# Tagged Questions

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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### 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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### 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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### 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 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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: ...
$F$ is an irreductible algebraic curve and we consider the application that sends a non-singular point $(a : b : c)$ in homogenius coordinates, to $\phi(a, b, c) = \bigl(f_x(a, b, c), f_y(a, b, c), ... 2answers 51 views ### Closed subset of$\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 \...
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): \$...