# Tagged Questions

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.

77 views

45 views

### Find the length of the piece of this curve where $x \geq \frac{3}{2}$

Consider the curve C which is the intersection of the two cylinders of equations $e^z=x$ and $x^2+y^2=2x$. Find the length of the piece of this curve where $x \geq \frac{3}{2}$ I have done the ...
287 views

### convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
72 views

### The heat equation shrinking convex plane curves

In the article "The heat equation shrinking convex plane curves" by M. Gage and R. S. Hamilton, I didn't understand the following theorem: ...
46 views

### The curvature blows up

A curve evolves according to the evolution equation $\displaystyle\frac{\partial X}{\partial t}= k \times N$ where $k$ is the curvature of the curve and $N$ is the inward unit normal vector. Then ...
55 views

### Why do left and right switch when direction is reversed? [closed]

If I make a left turn during a trip, it becomes a right turn on my way back. Why is this?
33 views

### Length of a curve in $\Bbb R^2$

How to compute the length of a curve given by the formula $$f: (0, \frac{\pi}{2}) \ni t \rightarrow ( \cos^3t,\sin^3t) \in \Bbb R^2$$ I know that the length of a curve in with image in $\Bbb R$ is ...
171 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 ...
222 views

### what would a planetary orbit look like if gravity had constant magnitude?

Consider a unit-mass particle that is always experiencing a single unit-magnitude force towards the origin. This is a central force, but it is not one of the familiar ones, e.g. gravity whose ...
284 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 ...
42 views

### Plane and Ellipse Intersection

Short Version: If some can solve the easier to read form as follows, I would be thankful. $$B = \frac{1 - d^{T}Bd}{ K_{1} } A$$ B^{T}d = \frac{1 - d^{...
155 views

### 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 ...
109 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 ...
441 views

Given implicit function $F(x, y) = 0$, how can I find its asymptotes? EDIT: Sorry, my calculations were wrong. Here is correct function: $F(x,y)=\sqrt{(x-a)^2 + (y-b)^2} - \sqrt{(x-c)^2 + (y-d)^2} ... 1answer 304 views ### From geometrical figures to function There's one basic mathematical thing that keeps bugging me: the fact that a really simple 2D geometrical figure (like a circle) might not be a function. I know what the definition of a function is. A ... 2answers 51 views ### Is circle the only Jordan curve with this property? When I was thinking about one problem that has to do with Jordan curves the problem which I am going to describe now, arose in my mind. And here it goes. It is known that for every$n\geq3$the ... 1answer 31 views ### Rectificable curve as a boundary of a convex set Let$K\subseteq\mathbb{R}^2$be a convex compact set. Is it true that$\partial K$(the boundary of$K$) is a rectificable curve (i.e. it has length)? 1answer 105 views ### Curvature and Circumference of Circle Theorem Let$\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 $$\int_{a}^b \kappa_N(s)ds=2\pi.... 3answers 803 views ### How to find the length of a curved path. We have to find a continuous model for a curved path which you then solve. A woman is running in the positive y-direction starting at x=50 (50,0) which is orthogonal to the x axis. At this point a dog ... 1answer 54 views ### If C \subseteq \mathbb{P}^2 is a plane curve, then genus(C)=\frac{1}{2}(d-1)(d-2). Compare with example in the notes In my Algebraic Geometry notes (see http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf) there is the following exercise: If C \subseteq \mathbb{P}^2 is a plane curve of degree ... 1answer 151 views ### what is the parametric form for “mystery curve”? Mystery curve found here looks like this : Was given by the complex formula :$$e^{it} – \frac{e^{6it}}{2} + i \frac{e^{-14it}}{3} $$Is the parametric form simpler or the polar form would be ... 1answer 87 views ### Tangent and normal vectors of a positively oriented curve Given the equation of positively oriented curve \mathbf{r}(s) = (x(s); y(s)), we obtain \mathbf{T} = ({dx \over ds}; {dy \over ds}) (tangent vector) \mathbf{N} = ({-dy \over ... 3answers 432 views ### Curves With Known Arc Length [closed] I would appreciate if you could list as many (planar) curves with known closed-form analytical expressions for the arc length as possible. Please include formulas for both the curve and the arc ... 2answers 66 views ### How can we define a 'periodic' parametric curve which is not closed? We all know that that the curve given by \gamma (t)=(t,\sin t) has a repeated pattern, even though it's not a periodic curve. Can we generalize somehow the definition of a periodic curve so that ... 3answers 106 views ### How to sketch -3x^2 - 8xy + 3y^2 = 1 [closed] The equation is as follows:$$-3x^2 - 8xy + 3y^2 = 1$$How to specify the axis of the given curve? How to as accurately as possible draw a curve defined by this equation? 1answer 52 views ### Carto-polar curve Is there any plane curve that has the same equation in cartesian as in polar coordinates? To be more specific, is there a function such that f(f(x)\cos x)=f(x)\sin x, \forall x? 1answer 62 views ### length of the curve y=x^n in the unit square Let l_n be the length of the curve y=x^n in [0,1]\times[0,1]. Then obviously \lim_{n\to\infty}l_n = 2. What about \lim_{n\to\infty}(n(2-l_n)) ? The formula l_n = \int_0^1\sqrt{1+n^2x^{2n-... 1answer 96 views ### How to create animated plot of curve depending on parameter in WolframAlpha? Is it possible to create in WolframAlpha an animated plot of a curve given by an equation, where some of the coefficients depend on the parameter (=on time)? For example if I would like to have a ... 2answers 155 views ### Solving non-linear second order differential equation: radius of curvature = k \theta I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates radius of curvature = k \theta$$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + ... 0answers 52 views ### 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 +(... 1answer 112 views ### 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 ... 1answer 46 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 ... 0answers 309 views ### 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 ... 1answer 133 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, \... 4answers 4k views ### 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 ... 1answer 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 ... 1answer 82 views ### 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. ... 0answers 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^{\... 3answers 984 views ### 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)},... 2answers 1k views ### 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)...
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 $P$
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}\...