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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2answers
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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 ...
0
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1answer
34 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 ...
2
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0answers
28 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 ...
2
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1answer
29 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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0answers
26 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
$$y^2 + (2-k)xz ...
0
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0answers
45 views
Generating an equation from an image I have
I am not exactly sure if this question belongs here but I could not think of a better place to ask.
So I recently discovered that various people on the internet have created equations for rather ...
0
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1answer
39 views
Finding a curve with a condition on winding numbers
I want to find a continuous and closed curve $\gamma$ so that the map $\nu_{\gamma}:\mathbb{C}$\Im$(\gamma)\to \mathbb{Z}$
takes infintely many values. Here $\nu_{\gamma}(a)$ is the winding number ...
5
votes
1answer
73 views
Problem with calculating a winding number
I have a problem with calculating the winding number $n\left ( \gamma ,\frac{1}{3} \right )$ of the curve $\gamma :\left [ 0,2\pi \right ]\rightarrow \mathbb{C}, t \mapsto \sin(2t)+i\sin(3t)$.
...
0
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1answer
41 views
Equation of a plane from 2 lines
I have two lines with the following equation
$$D1 : (x, y, z) = (2,0,0) + k(0,3,0)$$
$$D2 : (x, y, z) = (2,0,2) + k(0,0,1)$$
and I must find out the equation of the plane that they make. I ...
5
votes
1answer
110 views
Circumference of a superellipse?
Could someone help me formulate the circumference of a superellipse?
$$\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
22 views
Convex Curve Parametrization
How can I parametrize a convex plane curve using the angle $\theta$ between the tangent line and the $x$-axis?
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0answers
37 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 ...
0
votes
0answers
32 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 ...
4
votes
2answers
69 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 ...
1
vote
2answers
88 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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0answers
22 views
Equation of generating surface.
When it comes to word problems, math begins to elude me.
The question given to me is the following:
There is a hyperbola $4x^2-9y^2 = 36$ on the xy-plane
a. What is the equation of the generating ...
3
votes
1answer
169 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 ...
3
votes
3answers
113 views
How can I find the points of intersection between the curves $r=1+\sin\theta$ and $r=1-\sin\theta$?
Find the points of intersection for the curve $r=a(1+\sin\theta)$ and $r=a(1-\sin\theta)$
My book says the answer is $(0,0),(a,0),(a,\pi)$.
However I calculated $ (a,0),(a,\pi),(a,2\pi)$.
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votes
1answer
30 views
Understanding the Spiro Spline
My name's Wray. This is my first time here.
Firstly, I like curves. I've been keeping a pet project for a long time that would implement a delightful new curve-interpolation algorithm named the Spiro ...
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1answer
112 views
How to find the length of a curved path.
We have to find a continuous model for a curved path which you then solve. A woman is running in the positive y-direction starting at x=50 (50,0) which is orthogonal to the x axis. At this point a dog ...
0
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1answer
28 views
Plan curve with zero area has at least two points of zero curvature
Let $\alpha=(x,y)$ be a smooth closed plan curve defined on $[a,b]\subset \mathbb{R}$. We can define the oriented area of $\alpha$ by $A=\int_{a}^{b} x(s)y'(s)ds$. So, if A=0 then there exists $t_1 ...
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0answers
13 views
Name and parameterise this curve?
I have a curve of the form:
$$1 = (A+Bx)(Cx^2 + Dy^2)$$
What is this family called, is there a canonical parameterisation for it?
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3answers
43 views
Find the point where equations $x=t^2-t$ and $y= t^3 -3t-1$ cross itself.
Find the point where equations $x=t^2-t$ and $y= t^3 -3t-1$ cross itself.
This's the first time I meet this kind of problem, can someone give me some idea? Thank you.
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0answers
34 views
Curve is approximatable by function
I want to show that for $\gamma: [a,b] \subseteq \mathbb{R} \rightarrow V$ continously differentiable where V is a bounded subset of $\mathbb{R}^2$. There is always a sequence of functions ...
0
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0answers
30 views
Approximation theorem
I am looking for some theorem that gives me that each curve $x(t)=(x_1(t),x_2(t))$ that is continously differentiable and has $\dot{x_1}(t)\ge 0$ can be approximated by continously differentiable ...
0
votes
1answer
39 views
How To Write The equation for a line given a set of co-ordinates
I'm trying to learn how can I write equation for a line given all the points that belongs in the line.
I'm looking to find equation for a curve.
An example Set of points is:
{ (24,11) (25,11) ...
0
votes
2answers
49 views
Graph of a curve
Today in my test, there was a question which had contour C: $|z+\dfrac{1}{z}| = 2$. What does the curve represent? Is it a discrete set of points or really a curve?
3
votes
2answers
88 views
A problem on Residue Theorem
Today I had a problem in my test which said
Calculate $\int_C \dfrac{z}{z^2 + 1}$ where C is circle $|z+\dfrac{1}{z}|= 2$.
Now, clearly this was a misprint since C is not a circle. I tried to find ...
7
votes
1answer
96 views
Is it possible for a Jordan curve in the plane to enclose a set with area zero?
I read about the Isoperimetric Inequality the other day. It says that for any Jordan curve,
$$
\frac{4 \pi A}{L^{2}} \leq 1,
$$
where $ L $ is the length of the curve and $ A $ is the area of the ...
8
votes
1answer
118 views
Is there a $C^1$ curve dense in the plane?
Is there a curve $\gamma : \mathbb{R} \to \mathbb{R}^2$ injective and $\mathcal{C}^1$ whose range is dense in $\mathbb{R}^2$?
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0answers
30 views
boy surfaces and tetrahemihexahedron [duplicate]
Hey so I came across this problem and I wanted to know what the difference between Boy's surface which looks like
Boy surface
and The tetrahemihexahedron which looks like
tetrahemihexahedron
One ...
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vote
1answer
38 views
Radius function from length of curve
I have the following function definition for the length of a curve:
$$
l(\theta) = {K_0 \times \sin(\theta) \over \cos(\theta) + K_1}
\\ 0 \le \theta \lt \frac \pi 2
\\ K_0, K_1 \ge 0
$$
I would like ...
3
votes
1answer
44 views
Name for this problem regarding chords, and the area between two closed convex curves?
I want to read more about the amazing result that, when given a closed, convex curve in the plane that can be traversed internally by a chord of length $p$+$q$, and on that chord lying at p, a point, ...
2
votes
0answers
20 views
Characterize a large class of shapes using a finite number of parameters
I am doing some numerical computations searching for an optimal shape for a certain functional. In my particular case, the shape $\Omega$ is a 2 dimensional star shaped domain by the origin, which ...
2
votes
1answer
94 views
What does the definition of curvature mean?
First question, am I right in saying that curvature measure how quickly the direction of a cruve changes?
Also, we have been given the "definition":
$$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$
where ...
2
votes
1answer
173 views
Using equations to draw out complex objects
How do people come up with equations of curves to draw out complex objects?
Some popular examples would include:
batman curve &
PSY curve.
This stackexchange link explains the rationale for the ...
2
votes
1answer
88 views
fixed point projective geometry
I am thinking about the following:
Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
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votes
2answers
96 views
Change parabolic equation to canonical form
I have equation $y = -x^2 + 2x + 7$.
How can I change it to canonical form, which looks like $y^2 = 2px$ ?
($p$ will be parameter)
What i ve tried so far:
$$\begin{align}
y &= -x^2 + 2x + 7\\
y ...
3
votes
1answer
101 views
Normalisation of an algebraic curve.
I need to compute explicitly the normalisation of a singular algebraic curve $C$ which is given by an explicit equation in $\mathbb{A}^2$. This task is mostly reduced to finding the integral closure ...
2
votes
2answers
34 views
Parameterized curve describing trajectory of thrown object
We describe the trajectory of a thrown object (neglecting friction and similiar effects) with the curve
$$k(t) = \left(v_0\cos(\beta)t,\,v_0\sin(\beta)t-\frac{g}{2}t^2\right)$$
with ...
19
votes
0answers
353 views
Ambiguous Curve: can you follow the bicycle?
Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
3
votes
1answer
91 views
For any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.
Find the minimum possible value of $A$ such that for any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.
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0answers
36 views
Functionals defined on curves
I am looking for classical (real) functionals defined on rectifiable curves (neither necessarily simple nor closed) in either $\mathbb{R}^2$ or $\mathbb{R}^3$.
There's length, turning number, total ...
2
votes
1answer
53 views
Could you help me to find a model for this curve?
I am very bad in mathematics and I'm not able to find by myself the model corresponding to this kind of curve.
I wish to have a quick growth at the beginning, then it should increase slowly for a ...
0
votes
2answers
393 views
Finding a general equation for a quadratic curve passing through three points.
I have three points (250, 0), (500,500) and (750, 0).
To find a curve passing through these points all I have to do is plug-in these values into the general quadratic equation:
f(x) = ax^2 + bx + c
...
3
votes
1answer
142 views
What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid?
What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid?
If possible, show some reference please?
1
vote
0answers
55 views
Addition law on moduli space of curves
Dislaimer: I know very little about this, so if parts of my question don't make sense, please feel free to edit in a way that does, or ask me to clarify.
Let $\mathcal{M}_{1,2}$ be the moduli space ...
2
votes
2answers
327 views
Reconstructing space curves from its curvature and torsion
I have to write a program which gets two functions (curvature and torsion) and 3 vectors of the Frenet-Serret "frame" at the starting point - and I have to reconstruct the space curve from this given ...
1
vote
0answers
86 views
Plane curve: orhogonal projection and closest points
I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28:
"Let two curves $C_1$ and $C_2$ have a regular point $P$ ...
0
votes
1answer
47 views
Defining the movement of an object on a 2-dimensional plane
I am trying to define the movement of an object for a danmaku game I am making. Here is a link to some example gameplay (not my game, but a popular series in this genre made by Zun). Basically, I was ...
