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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Calculating the curvature of a tractrix

I'm trying to find the curvature of a tractrix expressed in the form $r(t)=(\sin{t},\cos{t}+\ln(\tan{(\frac{t}{2}))} $. From what I've found on the Internet it appears that people arrive at the ...
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1answer
781 views

s-shaped reverse logistic curve

Is there any curve that grows very slow at the beginning then growth picks up exponentially before hitting the wall. I need sort of reverse behavior of the logistic curve ...
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14 views

Plotting a Curved Plane

How would one go about plotting a curved surface? Can this be done if one were to obtain three equations, the same way a non curved plane is plotted?
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76 views

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
80 views

Metric on the space of plane curves

I am looking for a metric $d$ for smooth 2D curves. Hence $d(x,y)$ is the distance between the curves x and y. For the moment, we may assume that $x$ and $y$ are just directed line segments. Do you ...
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1answer
106 views

Piecewise Smooth Curve

Is the curve defined by: γ(t) = (t,t) for 0≤t≤1 and (2-t,2-t) for 1≤t≤2 piecewise smooth? My logic says yes because one can break it into a finite number of smooth curves (two in this case), but ...
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0answers
36 views

Green's theorem and 'simple regions'

I'm looking through at my notes and couldn't understand why, in the notes below, there is a need to compute the curve C2 and C4. It's hard to see why isn't computing C1 and C3 a sufficient ...
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4answers
76 views

Equation perpendicular to 2 non-parallel planes

Good day sirs! Can you help me with this questions? Find the general equation of the plane: (1) Through $(3,0,-1)$ and perpendicular to each of the planes $x-2y+z=0$ and $x+2y-3z-4=0$ (2) ...
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0answers
25 views

2D trajectory in minimum amount of time given min/max acceleration per axis

I am having a little problem with determining a trajectory. I have a 2D curve, say $\{x(\lambda), y(\lambda)\}$ where $\lambda$ is not time; it is only a parameter that describes the evolution of the ...
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1answer
91 views

Finding the equation for the tangent plane to earth given latitude and longtiude

I'm creating a program where I need to calculate the equation of the plane tangent to the earth at a given latitude and longitude. I used Projecting an Arbitrary Latitude and Longitude onto a Tangent ...
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0answers
301 views

How to tell whether a curve has a regular parametrization?

A parametrization of a 1-dimensional curve is called regular if its velocity is always positive. For example, the following parametrization: $$x(t)=t^3, y(t)=t^6$$ is not regular because its ...
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1answer
59 views

Curve with with curvature $k(s)\ge 1$ everywhere has diameter $\le 2$

Let $\alpha(s)$ be a simple closed plane curve. Define the diameter $d_\alpha$ of $\alpha(s)$ to be $$d_\alpha = \sup_{t,s\in\mathbb R} \| \alpha(s) - \alpha(t) \|.$$ Assume the curvature ...
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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 ...
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2answers
243 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 ...
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0answers
65 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: ...
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0answers
43 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 ...
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2answers
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?
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3answers
32 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 ...
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2answers
170 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 ...
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201 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 ...
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1answer
243 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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0answers
40 views

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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1answer
141 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 ...
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1answer
97 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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2answers
434 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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1answer
302 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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2answers
50 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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1answer
30 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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1answer
98 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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3answers
745 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 ...
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1answer
50 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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1answer
135 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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1answer
81 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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406 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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2answers
61 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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3answers
104 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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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 $f(f(x)\cos x)=f(x)\sin x, \forall x$?
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1answer
58 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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2answers
209 views

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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1answer
88 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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2answers
145 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 + ...
2
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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 ...
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1answer
97 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 ...
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1answer
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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293 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 ...
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1answer
109 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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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 ...
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1answer
94 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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1answer
80 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 $|F_1P|+|F_2P|=2a$. ...
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32 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 ...