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.
3
votes
1answer
188 views
Curve of a fixed point of a conic compelled to pass through 2 points
Suppose that in the plane a given conic curve is compelled to pass through two fixed points of that plane.
What are the curves covered by a fixed point of the conic, its center (for an ellipse), its ...
1
vote
1answer
198 views
What is a word to describe curves that have a tangent but are curved away from each other?
I'm writing a description that involves two curves behaving (approximately) as shown below. There aren't actually two intersections: they are mutually tangential. Also, I have many such curves, this ...
1
vote
2answers
56 views
Determining distance across face of bars following a curve
So I'm not much of a mathematician, and I've been trying to figure out how one would solve this real world problem I have.
I have 2"x4" wood cut into squares (2x4x4).
I am trying to figure out at ...
3
votes
1answer
221 views
Fourier-like expansion of a closed curve in 2D
Fourier expansion can be used to represent any periodic function in one variable.
Closed surfaces in 3D can be built out of spherical harmonics.
Is there a similar expansion to represent a curve of ...
3
votes
1answer
51 views
Segments on a plane, what curve do the intersections tend to?
In a Cartesian diagram, given a size $s$, suppose I create $m$ segments as such:
I connect $(0,s/m)$ with $(s,0)$; $(0,2s/m)$ with $(s-s/m,0)$; ... ; $(0,s)$ with $(s/m,0)$.
For example, if $s=4$ ...
1
vote
4answers
194 views
Difficult equations to rewrite as ellipses
I have this equality that defines an elliptic boundary. I am trying to rewrite it in the form of the equation of an ellipse, but I am having trouble doing that. How would I go about rewriting this ...
1
vote
1answer
176 views
How to place objects equidistantly on an Archimedean spiral?
To place objects equidistantly on an Archimedean (arithmetic) spiral, the arc length of the spiral has to increase linearly between the objects.
This is what I have so far: The length of a spiral is ...
1
vote
0answers
69 views
Approximation in the Plane of Constant Curvature
In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve.
Now, The ...
2
votes
0answers
31 views
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 ...
5
votes
2answers
587 views
Tractrix-like curves
Is there a common name for curves, obtained from dragging a point along another curve, similar to how tractrix is obtained by dragging a point along a line?
What is a parametric equation of such ...
2
votes
1answer
175 views
Polar form of a superellipse?
What is the polar form for a superellipse with semidiameters $a$ and $b$, centered at a point $(r_0, θ_0)$, with the $a$ semidiameter at an angle $\varphi$ relative to the polar axis?
1
vote
2answers
492 views
What does “the circle is tangent to the curve” mean?
I've got a math exercise where it's said I have to prove that a circle is tangent to a curve (described by a parametric plot). Here's the graph :
So we can see that when $y=0$, the circle is really ...
1
vote
2answers
137 views
Is there a generalized method of rotation for curves?
I know that we can rotate a curve in $R^2$ about a linear axis, as is common for first year calculus problems involving solids of revolution. But has anyone come up with a general method to take a ...
1
vote
0answers
103 views
Vector equation and curvature
Vector equation
$r(t)=2\cos(t)\mathbf i + 3\sin(t)\mathbf j\ \ (0 \le t \le 2\pi)$
represents ellipse.
I need to find curvature of this ellipse on endpoints of x and y axis that are given with ...
1
vote
1answer
110 views
How to determine a point is inside a superellipse?
I have a superellipse which I will rotate it and then translate it. My question is that how to determine the origin will still be inside the superellipse after all the action?
Thanks
16
votes
2answers
1k 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:
...
1
vote
2answers
472 views
Tractrix tangent segment (from Baby Do-Carmo)
I need some help with question 4 in section 1.3 in Baby Do-Carmo textbook in DG.
The question asks:
Let $\alpha(t):(0,\pi)\rightarrow R^2 $ be given by:
$$ \alpha(t)= (\cos(t), \cos(t) ...
1
vote
2answers
391 views
Express this curve in the rectangular form
Express the curve $r = 9/(4+\sin \theta)$ in rectangular form.
And what is the rectangular form?
if I get the expression in rectangular form, how am I able to convert it back to polar ...
0
votes
1answer
116 views
Can every closed curve be modified in the following way to produce a simple closed curve?
Is there a sequence of the following operation that change a closed curve with finite number of self-intersections to a simple closed curve?
Also, every self-intersection differs at least $\epsilon$ ...
0
votes
0answers
82 views
How can Gauss maps help me with plane curves intersection?
I read somewhere that Gauss maps help to discover if two plane curves intersect without converting them in piecewise linear segments. Where can I learn more?
3
votes
1answer
253 views
Why do definitions of distinct conic sections produce a single equation?
I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
2
votes
0answers
326 views
How to find all intersection points of two splines?
2D-Cubic splines are given in parametric form (X(t), Y(t) and X(s), Y(s)). Every segment has it's own X and Y expression. And I want to find all intersection points. Some segments are intersecting ...
1
vote
1answer
190 views
Why is this curve convex?
I am considering the curve traced by the equation $r=a\sin 3\theta$. Specifically as $\theta$ varies from $0$ to $\frac{\pi}{6}$, $r$ varies from $0$ to $a$. How do I conclude that the curve is convex ...
1
vote
2answers
503 views
How Can I Calculate Area of Astroid Represented by Parameter?
Let $x=2\cos^3\theta$ and $y=2\sin^3\theta$ known as the astroid.
In this case, radius $r=2$.
and gray part's $x$ range is $1/\sqrt{2}\leq x\leq 2$. this deal with $0\leq\theta\leq \pi/4$.
...
2
votes
1answer
462 views
The signed curvature of the Catenary
Now I want to show that the signed curvature of the catenary, with parameterization
$$(t,\cosh(t))$$
is $k(t)=\frac{1}{\cosh^2(t)}$
Now what I have done (and presumably went astray), is first ...
2
votes
1answer
103 views
Does using an ellipse as a template still produce an ellipse?
Suppose I have a (physical) template, consisting of a piece of stiff sheet plastic with a hole cut in the middle. Suppose the hole is in the shape of an ellipse, say, 8 x 12 inches. Suppose I then ...
1
vote
0answers
152 views
Is the extra condition in this definition superfluous?
I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
4
votes
1answer
693 views
Why does the focus of a rolling parabola trace a catenary?
I keep hearing that when one rolls a parabola on a straight line, the focus traces a catenary. I kept trying to find a proof on the Internet, but no dice. How does one prove this to be true?
1
vote
1answer
215 views
rolling wheel problem
To achieve this: http://en.wikipedia.org/wiki/Square_wheel, what should $L$ be?
2
votes
1answer
217 views
Finding a homeomorphism guaranteed by Schoenflies Theorem
Assume I have a Jordan curve $C \subset \mathbb{R}^2$. Then by Schoenflies Theorem there exists a homeomorphism $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(C)$ is the unit circle. Is ...
7
votes
1answer
429 views
Comprehensive compilation of conic section formulae
My frustration started after hours of searching failed to turn up a formula for the vertex of a parabola in the general form
$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$
As is already well known, the discriminant ...
3
votes
1answer
370 views
Finding Simply Connected Open Sets in a Connected Set?
I believe that the following statement is true:
Let $E$ be a connected open subset of $\mathbb{R}^2$. For any $n$ distinct points in $E$, there exists a connected and simply connected open set $G ...
1
vote
1answer
154 views
Question on the catenary
The catenary minimizes the potential energy of a cable and has equation $y - y_0 = A \cosh (\frac{x-x_0}{A})$. It is physically intuitive that the catenary is unique, but is there a mathematical ...
1
vote
0answers
47 views
References for implicit co-ordinates
When I was at school I learned about "implicit coordinates" for curves in a plane. Essentially these mapped the path in terms of arc length $s$ from a fixed point, and direction of travel $\psi$ ...
0
votes
3answers
258 views
points toward the center of the osculating circle (second derivate in a arc length parameter curve)
Is it true that if I have a planar curve parametrized by its arc length, then the second derivative points towards to the center of the osculating circle?
I can´t see it, but the book says that it´s ...
2
votes
2answers
372 views
Direction of the second derivative of an arclength parametrized curve
I have a question, it's so simple and stupid ._. If I have a planar curve parametrized by arc length, it's easy to show that the second derivate is orthogonal to the first derivate vector (tangent ...
7
votes
1answer
988 views
Parametrizing implicit algebraic curves
Back in the day, I was absolutely enthralled by the study of plane curves and their properties (I have Lockwood and Zwikker to thank). I learned early on that for the purposes of generating plots on a ...
0
votes
4answers
176 views
how can I graph a bicorn given only its equation?
what are the parts or the variables present in the bicorn equation?
2
votes
1answer
256 views
Does anyone know the name of this curve?
I have come upon
the curve with the following parametric equations:
$$x(t)=\log(2+2\cos(t))/2$$
$$y(t)=t/2$$
for $-\pi<t<\pi$. It gives the image in the complex plane under $\log(1+z)$ of the ...
0
votes
1answer
198 views
How do I calculate $t$ in the general parametric equation of an ellipse when the point $(x,y)$ is known?
I have the general parametric equation of an ellipse.
$$\begin{align*}x&=c_x+a\cos{t}\cos{\alpha}-b\sin{t}\sin{\alpha}
\\
y&=c_y+a\cos{t}\sin{\alpha}+b\sin{t}\cos{\alpha}\end{align*}$$
I ...
8
votes
2answers
494 views
What type of curve is $\left|\dfrac{x}{a}\right|^{z} + \left|\dfrac{y}{b}\right|^{z} = 1$?
In order to fit experimental data, I want to use a Cartesian equation which looks like:
$\left|\dfrac{x}{a}\right|^{z} + \left|\dfrac{y}{b}\right|^{z} = 1$
$a$, $b$, $c$, and $z$ can take any real ...
2
votes
2answers
176 views
Looking for the name of a Rising/Falling Curve
I'm looking for a particular curve algorithm that is similar to to a bell curve/distribution, but instead of approaching zero at its ends, it stops at its length/limit. You specify the length of the ...
9
votes
3answers
232 views
Hilbert curve, understanding the original article
I'm trying to read and understand the article in which Hilbert gave an illustration of a space filling curve, namely "Ueber die stetige Abbildung einer Linie auf ein Flächenstück". It's only a short 2 ...
2
votes
1answer
361 views
Property of an ellipse
I need proof for the following question. Also, I want to know, can we apply the same for other conics. If yes, where and when... Please explain.
Show that there exists a point K on the major axis of ...
6
votes
3answers
227 views
Can closed curves have small curvature?
Let $\gamma$ be a smooth curve in Euclidean space of length $2\pi$ whose curvature function satisfies $-1 < k(t) < 1$. Can $\gamma$ be closed?
This seems like it should be an easy exercise, at ...
2
votes
4answers
409 views
Sketch a curve given parametrically by $x=2t-4t^3$ and $t^2-3t^4$
I am unable to see how to eliminate $t$. Wolfram Alpha fails at it too.
$$x=2t-4t^3$$ $$y=t^2-3t^4$$
I can guess that the curve is a polynomial equation so in principle I can write this as
$$w_1 ...
2
votes
4answers
587 views
Calculate area under a curve
How do I analytically calculate using integration the area under the following curve?
$$x^2+ xy + y^2= 1$$
Its some ellipse and I see it might help that it's symmetric in exchange of x and y, so ...
0
votes
2answers
73 views
Curve intersection criteria
I have two curves, which are given by sets of values:
$C = [( x{_1} ,y{_1}),(x{_2},y{_2}),(x{_3},y{_2}),...,(x{_n},y{_n})]$
$C^' = [( x^'{_1} ...
4
votes
2answers
169 views
Components of algebraic varieties
Sorry, but I have to ask a dumb question:
Algebraically, a hyperbola has only one irreducible component (given by an irreducible polynomial).
Why, then, does the real image of a hyperbola show two ...
4
votes
1answer
280 views
What constants do I need to create this specific logarithmic spiral?
please bear with me as I'm not a mathematician and this is difficult to word properly. :]
I need the equation for a logarithmic spiral (let's call it $S(\theta)$) that meets certain constraints for a ...
