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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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 ...
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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 $0\leq\theta\leq \pi/4$. ...
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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 ...
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209 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 ...
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0answers
175 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 ...
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4k 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?
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424 views

rolling wheel problem

To achieve this: http://en.wikipedia.org/wiki/Square_wheel, what should $L$ be?
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333 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 ...
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652 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 ...
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1answer
577 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 ...
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457 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 ...
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1k 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 ...
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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 ...
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324 views

how can I graph a bicorn given only its equation?

what are the parts or the variables present in the bicorn equation?
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1answer
307 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 ...
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1answer
363 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 ...
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554 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 ...
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2answers
307 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 ...
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664 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 ...
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1answer
580 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 ...
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344 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 ...
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4answers
908 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 ...
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4answers
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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 ...
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77 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} ...
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313 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 ...
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1answer
499 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 ...
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2answers
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How do I draw an elliptic curve?

I can draw a circle using a compass. I can draw an ellipse using two focal points and a loop of string. I think that you can draw an arbitrary conic with a "generalized" compass for which the pencil ...
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definition of length of non-rectifiable curves

Following the usual definition of length of a curve, the curve have to be rectifiable to have finite length. However for example the curve $ \gamma \colon [0,1] \to \mathbb{R}, t \to t \cos^2(\pi/t) ...
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2answers
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On deriving the arclength of a hyperbola

In my attempts to derive the closed form for the arclength of the hyperbola, I wound up with the following integral: $$\int\frac{\sqrt{1-m\;\sin^2 u}}{\sin^2 u}\mathrm{d}u$$ I am aware that such ...
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1answer
271 views

every divisor of degree $0$ on a smooth cubic curve $\mathcal{C}\subseteq\mathbb{P}^2$ is equivalent to $A-A_0$ for a fixed $A_0\in\!\mathcal{C}$

NOTATION: Let $\mathcal{C}=\mathcal{V}(F)\subseteq\mathbb{P}^2$ be a curve of degree $3\!=\!deg(F)$ with no singularities and let $A_0\!\!\in\!\mathcal{C}$ be fixed. Let $Div(\mathcal{C})$ denote the ...
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581 views

The Dido problem with an arclength constraint

It is well known that the solution to the classical Dido problem is a semicircle, and that the solution to the classical isoperimetric problem is a circle. It's also reasonably obvious that the ...
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1answer
74 views

How can I proceed with this exercise?

It asks me to graph the region in the complex plane. $Re(z+iz) \le 1$
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412 views

Parametric equations of curves

Is there a way to produce parametric equations for a curve?(If we do know cartesian coordinates of course)
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2k views

How many cubic curves are there?

It is well-known that there is only one "kind" of line, and that there are three "kinds" of quadratic curves (the nature of which depends on the sign of a so-called "discriminant"). It is noteworthy ...
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747 views

Finding standard ellipse characteristics from specific ellipse parametrisation

I have found the following ellipse representation $(x,y)=(x_0\cos(\theta+d/2),y_0\cos(\theta-d/2))$, $\theta \in [0,2\pi]$. This is a contour of bivariate normal distribution with uneven variances and ...
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1answer
171 views

Continuous curve interpolating a list of points

I need a function (a curve -- preferably a simple one) that, given $n$ points of a 2D space ($R^2$) passes (interpolates) through all points in a smooth/continuous way. Found out that what I need is ...
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1answer
1k views

Determining the symmetry of a plane algebraic curve from its Cartesian equation

This is a bit of a silly question, but I've been puzzled on how this can be done, so I ask here. You are given an implicit Cartesian equation like $$4 x^4-4 x^3+8 y^2 x^2-27 x^2+12 y^2 x+4 y^4-27 ...
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3answers
2k views

The function that draws a figure eight

I'm trying to describe a counterexample for a theorem which includes the figure eight or "infinity" symbol, but I'm having trouble finding a good piecewise function to draw it. I need it to be the ...
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3answers
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A plane algebraic curve with all four kinds of double points

During my study of plane algebraic curves, I got curious if there is a nontrivial example of a plane algebraic curve that has a node, a cusp (for my purposes I do not care which of the two kinds of ...
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2answers
909 views

Deriving an implicit Cartesian equation from a polar equation with fractional multiples of the angle

Usually, applying the conversion formulae $r^2=x^2+y^2$ $\cos\;\theta=\frac{x}{r}$ $\sin\;\theta=\frac{y}{r}$ to transform an equation in polar coordinates to an implicit Cartesian equation is ...