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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Area of intersection of two Annulus

Given two separate annulus with centers $[C_1,C_2]$ and their corresponding radii being $[R_1,r_1]$ and $[R_2,r_2]$ respectively, larger radius being $R$. There are methods to look at whether they are ...
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439 views

Finding asymptotes of hyperbola

Given implicit function $F(x, y) = 0$, how can I find its asymptotes? EDIT: Sorry, my calculations were wrong. Here is correct function: $F(x,y)=\sqrt{(x-a)^2 + (y-b)^2} - \sqrt{(x-c)^2 + (y-d)^2} ...
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303 views

From geometrical figures to function

There's one basic mathematical thing that keeps bugging me: the fact that a really simple 2D geometrical figure (like a circle) might not be a function. I know what the definition of a function is. A ...
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51 views

Is circle the only Jordan curve with this property?

When I was thinking about one problem that has to do with Jordan curves the problem which I am going to describe now, arose in my mind. And here it goes. It is known that for every $n\geq3$ the ...
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31 views

Rectificable curve as a boundary of a convex set

Let $K\subseteq\mathbb{R}^2$ be a convex compact set. Is it true that $\partial K$ (the boundary of $K$) is a rectificable curve (i.e. it has length)?
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Curvature and Circumference of Circle

Theorem Let $\gamma\colon [a,b]\rightarrow \mathbb{R}^2$ be a unit speed simple closed curve, with $\gamma'(a)=\gamma'(b)$ and $N$ is the inward-pointing normal. Then $$ \int_{a}^b \kappa_N(s)ds=2\pi....
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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 ...
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If $C \subseteq \mathbb{P}^2$ is a plane curve, then $genus(C)=\frac{1}{2}(d-1)(d-2)$. Compare with example in the notes

In my Algebraic Geometry notes (see http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf) there is the following exercise: If $C \subseteq \mathbb{P}^2$ is a plane curve of degree ...
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148 views

what is the parametric form for “mystery curve”?

Mystery curve found here looks like this : Was given by the complex formula : $$e^{it} – \frac{e^{6it}}{2} + i \frac{e^{-14it}}{3} $$ Is the parametric form simpler or the polar form would be ...
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85 views

Tangent and normal vectors of a positively oriented curve

Given the equation of positively oriented curve $\mathbf{r}(s) = (x(s); y(s))$, we obtain $\mathbf{T} = ({dx \over ds}; {dy \over ds})$ (tangent vector) $\mathbf{N} = ({-dy \over ...
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Curves With Known Arc Length [closed]

I would appreciate if you could list as many (planar) curves with known closed-form analytical expressions for the arc length as possible. Please include formulas for both the curve and the arc ...
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How can we define a 'periodic' parametric curve which is not closed?

We all know that that the curve given by $\gamma (t)=(t,\sin t)$ has a repeated pattern, even though it's not a periodic curve. Can we generalize somehow the definition of a periodic curve so that ...
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105 views

How to sketch $-3x^2 - 8xy + 3y^2 = 1$ [closed]

The equation is as follows: $$-3x^2 - 8xy + 3y^2 = 1$$ How to specify the axis of the given curve? How to as accurately as possible draw a curve defined by this equation?
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52 views

Carto-polar curve

Is there any plane curve that has the same equation in cartesian as in polar coordinates? To be more specific, is there a function such that $f(f(x)\cos x)=f(x)\sin x, \forall x$?
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length of the curve $y=x^n$ in the unit square

Let $l_n$ be the length of the curve $y=x^n$ in $[0,1]\times[0,1]$. Then obviously $\lim_{n\to\infty}l_n = 2$. What about $\lim_{n\to\infty}(n(2-l_n))$ ? The formula $l_n = \int_0^1\sqrt{1+n^2x^{2n-...
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How to create animated plot of curve depending on parameter in WolframAlpha?

Is it possible to create in WolframAlpha an animated plot of a curve given by an equation, where some of the coefficients depend on the parameter (=on time)? For example if I would like to have a ...
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152 views

Solving non-linear second order differential equation: radius of curvature $= k \theta$

I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates radius of curvature $= k \theta$ $$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + ...
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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 +(...
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Proving a corollary to the Jordan Curve Theorem

A Jordan curve is a continuous closed curve in $\Bbb R^2$ which is simple, i.e. has no self-intersections. The Jordan curve theorem states that the complement of any Jordan curve has two connected ...
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43 views

Definition of angle between non-differentiable curves

(Background: I am trying to understand the definition of angle-preserving function..I posted a question earlier but I still have doubts) My question is:how is the angle between two curves defined if ...
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Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are some ...
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126 views

Curvature and convexity of a plane curve

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2$ be a $C^2([a,b])$ regular curve. Is it true that $\mathbf{r}$ is convex if and only if its curvature $\kappa(t)\leq 0, \forall t\in [a,b]$ or $\kappa(t)\geq 0, \...
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Adjustable Sigmoid Curve (S-Curve) from $(0,0)$ to $ (1,1)$

I feel like this is such a simple question but I am at such a loss. I currently have a set of values that I would like to weigh by an S Curve. My data ranges from $0$ to $1$ and never leaves those ...
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107 views

Give an example of three different points in $\mathbb R^3$ such that there are infinitely many planes in $\mathbb R^3$ passing through all of them.

A past exam question. I'm not certain on the meaning. I assume it wants a $3$ points on a straight line, one which case there would be infinitely many planes passing through all of them. But that ...
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Equivalence of focus-focus and focus-directix definitions of ellipse without leaving the plane

Take a look at the following two definitions of ellipse: For some fixed points $F_1,F_2$ and real number $2a>|F_1F_2|$ an ellipse is the locus of points $P$ such that $|F_1P|+|F_2P|=2a$. ...
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42 views

Unique solution of a simple functional equation

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in C^{\...
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Parametrization of the lemniscate

All over the net, it is stated that the parametrization of the lemniscate with Cartesian equation $(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)$ is: $$\varphi: t \mapsto \left(\frac{a\sqrt{2}\cos(t)}{1+\sin^2(t)},...
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Solving (Frenet-Serret) differential equation system in Matlab

I'm about (trying) to solve the Frenet-Serret equation given by the known formulas, finding $e(s)$, $n(s)$, $b(s)$, where $e'(s) = \kappa(s)v(s)n(s)$ $n'(s) = -\kappa(s)v(s)e(s) + \tau(s)v(s)b(s)$ $...
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Planes And Lines

Given :Point $A(1,2,4)$ and plane $P: x-y+z+2=0$ How to find coordinates of point $A'$ the symmetric of point $A$ with respect to plane $P$
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Reparametrization with non-vanishing lateral derivatives

Let $\mathbf{r}:I\subseteq\mathbb{R}\to\mathbb{R}^2$ be a $C^{1}(I)$ function, non-constant on any subinterval, such that $\forall\ t_0\in I$, the following limits exist: $\lim\limits_{t\nearrow t_0}\...
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Curvature in $\mathbb{R}^2$

Let $f(t) = (x(t),y(t))$, not necessarily parametrized by arclength. We define the unit tangent vector, $T(t) = (1/|f'(t)|)(x',y')$. Also the normal vector, $N(t) = (1/|f'(t)|)(-y',x')$, which is ...
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Image of any curve can be parametrized without zero derivative [closed]

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a $C^{1} ([a,b])$ injective application. Prove that there is another continuous parametrization $\rho:[c,d]\to\mathbb{R}^2$ such that the following two sets are ...
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Fourier series for convex plane curves.

The following problem is from Stein's Fourier analysis. This problem explores another relationship between the geometry of a curve and Fourier series. The diameter of a closed curve $\Gamma$ ...
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What's the name of this wavy curve?

What's the name of the curve you get from changing the x or y frequency on what was previously a path around an ellipse? The equation would be: f(t) = (Acos(ut), Bsin(vt)) And it looks like a wavy ...
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37 views

Showing a limit is equivalent to the curvature of a planar curve

I've been working on this limit for a long time and just don't know how to show this. I tried representing h and d as vectors, but I cannot seem to accurately describe them. What would be a good ...
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Intersection of point normal to plane

Hi if I have a point p (red dot) and I have a plane P (yellow) how do I find the intersection of the point normal to the plane? thanks! enter image description here
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A 4th grade curve meets a line in one point with multiplicity 4

Suppose a 4th grade curve meets a line in one point with multiplicity 4. Example: the lemniscate $(x^2 + y^2)^2 = y^2 - x^2$ meets the line $x=y$ when the condition $x^4=0$ holds. This shows that ...
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Finding the directional derivative parallel to an intersection of planes

We must find the directional derivative of $ f(x,y,z) = x^2 + 2xyz -y^2 $ at $ (1,1,2) $ in a direction parallel to the straight line $ \frac{x-1}{2} = y-1 = \frac{z-2}{-3} $ The straight line is an ...
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82 views

Find the condition for a center of a circle with exactly one lattice point on its circumference

Statement Find the condition for a center of a circle with exactly one lattice point on its circumference (this lattice point must not be the only one lattice point of the disk) What I have ...
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length of path travelled on $(t, \cos t, \sin t)$ from times $t = 0$ and $t = 2\pi$

Let the position of a particle in three dimensional space at time t be $(t, \cos t, \sin t)$. Then the length of the path traversed by the particle between the times $t = 0$ and $t = 2\pi$ is (A) $2\...
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Finding a local parameterization of a plane curve

I'm attempting to find a parameterization of $\frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} = 1$. I find a tangent vector field: $X = \left( \frac{2x_2}{b^2}, -\frac{2x_1}{a^2} \right)$ (by taking the ...
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Determine the Winding Numbers of the Chinese Unicom Symbol

I'm practicing with Winding Numbers, and encountered an interesting example. You might be familiar with this liantong symbol, the logo of China Unicom: Suppose we make this into a fully closed and ...
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Equation of a plane that crosses the axes at points equidistant from the origin.

Give the equation of a plane that crosses the axes at points equidistant from the origin. Explain your reasoning. I know the equation should be on a 45 degree angle looking towards the axis. I have ...
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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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Is it possible to join a set of points using multiple circular arcs to get a smooth curve?

I have a set of points on a plane, and I want to join these points using a circular arc between consecutive points such that the final curve I get is smooth (no sharp edges). Is this possible? If so, ...
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Finding $y$ In Calculus(Area) Problem? [duplicate]

Find the number b such that the line $y=b$ divides the region bounded by the curves $y = x^2$ and $y = 4$ into two regions with equal area.
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If $0 < \theta < \frac{\pi}{2}$, then $\gamma$ is a logarithmic spiral

Let $\gamma$ be a plane curve parametrized by the arc length, having the property that its tangent vector $T(t)$ forms a fixed angle $\theta$ with $\gamma(t)$. Before explaining where I am, let ...
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Direction of outward normal differential $\hat n\ ds$

In vector calculus, why is the outward normal differential $\hat n\ ds$ around a closed curve $dy\ \hat i - dx\ \hat j$? Why isn't it $-dy\ \hat i + dx\ \hat j$ or something else?
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find travel time given path and velocity field

As I was studying refraction, I began wondering what path would light take when entering a non-homogeneous transparent medium, i.e. a certain material in which the refraction index $n$ varies (...
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Can we classify plane cubics, What are they?

There are four qualitatively distinct pictures of the plane cubics. What are the polynomials corresponding to them? I know two of them have special names: nodal cubic and cuspidal cubic with ...