of smooth fractals

A classic example of a fractal curve is the Koch Snowflake. This is a topological manifold (as opposed to many other fractals which are not), but it also clearly not smooth. Question: Are there ...
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689 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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36 views

Proof of Netto's theorem

I am trying to show any bijective mapping $f:I \to \mathcal{Q}$, where I is the unit interval in $\mathbb{R}$ and $\mathcal{Q}$ is the unit square, is necessarily discontinuous. How do I go about ...
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1answer
30 views

Reference request: an analytical proof the Hilbert space filling curve is nowhere differentiable

I am studying space filling curves and I am using the Hans Sagan book. I am trying to understand the nowhere differentiability of the Hilbert curve presented in this book but it does not seem to ...
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21 views

Proving the Hilbert space filling curve is nowhere differentiable. [duplicate]

I am trying to understand the proof the Hilbert curve is nowhere differentiable. The Hilbert curve is defined as the mapping $f_h: I \to \mathcal{Q}$ where I in the unit interval in $\mathbb{R}$ and ...
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25 views

Text explanation: Ellipses and their intersection points

Given two ellipses $E_1,E_2$ of equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=4$$ prove that for all $p\in E_1$ there exist a unique ellipse $F_p$ that meets $...
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53 views

Looking for a family of astroids

I'm wondering what's the formula for a family of curves. Specifically the astroid. A few requirements: There should be one main one and then a bunch of them nestled inside. At each of the cusp-...
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1answer
27 views

Exercise $1.20$ from Montiel and Ros: Curves and Surfaces

Let $\vec{\alpha}:I\longrightarrow \mathbb{R^2}$ be a curve parametrized by arc lenght. If there is a differentiable function $\theta:I\longrightarrow \mathbb{R}$ such that $\theta(s)$ is the angle ...
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45 views

Tangent vector of a curve

Let $\vec{\sigma}:[a,b]\longrightarrow \mathbb{R}^2$ be a regular and closed curve of class $C^1$, parametrized respect to the arc lenght. Is true that the map $\vec{\sigma}':[a,b]\longrightarrow S^1$ ...
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13 views

Geometric generation of the Peano curve

Peano defined a continuous mapping from $I$ to the square $\mathcal{Q}$ defined by $f_p : I \to \mathcal{Q}$ This mapping is defined by the continuous and surjective operator \begin{equation} kt_j=...
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What does it mean to say a point is uniquely mapped?

I am looking at space filling curves. Essentially their is a mapping $f: I \to \mathcal{Q}$ where I is an interval in $\mathbb{R}$ such as $[0,1]$ and $\mathcal{Q}$ is a square $[0,1]^2$. For the ...
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1answer
42 views

Hypocycloid - Direction of circle's rotation and revolution

Ive been trying to derive the equation of a hypocycloid. I am confused with one thing, in the hypocycloid is there a define direction of rotation and revolution of the smaller circle? (by direction I ...
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29 views

Homotopy of closed curves is also a closed curve?

I'm trying to prove the following statement: Let $\gamma_1$ and $\gamma_2$ be two closed curves from $[a, b]$ to $\mathbb{C}$, and let $h: [a,b]\times[0,1] \to \mathbb{C}$ be a homotopy between ...
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1answer
21 views

Arcwise and pathwise connectivity in space filling curves

We know that a space filling curve is not injective from Netto's theorem. We know that a Peano space is a compact, connected, locally connected metric space. Essentially in pathwise connectivity ...
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1answer
29 views

How to determine if the implicit curve is closed?

Let the implicit equation $$F(x,y)=0, \quad (x,y)\in\mathbb{R}^2$$ defines a curve $\gamma$. The question is what properties must have the function $F$, s.t. the curve $\gamma$ be topologically ...
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1answer
25 views

Is there a difference between arc-wise connectivity or path-wise connectivity?

When authors refer to arc-wise connectivity, do they mean path-wise connectivity? I am studying space filling curves and when reading books, I either come across the concept of arc-wise connectivity ...
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1answer
36 views

Continuous image of a cantor set and other space filling curves

I am studying the theory of space filling curves, more specifically looking at the continuous image of a Cantor set. I have been given this definition to characterise the continuous image of a Cantor ...
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51 views

The shapes of general lemniscates (i.e., Cassinian curves) on the complex plane

On the complex plane, curves given by an equation of the form: $$ |z-z_1|\cdot |z-z_2| \cdots |z-z_n| = C $$ with $ C \gt 0$, are known as general lemniscates, or Cassinian curves with $n$ foci. I ...
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1answer
48 views

Approximation of Jordan curves

Let $J$ be a Jordan curve in the plane, looked upon as the homeomorphic image of the unit circle T. Suppose that for some $\epsilon>0$ there is another Jordan curve $K$ such that $K\subseteq V:= \...
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31 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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1answer
40 views

Algebraic Curve

I know that a curve of the type $$\vec{\sigma}(t)=\cos(mt)\hat e_1+\cos(nt)\hat e_2$$ with $m,n\in\mathbb{Z}$ is algebraic. My question is: what is the polynomial that define this curve?
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20 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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2answers
40 views

Can a curve be rotated and translated to 'fit inside itself'?

Working in $R^2$, consider any continuous curve $C$ with endpoints $a$ and $b$, such that the curve does not intersect the line formed by $a$ and $b$. Then the line $ab$ and $C$ form an enclosure, $E$....
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2answers
126 views

3-D Geometry Problem. Find a curve which touches the straight line.

If two perpendicular tangent planes to paraboloid $x^{2}+y^{2}=2z$ internsects in a straight line in the plane $x=0$, obtain the curve to which the straight line touches. I don't know how to ...
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36 views

Is the plane curve $y^3=x^4+x^3$ an irreducible algebraic affine set?

I'm dealing with the plane curve $C=\{(x,y)\in k^2:y^3=x^4+x^3\}$. I want to know if this curve is irreducible, where $k$ is a commutative field. I know this is equivalent to the ideal $\sqrt{I}$ ...
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19 views

Direction of a curve given by ODE

Let $a,b \in C(\mathbb{R^2})$ be bounded and $(x_0,y_0) \in \partial B_1(0)$. Consider the ODE system $$ \begin{cases} x'(t)=a(x,y) \\ y'(t)=b(x,y) \\ x(0)=x_0 \quad y(0)=y_0 \end{cases} $$ We know ...
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1answer
133 views

Envelope of family of curves $x(u,v)=\cos^2(u)\cos(v)+\cos(u)\sin(u)\sin(v)$, $y(u,v)=\cos^2(u)\sin(v)-\cos (u)\sin(u)\cos(v)$

How to generally find singular solution or envelope of a two parameter family of curves $ x(u,v),y(u,v) $ in the plane? The parametric equations $$x(u,v) = \cos^2 (u) \cos (v) + \cos (u) \sin (u) \...
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1answer
26 views

Curve in a disk

I take a curve $\vec\gamma:[a,b]\longrightarrow \Delta$, where $\Delta \subseteq \mathbb{R}^2$ is the disk of radius $r>0$. If the curve has length $L>0$ does exist an upper bound (in terms of $...
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Deviation of a plane curve from the $x$-axis

I have a smooth plane curve $\gamma$ enclosed in a circle of radius $R$. See Figure 1. I'm interested in measuring the deviation of this curve from the $x$ axis, denoted $\Delta(t)$ in the figure, as ...
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Determine equation of tangent plane?

Determine equation of tangent plane in points $(\frac{1}{2},1,f(\frac{1}{2},1) )$ $f(x,y)=x^{4}-x^{2}+y^{2}$ I know usually how these examples work, but I am confused with these $3$ points. I have ...
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1answer
91 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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1answer
46 views

The plane curve of maximum average speed under constant gravitational force

A line gives us the minimum distance from $A$ to $B$. A cycloid gives us the minimum traveling time of a point mass from $A$ to $B$ (under constant gravitational acceleration $g$). What about the ...
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36 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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2answers
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What is the definition of a path along a multivariable function?

I'm taking a class equivalent of Calculus III, and we saw how to prove continuity of a multivariable function. Recently we looked at the following example: \begin{align} f(x,y) = \begin{cases} \frac{...
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1answer
957 views

S-shaped Reverse Logistic Curve

Is there any curve that grows very slow at the beginning then growth picks up exponentially before hitting the wall. I need sort of reverse behavior of the logistic curve. Here's raw idea how it ...
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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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1answer
97 views

How To Write The equation for a line given a set of co-ordinates

I'm trying to learn how can I write the equation for a line given all the points that belongs in the line. I'm looking to find the equation for a curve. An example set of points is: $$\{ (24,11), (...
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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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21 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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1answer
36 views

Level curve of the function $f(x,y)=\min\{x^2+y^2,xy\}$ [closed]

How I can find the level curve of the function $$f(x,y)=\min\{x^2+y^2,xy\}$$ From where I need to start to solve this problem? Thank you!
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1answer
24 views

Path from a start point at a certain heading to an end point at a certain heading while obeying a minimum turn radius

So not sure if my title is clear (also no idea what to tag, because you need 1000 rep to add tags) so I will try my best to explain the problem. I'm working in 2D space and to simplify the problem, I'...
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How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
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Injective curve

How can i show that the curve $\gamma:\mathbb{R}\to\mathbb{R}^2$ defined by \begin{equation} \gamma(t)=\left(\frac{t}{1+t^2},\frac{t}{1+t^4}\right) \end{equation} is injective (without using algebraic ...
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1answer
41 views

Does there exist a curve with non zero area?

Does there exist a curve with a bounded infinitely diffrentiable derivative (i.e. has a minimum |curvature|) of hausdorf demension 2? Or even a diffrentiable curve of hausdorf demension 2?
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Index of a Jordan curve

Winding number theorem: If $J\subset \mathbb{C}$ is a Jordan curve and a point $z$ lies in its interior domain, then the winding number $n(J,z)=\pm 1$. Now suppose that $J$ is smooth and we have the ...
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Blowing up a model for a plane curve

Let $R = \mathbb{C}[[t]]$ and let $\mathcal{X} \hookrightarrow \mathbb{P}_R^2$ be the arithmetic surface defined by the equation $$(X^2 - 2Y^2 + Z^2)(X^2 - Z^2) + tY^3Z = 0.$$ The generic fiber ...
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How to prove this inequality about the arc-lenght of convex functions?

Let be $f,F:[0,1] \to \mathbb{R}$ with $f,F \in C^2([0,1])$ two convex functions such that $f \le F$ in all points and $f(0)=F(0)$, $f(1)=F(1)$. Considering the plane-curves $\gamma(t)=(t,f(t))$ and $...
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135 views

converse to the jordan curve theorem

Suppose $K\subset \mathbb{R}^2$ is compact and locally connected, and does not contain 0. Let $A$ be the component of $\mathbb{R}^2-K$ containing 0, and let $B$ be the unbounded component. Assume ...
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1answer
36 views

find a closed curve with given winding numbers

Is there a closed (piecewise) $C^1$ curve $\gamma: [a,b]\rightarrow\mathbb{C}$, so that $\mathbb{C}\setminus\gamma([a,b])$ has four components with winding numbers -1, 0, 2, 3? Thanks!
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Parametric equation of $x^y=y^x$ curve

What is the easiest/most natural way to parametrize the following curve? $$\{(x,y)\in\Bbb R^{+2}\mid x^y=y^x, x\neq y\}\cup\{(e,e)\}$$ The best I could do was taking it apart, and for $x>y$ use $...