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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Show that $r=\langle x,y\rangle$ satisfies $(r-a)\cdot (r-b)$ iff $(x,y)$ is on a circle

Given that $a=\langle a_1,a_2\rangle$, $b=\langle b_1,b_2\rangle$. How can we show that $r=\langle x,y\rangle$ satisfies $(r-a)\cdot (r-b)=0$ iff $(x,y)$ is on a circle? What would be the center and ...
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Plane and Ellipse Intersection

Short Version: If some can solve the easier to read form as follows, I would be thankful. \begin{equation} B = \frac{1 - d^{T}Bd}{ K_{1} } A \end{equation} \begin{equation} B^{T}d = \frac{1 - ...
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How to calculate the area of a region with a closed plane curve boundary?

Under the conditions of Green’s Theorem, the area of a region $R$ enclosed by a curve $C$ is $$\oint_C x \, dy=-\oint_C y \, dx=\frac{1}{2}\oint_C (x \, dy - y \, dx)$$ I tried to use the result to ...
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51 views

How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
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Self dual curves

Can you please tell me some of the applications of self dual curves or duality of any curves as a whole,because I can't find any of these, even though I've been looking very hard.
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26 views

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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37 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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24 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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56 views

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 ...
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39 views

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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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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34 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 ...
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293 views

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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35 views

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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82 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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100 views

Laplace-Beltrami on a Curve

Is there a way to write out Laplace-Beltrami operator explicitly for a sufficiently smooth plane curve given by implicit equation $s(x,y)=0$? I know that if we knew the parametrization of the curve, ...
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47 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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53 views

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 = ...
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Are closed simple curves with that property necessarily circles?

This is a more interesting follow-up to the question Are closed simple curves with this property necessarily circles? Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple $C^1$ convex curve and ...
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1answer
43 views

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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How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
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76 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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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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52 views

Roll one ellipse on another: Locus of center ever a circle?

Let $E_1$ be an ellipse fixed in the plane. Let $E_2$ be a second, possibly different ellipse, which rolls around without slippage outside $E_1$, touching perimeter-to-perimeter. Let $c_2(t)$ be the ...
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31 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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1answer
42 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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55 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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22 views

Consider the Plane Curve?

Consider the plane curve $$\gamma(t) = \left( \cosh(t) \cos(t), \cosh(t) \sin(t) \right), \;\; t \in \mathbb R.$$ Is $\gamma$ regular? If $\gamma$ is not regular, can you restrict the parameter ...
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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 ...
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38 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 ...
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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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25 views

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 ...
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1answer
64 views

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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Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
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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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49 views

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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41 views

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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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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31 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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21 views

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 curves with special constraints. How do i verify the answer?

Let $O$ be a fixed point on the plane. Find all curves $\gamma : \mathbb{R} \to \mathbb{R}^2$ such that if $C(t_0)$ is the centre of the curvature of $ \gamma$ at $t_0$ then the angle $\widehat{ ...
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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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How to find tangent line given several variables

I have a question that I'm having difficulty on. I can solve these normally, but I'm having a bit of a challenge dealing with these extra terms: "Find the equation of the tangent line to the ellipse ...
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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) ...
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54 views

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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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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36 views

A point of infinite curvature on a curve

Let $\gamma(t)$ be a $C^r$ smooth curve in the plane. Suppose $r\ge 2$, so that one can define the curvature $\kappa(t)$ at $\gamma(t)$. For example, $\kappa(t)=0$ means that the curve is kind of flat ...
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66 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 ...