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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Counting points on curve

Given degree $2$ $f(x)\in\Bbb F_p[x]$, what is the maximum number of $\Bbb F_p$ points $(x,y)\in\Bbb F_p^2$ on plane curve $$f(x)+x(x-1)y=0\in\Bbb F_p[x,y]?$$
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22 views

How to find a plane that is tangent to 3 spheres?

So there are spheres with radius of 1 centered at (1,2,0), (4,5,0) and (1,3,2). How can one find a plane that is tangent to all 3 spheres? Visually, it looks like as if the spheres are sitting on a ...
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1answer
31 views

Fitting an ellipse to a point with the first and second derivatives specified

I am trying to fit an ellipse to the end of a curve, $y(x)$, such that first and second derivatives of the curve, $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$, are preserved at the contact point, $(x,y)$, ...
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13 views

move curve normal to itself

I have a plane curve given by $y = f(x)$. At every point on this curve, I construct a normal direction to this curve and move the point a fixed distance $s$ along the normal to a new point. What is ...
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1answer
22 views

bending an arc to accommodate a constraint

I'm working with piecewise polynomial spirals: curves of the form $z(t) = z_0 + \int_0^t e^{i f(s)} ds$ where $f$ is a quartic polynomial determined by the tangent angles and curvatures at given ...
2
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0answers
64 views

Existence of a Rectifiable Piecewise Smooth Path

Suppose you have $\gamma(t):[0,1]\rightarrow \mathbb{C}$ simple piecewise smooth, $\gamma(0) = 0$ and $\gamma(1)=1$. Does there exist $\eta:[0,1]\rightarrow \mathbb{C}$, another simple piecewise ...
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24 views

Find the maximum volume of the pyramid bounded by the plane and the coordinate planes?

Surface $\sqrt{c}=\sqrt{x}+\sqrt{y}+\sqrt{z}$ , $(c>0)$ I found that at $(x_{0},y_{0},z_{0})$ a tangent plane to the surface is : ...
54
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2answers
5k views

Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
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1answer
121 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 $\,s(x,y)=0\,$? If I knew knew the parametrization of the curve I ...
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12answers
15k views

Derivation of the formula for the vertex of a Parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $y = a x^2 + b x + c$ My teacher gave me the formula: $x = -\frac{b}{2a}$ as the $x$ ...
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1answer
61 views

Not all tangents to a plane curve are bitangents

I'm struggling on a question from a previous qualifying exam, and I don't see a clean way to do it. Let $X\subset \mathbb{R}^2$ be a connected, 1-dimensional, real analytic submanifold, not contained ...
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1answer
526 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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49 views

Intersecting convex figures

Take in the real plane a finitely long horizontal line segment and connect the two endpoints by a convex path, above the segment, with the property that the only extreme points of the convex hull of ...
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1answer
58 views

Curvature of plane parametric curves

What is the neatest way to derive the following formula for the curvature of a parametric curve? $$\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $$
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44 views

A nonplanar closed curve such that the plane curve with the same curvature as function of the arclength is not closed

Find a non plane, closed curve such that the plane curve with the same curvature as function of the arclength is not closed. Been thinking a lot in this problem and haven't got a clue. Any ...
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4answers
43 views

Find the equation $ax + by + cz = d$ of the plane which has equal distance to the points $A(1, 2, 3)$ and $B(4, 5, 6)$

I was just wondering if anyone has any suggestions as to how to compute this equation? Find the equation $ax + by + cz = d$ of the plane for which every point has equal distance to the points ...
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1answer
364 views

fixed point projective geometry

I am thinking about the following: Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
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2answers
31 views

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
394 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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11 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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1answer
68 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
79 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
303 views

arithmetic and geometric genus for a reducible plane curve

If $C$ is an irreducible plane curve we have the well known formula relating the airthmetic (obtained via the degree-genus formula) and the geometric genus $$\frac{(d-1)(d-2)}{2} - \sum ...
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1answer
35 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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22 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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2answers
41 views

Characterizations of cycloid

There are several motions that create a cycloid. I have some examples here. Are there any others? Trace of a fixed point on a rolling circle Evolute of another cycloid (the locus of all its centers ...
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4answers
35 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
10 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
20 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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48 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
49 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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36 views

What's the area of the shape defined by all points whose distances from two focal points multiply to give the same product?

This shape, which I call the multiplicoid, is the equivalent of, and very similar to, an ellipse. However, instead of the distance between each point and the two focal points summing to a constant, ...
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1answer
33 views

The outer parallel (offset) curve of an ellipse [closed]

The inner edge of a track has equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. The track has uniform width $d$. What is the equation of the outer edge?
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2answers
42 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
135 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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28 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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38 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
54 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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2answers
59 views

Measure of curve smoothness

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness? I was thinking that since curvature measures how fast a curve changes, ...
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3answers
30 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
164 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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2answers
143 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
76 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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27 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
72 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
40 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
406 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
281 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
41 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
25 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)?