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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103 views
Parametrise curve by angle and convex curves
Can one parametrise any closed curve by the angle its tangent makes to the $x$-axis? I seem to remember that this is only possible for convex curves. Could anyone tell me why, please?
Also is ...
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2answers
81 views
What is the limit distance to the base function if offset curve is a function too?
I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
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2answers
359 views
Parallel functions.
In 2 dimensions, we can draw 2 parallel lines that have the same distance from a line.
I wanted to find parallel functions of a function and their distance is $d$ to the function for all inputs and ...
5
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3answers
154 views
Level sets of convex functions
Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a convex function. For $t\in\mathbb{R}$, consider the corresponding level set
$$f^{-1}\{t\}=\{(x,y)\in\mathbb{R}^2: f(x,y)=t\}.$$
For the application I ...
2
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1answer
88 views
Divisor of degree 2 on a smooth plane curve
Let $X$ be a smooth plane curve of genus $3$ (assume a smooth plane quartic) and $D$ a divisor of degree $2$ on this curve.
Assume that $\mathcal{l}(D)>0$. It means that there exists a rational ...
8
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2answers
155 views
Where do people learn about things like caustics, evolutes, inverse curves, etc.?
When I look up a curve on Wikipedia, I'll often see a lot of properties along the lines of "you can generate curve X by rolling a circle along curve Y and tracing the trajectory of a single point," or ...
2
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2answers
104 views
Describing multivariable functions
So I am presented with the following question:
Describe and sketch the largest region in the $xy$-plane that corresponds to the domain of the function:
$$g(x,y) = \sqrt{4 - x^2 - y^2} \ln(x-y).$$
...
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1answer
73 views
Computing the gradient of the function $\psi(u) = f[ \phi(u)]$
This question is based on section 6 of the paper Kriging and splines with derivative information.
A parametric curve $\phi(u)$ in three dimensions is deformed by the function $f$ to a new curve ...
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0answers
59 views
How to fit a convex curve to set of data points
have a set of data points (x1,y1) (x2,y2) (x3,y3) etc. The data is such that slope of successive points are increasing most of the time. But there are a few exceptions. The software I load the data ...
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0answers
94 views
How to define the transform?
$y=f(x)$ is continuous and defined for all $x$ real numbers.
Point $A(0,f(0))$ is to be moved to on x axis while $f(x)$ is rigid curve and the rigid curve always passes on point $B (x_1,f(x_1))$ ...
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1answer
49 views
Equivalence of two definitions of path (in $\mathbb{R}^3$) length
In a previews question I asked here I used the following definition of path length:$\gamma=(x(t),y(t),z(t))$ : $L(\gamma)=\intop_{a}^{b}\sqrt{(x'(t))^{2}+(y'(t))^{2}+(z'(t))^{2}}$.
In the answer ...
3
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1answer
256 views
Name that curve!
What curve will a kayak describe if the paddler aims her bow at an object on a distant shore ahead and keeps the bow pointing to that object as she paddles toward it with constant velocity, in the ...
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1answer
65 views
Finding POI's for the follow two curves
So I need to find the POI (point of intersection) of the following two curves:
\begin{align*}
r & = 1 + \cos \theta, \\
r & = 2 - 2\cos \theta.
\end{align*}
What I did was I just set both the ...
269
votes
9answers
294k views
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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4answers
555 views
Is there such a function?
Does there exist a continuous function $f:[0,1]\rightarrow\mathbb{R}$ such that for any two points P,Q on the curve, there exists a point R on the curve such that PQR is an equilateral triangle? If ...
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1answer
87 views
Non-intersecting smooth paths in the plane and the relation to curvature
I'm interested in a problem and have no idea how to approach solving it. Could you please point me in the right direction.
Given 3 smooth paths $\varphi_{1}:[0,1]\to \mathbb{R}^{2}$, ...
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2answers
111 views
How to mathematically color the regions bounded by a parametric curve?
Usually, if an implicit equation F(x,y) = 0 defines a curve (or curves) on the x-y plane, then we can use the inequalities F(x,y) < 0 or F(x,y) > 0 to color the regions bounded by the curve (or ...
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1answer
85 views
Gaps in the Genera of Space Curves
We learned the following relationship between the degree and genus of plane curves in my algebraic geometry course:
\begin{array}
a
\text{degree} &d &1 &2 &3 &4 &5 &6 ...
3
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3answers
1k views
How to find the parametric equation of a cycloid?
"A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia
In many calculus books I have, the cycloid, in parametric form, is ...
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2answers
365 views
How to draw a family of curves and its envelope?
Given a family of curves $$F=(x-t)^2+y^2-\frac{1}{2}t^2,$$ I am trying to compute the envelope of this family.
The envelope is described by the equations
$$F=0, \\
\dfrac{\partial F}{\partial t} ...
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1answer
95 views
Point travels around curve
I wonder what does this mean: Point travels around curve. I try to figure out some math explanation in the book and I can't move forward because I can't understand these words.
I can understand when ...
2
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0answers
128 views
Reversing a roulette on a straight line - solving for a parameterization?
(See below for update.)
I would like to reverse the following equations. Here, the path is traced out by the polar origin of the wheel. You can put the trace point anywhere on or off of the wheel ...
3
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0answers
95 views
Showing that the cardioid is or isn't the caustic of the truncated cone.
Today I had a nice breakfast, but instead of using the usual cylindrical cup and admiring the nephroid inside it, I chose a cup in the form of a truncated cone. It seemed that this cup produced a nice ...
5
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1answer
248 views
What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse.
Ellipse equation is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ and the length of line segment is $2k$, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ...
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1answer
335 views
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2answers
66 views
Is there plane curves with limit number of operations in which is non-constructible and how do we prove it
Is there plane curves with limit number of operations in which is non-constructible and how do we prove it is non-constructible, i call it non-constructible if we have to plot infinity number of point ...
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2answers
2k views
Difficult conversion from polar equation to rectangular equation.
How do we convert this into rectangular equation?
$r=5\theta$
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2answers
125 views
How do we get the rectangular form of this?
I know if $\sqrt{x^2+y^2} = x$, then the polar equation of this is $r=cos\theta$
So,how to get the rectangular form of this polar equation, is it complicate:
$r=cos(10\theta)$
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1answer
274 views
What are curves (generalized ellipses) with more than two focal points called and how do they look like?
An ellipse is usually defined as the locus of points so that sum of the distances to the two foci is constant. But what are curves called which are defined as the locus of points so that the sum of ...
1
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3answers
268 views
How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?
Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on ...
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0answers
251 views
Distance between two parametric curves
I have two parametric planar curves.
The curves are not self-intersecting.
Curve $C_0$ is inside $C_1$.
With $t \in [0..1]$
$ C_0:x = f_0(t); y = g_0(t) $
$ C_1:x = f_1(t); y = g_1(t) $
Now ...
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1answer
133 views
A question about geometry of plane curve books
I can't study algebraic geometry yet, so before i learn that, yet i need a comprehensive treatment to the geometric theory of plane curves. i notice that the books refer to this subject either out of ...
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1answer
663 views
Making a circle with paper folding, scissors, pencil, and a straightedge
Can we make a circle using paper folding, scissors, straightedge, anda pencil, allowing an infinite number of operations?
I think my chemistry teacher have show me once how to make it during the ...
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1answer
100 views
Prove if a polar function involves only the rational numbers and sin, cos, tan functions, it can be written in rectangular form.
Prove if a function only have including the rational numbers and sin, cos, tan function and $r$, you always could write it in rectangular form. Ex. For $r=2/(2+2\cos\theta)$ it could be represent in ...
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1answer
342 views
Tractrix Tangent Length Problem
I am having trouble with a problem I am working on
The trace of $\vec{r}(t):=\sin(t)\vec{i}+[\cos(t)+\ln[\tan(t/2)]]\vec{j}$ where $t\in(0,\pi)$ is called a tractrix. Show the length of the line ...
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5answers
4k views
Drawing heart in mathematica
It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer ...
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votes
3answers
88 views
Non-linear function on $\mathbb{R}^2$ preserving the origin and maps lines onto lines?
Is there an $f:\mathbb{R}^2 \to \mathbb{R}^2$ such that:
$(0,0)\mapsto (0,0)$; and
for any $a,b,c$ with $a^2 + b^2 >0$, the set $A=\{(x,y):ax+by=c\}$ is mapped onto ...
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votes
1answer
166 views
what is the total curvature of the logarithmic spiral?
given the parameterization: exp(t)*(cos(t), sin(t))
t $\in [0, 2\pi$]
how do I calculate the total curvature?
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0answers
145 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: ...
3
votes
3answers
1k views
Writing a Polar Equation for the Graph of an Implicit Cartesian Equation
If $(x^2+y^2)^3=4x^2y^2,$ then $r=\sin 2\theta$ for some $\theta$.
Using $r^2=x^2+y^2, x=r\cos\theta,y=r\sin\theta$, it's easy to get $r^2=\sin^22\theta$.
But I don't know what to do next, since ...
8
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3answers
735 views
Minimal Ellipse Circumscribing A Right Triangle
Find the equation of the ellipse circumscribing a right triangle whose lengths of it's sides are $3,4,5$ and such that its area is the minimum possible one.
You may chose the origin and orientation ...
3
votes
1answer
99 views
when the curve $\mathbb{r=a\sin(b\theta)}$ is algebraic?
A need to show that the curve given in polar equation $\mathbb{r=a\sin(b\theta)}$ is an algebraic curve if $b=\frac{m}{n}$, $m,n\in \mathbb{N}^{*}$ and $(m,n)=1$. Also I am supposed to find the ...
2
votes
1answer
516 views
Polynomials, Rouche's theorem and index of vector fields
In the proof of Rouche's theorem I saw in a book, there are two points I failed to understand, or failed to prove myself. (if you aren't familiar with the theorem, please try to look at the two ...
5
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3answers
269 views
Division of Other curves than circles
The coordinates of an arc of a circle of length $\frac{2pi}{p}$ are an algebraic number, and when $p$ is a Fermat prime you can find it in terms of square roots.
Gauss said that the method applied to ...
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1answer
274 views
definition of sinusoidal curve
I have question related with these two definition:
In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates
$$r^n = a^n \cos(n \theta)$$
where $a$ is ...
1
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1answer
344 views
area bounded by spirogram
A circle of radius r rolls without slipping inside an n-gon of side length l. A curve C is traced out by a pencil through a hole a distance d from the centre. Initially the circle is in a corner with ...
1
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1answer
194 views
Where is my (algebra) mistake? Converting parametric to Cartesian equation
I'm having a problem with my solution to a textbook exercise:
Find the Cartesian equation of the curve given by this parametric equation:
$$x = \frac{t}{2t-1}, y = \frac{t}{t+1}$$
The textbook's ...
3
votes
1answer
566 views
Can you write a non-piecewise equation that describes an arbitrary shape?
This batman equation thing got me thinking: for an arbitrary curve drawn on the Cartesian plane, can you write a corresponding equation which is not piecewise? What about closed shapes, a la the ...
39
votes
1answer
652 views
Fractal behavior along the boundary of convergence?
The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
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2answers
2k views
Polar to Parametric Equation?
I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right.
Curve C has polar equation ...
