Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [plane-curves]

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.

Plane curves are continuous (or smooth) functions $\gamma\colon I\to\mathbb R^2$ from a real interval to the plane. Sometimes the image $\gamma(I)$ is also called a curve.

1306 questions
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A curve is a circle or a line

Let $\gamma(t):\mathbb{R}\to\mathbb{R}^2$ be a continuous curve in the plane such that for every $t_1,t_2\in\mathbb{R}$ the euclidean distance $d(\gamma(t_1),\gamma(t_2))$ depends only on $|t_1-t_2|$. Must the curve be a circle or a line? I believe…
Tommy1234
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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 $y(x)=\sqrt[x]{x}^{\sqrt[x]{x}^{\sqrt[x]{x}^{.....}}}$,…
Dave
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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 of the other. $\mathbb{R}$ cannot be partitioned into…
Angela Pretorius
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Space-Filling Jordan Curve

My question is about a simple closed curve that is also a space-filling curve. The figure shows 6 iterations of the formation of a Hilbert curve (limit), whose trace is a solid square. I think we may, at each iteration, connect the endpoints of the…
João Alves Jr.
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Can every possible curve be expressed mathematically

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?
Vent
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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 an egg-shape. (Please correct me if this…
kdaquila
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A plane curve without self-intersection has empty interior

Let $\alpha:[0,1]\rightarrow \mathbb{R}^2$ be continuous and injective. I want to prove that $\alpha([0,1])$ has empty interior. One way to show this I have seen in a question here in MSE (but I can't find it now to link it to the post) is to…
Curious
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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-points, all of them are exactly at the (0,1),…
user3832863
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What is the name of this shape?

What is this shape called?
TheBigO
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What curve is always perpendicular to a constant force under rotation?

[I have created a GeoGebra link to the system I describe below. It helps a lot in understanding.] Assume I have a 2D body whose border is a curve. This body is pivoting about a point $(0,0)$. A constant vertical downward force is applied on …
Disousa
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2D trajectory in minimum amount of time given min/max acceleration per axis

I am having a little problem with determining a trajectory. I have a 2D curve, say $\{x(\lambda), y(\lambda)\}$ where $\lambda$ is not time; it is only a parameter that describes the evolution of the curve. The point can move with a max acceleration…
Jepessen
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How do I use k-dimensional planes as bounds for generating k-dimensional vectors?

I am an essentially self-trained programmer with little mathematical background. I do not quite know where to start for this problem, and do not know the terminology to help me get conclusions researching the answer myself. My efforts have lead me…
sadakatsu
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Parameterization of simple closed curve

A curve $Z$ in a two dimensional space is parametrized by $0\leq t < 1$ , and satisfies $Z(t) = Z(t+1)$. If it is sufficiently well behaved, it can be represented using a Fourier series with a basic frequency of $1$. So far so good, but this…
user1834069
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When is a curve smooth at points where $dy/dx$ does not exist?

The curve $y=x^{1/3}$ is smooth everywhere even though $dy/dx$ does not exist at $x=0$. Why? In general; Wherever $dy/dx$ does not exist on a curve, how can I show that it could still be smooth at those places?
Jostein Trondal
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Self intersection of the implicit curve $x^y-y^x=0$

Watching the graph of the curve defined by $x^y=y^x$, which contains the line $y=x$, I noticed that the line intersects the curve itself only at one point that looks to be $(e;\;e)$. How can I formally prove this?
Raffaele
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