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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52 views

Rotations around the origin

The problem is to find the number of rotations around the origin for the function $$f(z)=z^{2013}+2z+1 $$ when $z$ moves through $\left\{|z|=1\right\}$. I tried to solve it with the help of argument ...
2
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2answers
930 views

Adjustable Sigmoid Curve (S-Curve) from (0,0) to (1,1)

I feel like this is such a simple question but I am at such a loss. I currently have a set of values that I would like to weigh by an S Curve. My data ranges from $0$ to $1$ and never leaves those ...
2
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1answer
133 views

Support function of a convex domain

Let $Q$ be a compact, convex domain in the plane, with smooth boundary $\partial Q$. We further assume that the origin is contained in $Q$. For a concrete example, let's take an ellipse ...
2
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1answer
131 views

Equation for a distorted circle

When you view a circle posted on a wall at a distance and at a glancing angle, the circle elongates. However, I don't think it is just an ellipse because it will also become asymmetric. It has more of ...
2
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2answers
118 views

Self-intersection of vector valued function

A vector valued function $r(t)$ has the following coordinates: $$x = 4\cos\left(\frac12t\right)+2\cos(2t)+\cos(4t)\\ y = 4\sin\left(\frac12t\right)+2\sin(2t)+\sin(4t)$$ I have to find the $t$-values ...
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0answers
46 views

Approximately space-filling parametric curves from unit line to n-cube

I am trying to find the solution of TSP in 2D Euclidean space using parametric curves of two interpolation polynomials constructed so that at times $t_k$ the curve passes through point $p_t$. See my ...
5
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0answers
114 views

curve which crosses itself at every point

Does there exist a continuous curve $C$ on $\mathbb{R}^2$ that crosses itself at every point on $C$? For two pieces of curve to cross and not just touch, each must have some length either side ...
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1answer
46 views

Minimum/Maximum Extremas in a Plane

Explain one way in which extrema of a function of two variables f(x,y) defined on a closed, bounded region of S of plane differ from extrema of a function of one variable f(x) defined on a closed ...
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0answers
129 views

What is the parametric equations for the following closed curves?

First case Second case For the sake of simplicity, let the circle of radius $R$ be at the origin, the rectangle width and height be $w$ and $h$, respectively.
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2answers
2k views

LOVE +MATH = can you read this formula?

i don't remember where exactly, i found in internet this image: i tried to replicate the formula with python and i tried this: ...
2
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1answer
93 views

Constructing shortest interpolation curve from points in $\mathbb R^2$ with parametric equations.

Assume we are given a set of $n$ points from $\mathbb R^2$, $(x_1,y_1),(x_2,y_2)\dots(x_n,y_n)$. We want to construct a path connecting all these points using a pair of parametric equations ...
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0answers
52 views

Evaluate $\|f\|$ on complex domain

Given a function $ f\colon \mathbb{C} \rightarrow \mathbb{C}$, how do I find $||f||_{\operatorname{Trace}(\gamma)}$, where $\gamma$ is a curve in the complex plane? In particular, I need to do this ...
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2answers
245 views

General way to find out whether a curve is positively oriented

I have a general question, that deals with the question how I am able to find out whether a particular curve(in $\mathbb{C}$) is positively oriented? Take e.g. $ y(t)=a+re^{it}$. Obviously this one ...
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1answer
116 views

What is the general formula for NURBS curves?

Give me the general mathematical formula for NURBS curves, with special cases (B-spline and Bézier curves)
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1answer
96 views

Fourier transform for a 2D curve

I am stuck on the following problem about a Fourier transform of a 2D curve: I have to calculate the Fourier transform (using 1D complex FT) (and the opposite of it) for a 2D curve z(t). The curve is ...
4
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3answers
437 views

Where is Greens theorem used?

Where is Greens theorem used? I think it's weird going from a vector field to calculating a volume on a scalar field, where do we use this kind of calculation?
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0answers
18 views

How can I extend Moore curves to more than $d\geq3$

I know the Moore curves are similar to Hilbert curves, but I cannot find an extension for $d\geq 3$. If there is such an extension. In fact, I want a quick indexing solution for $d\geq 3$. Any ...
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1answer
115 views

Parallel to line on $f(x)=1+\sin(x)/x$

I want to draw a curve on the top of the function $f(x)=1+\dfrac{\sin(x)}{x}$, but the curve should be equidistant (perpendicular distance from any point of the function $f(x)=1+\dfrac{\sin(x)}{x}$) ...
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3answers
147 views

Arc Length Formulas

Use the arc length formula to find the arc length of the upper half of the circle with center at $(0,0)$ and radius $3$. Also, find the arc length of the curve in the first question by using ...
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2answers
96 views

Can a set of non self-intersection points of a space-filling curve contain an arc?

Consider a continuous surjection $f:[0,1]\to[0,1]\times[0,1]$. It can be proved that set of self-intersection points must be dense. In the Hilbert curve, the set of self-intersections are points ...
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1answer
99 views

Is each space filling curve everywhere self-intersecting?

Consider a continuous surjection $f:[0,1]\to [0,1]\times[0,1]$. Is $$\{x:\exists(t_1\not=t_2) f(t_1)=f(t_2)=x\}=[0,1]\times[0,1]?$$
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1answer
160 views

Pursuit curves and arc length question

I am studying pursuit curves where a fast pirate ship which pursues a heavily laden treasure ship which tracks along a straight line. The ratio of the speeds of the ships is r > 1 (which is fixed) and ...
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0answers
104 views

Canonical form of a curve (geometry)

I am bothering with this geometric problem more than half a day and couldn't understand it yet. Here it is: In orthonormal coordinate system K=Oxy we have a curve C: $9x^2 - 4xy + 6y^2 + 6x - 8y + ...
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1answer
195 views

polar graphs and investigation

I am new to polar graphs and I am trying to investigate some certain cases: What happens when you change the $b$ value to different positive integers in polar equations of the forms: ...
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12answers
9k views

Derivation of the formula for the vertex of a Parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $y = a x^2 + b x + c$ My teacher gave me the formula: $x = -\frac{b}{2a}$ as the $x$ ...
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0answers
86 views

Curve genereated by a hanging rope with both ends hold together

So a hanging rope assumes the form of a catenary, but if the ends are together it takes some kind of "drop" shape. I think that if the rope is perfectly flexible the shape is a vertical line but what ...
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0answers
71 views

How to extend an interval to a circle in $\mathbb{R}^2$

Let $$\gamma : [0,1] \rightarrow \mathbb{R}^2$$ be a $C^0$ imbedding. How can I show that there exists another imbedding $$\eta : [0,1] \rightarrow \mathbb{R}^2$$ with $\eta ((0,1)) \subset ...
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2answers
122 views

Proof of Astroid?

How can I prove that an astroid is an envelope of all line segments of length 1 from the x-axis to the y-axis? I read one proof of this online at the link ...
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6answers
1k views

Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?

Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
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0answers
47 views

Jordan curves, its interiors and the existence of a continuous function.

Let $L_t(s):S^1 \rightarrow \mathbb{R}^2(t\in [0,1])$ is a Jordan curve, $O(t)$ is its interior and $H(t,s)=L_t(s)$. If $H$ is a homotopy from $L_0(s)$ to$L_1(s)$, is there exists a continuous ...
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2answers
248 views

Fitting a set of points in a plane to a smooth curve obtained by joining a half-line and an arc of a circle

I have a set of points in the plane and I want to find a curve that best fits these points (e.g., in a least squares manner, or using some other convenient "measure"). I want that the curve be either: ...
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3answers
121 views

when is the region bounded by a Jordan curve “skinny”?

How can I formalize and prove the following intuition?: Picture a very skinny rectangle, one with base length 1 and sides length $\epsilon$. Or imagine a very flattened ellipse. The interiors of ...
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2answers
86 views

why is an annulus close to it's boundary when it's boundary curves are close?

This is the motivating question for the rather vague question here: when is the region bounded by a Jordan curve "skinny"? Suppose we are given two Jordan curves in the plane, one inside ...
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1answer
107 views

Relating the curvature of a plane curve to the curvature of a stretched version

Let $\theta : I \to \mathbb{R}^2$ be a regular plane curve with curvature $ |k_{\theta}|\leq1$ everywhere. We now define a curve $\theta_{d}$ by stretching $\theta$ in one direction, i.e., $\theta = ...
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1answer
234 views

Problem with calculating a winding number

I have a problem with calculating the winding number $n\left ( \gamma ,\frac{1}{3} \right )$ of the curve $\gamma :\left [ 0,2\pi \right ]\rightarrow \mathbb{C}, t \mapsto \sin(2t)+i\sin(3t)$. ...
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1answer
57 views

Definition of regular point of a boundary with planar brownian motion

This is an exercise in G.Lawler's book Conformally invariant processes in the plane. First he defined regular point of a boundary using brownian motion: Suppose $D$ is a domain in $\mathbb{C}$ with ...
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2answers
61 views

Prove using an example that there is no plane on R3 that contains every group of 4 points

Well, this is a homewrok question (which I know I should not be asking, but I cannot find an answer to this anywhere): The exercise is as follows: i) Find the equation of the plane of R3 that ...
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1answer
39 views

Define together with the $x$-axis an area.

The curves $y = \sqrt{2x+3}$ and $y = x$ define together with the $x$-axis an area. Determine the exact value of the specific area. How do you solve?
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53 views

Why can I write the curve shortening flow system as..

Consider the family of curves $$\gamma:S^1\times [0, T)\rightarrow \mathbb R^2, \gamma(u, t)=(x(u, t), y(u, t)),$$ where $u$ parametrizes the trace of the curve and $t$ parametrizes which curve is ...
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0answers
38 views

Solving the algebraic equations .

I am working with an equation to find the singular points in $\mathbb P^2 (\mathbb C)$ . Basically after taking the partial derivatives and doing some manipulations it reduces to $$y^2 + (2-k)xz ...
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197 views

Generating an equation from an image I have

I am not exactly sure if this question belongs here but I could not think of a better place to ask. So I recently discovered that various people on the internet have created equations for rather ...
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1answer
50 views

Finding a curve with a condition on winding numbers

I want to find a continuous and closed curve $\gamma$ so that the map $\nu_{\gamma}:\mathbb{C}$\Im$(\gamma)\to \mathbb{Z}$ takes infintely many values. Here $\nu_{\gamma}(a)$ is the winding number ...
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2answers
515 views

A hyperbola as a constant difference of distances

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, the distance between the two ...
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1answer
951 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
326 views

Circumference of a superellipse?

Could someone help me formulate the circumference of a superellipse? $$\frac{x^n}{a^n} + \frac{y^n}{b^n} = 1$$ If it makes things easier, I'm considering only the cases $n>2$, and ...
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1answer
53 views

Convex Curve Parametrization

How can I parametrize a convex plane curve using the angle $\theta$ between the tangent line and the $x$-axis?
4
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2answers
108 views

Circumference parametrization

Let $C=\{(x,y)\in \Bbb R^2: (x-x_0)^2+(y-y_0)^2=r^2\}$ and let $\varphi :[0,2\pi]\to \mathbb{R}^2$, $\theta \mapsto (x_o+r\cos \theta, y_0+r\sin \theta)$, with $r>0$. I'm trying to prove that ...
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0answers
168 views

Symbolic integration of vector norm

I'd like to symbolically integrate the expression $\int_0^1{\|r'\left(t\right)\|_2\,dt}$ where $r$ is a function $\mathbb{R} \rightarrow \mathbb{R}^2$ (so the expression is the arc length of the curve ...
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2answers
616 views

Find area of a simple, smooth, closed curve lying in a plane

I was given this question in class and I assume it is a spin off of Green's theorem for finding the area of a closed curve $\lambda$ in 2D but expanded to 3D I believe. Anyways I am pretty confused ...
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1answer
804 views

How to calculate the area between 2 polar curves: $r=\frac{4}{2}-\sin\theta$ and $r=3\sin\theta$?

How to calculate the area between 2 polar curves: $r=2-\sin\theta$ and $r=3\sin\theta$? I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ...