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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Limit of an expression as $x$ tends to a particular quantity on a curve

Say we have a curve like this: $$ xy + y^2x^2 = x.$$ Let's call it the curve $\mathcal{C}$. Say then that we want to find the limit of the quantity $x^2/y$ (if it exists) as $x \to - \infty$ on the ...
4
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1answer
86 views

Degree of ramification divisor and number of fixed points under a group action.

Let $C$ be a projective plane nonsingular complex curve and a finite group $G$ acts on $C$. Consider the quotient $f: C\rightarrow C/G=:C'$. Then, by Riemann–Hurwitz $2g_C-2=|G|(2g_{C'}-2)+\deg R,$ ...
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0answers
67 views

Parameterization of simple closed curve

A curve $Z$ in a two dimensional space is parametrized by $0\leq t < 1$ , and satisfies $Z(t) = Z(t+1)$. If it is sufficiently well behaved, it can be represented using a Fourier series with a ...
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2answers
41 views

Find the distance of the point $( 1,2,3 )$ to ,,,,,,,,,,,,,,

Find the distance of the point $( 1,2,3 )$ of the plane $3x-2y+5z+17=0$ .
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2answers
105 views

Curve parametrization

When plotting following implicit function $x^4 + y^4 = 2xy$, what is the best way to parametrize it? I tried $y=xt$, but then I get $x=\sqrt{2t\over {1+t^4}}$ and $y=\sqrt{2t^3\over {1+t^4}}$, which ...
3
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2answers
243 views

find maximum area

Consider a problem here : There is a wall in your backyard. It is so long that you can’t see its endpoints. You want to build a fence of length L such that the area enclosed between the wall and the ...
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1answer
47 views

Real valued function associated with the Diophantine equation $a^2(2^a-a^3)+1=7^b$

The parent question that maybe still remains to be answered at this moment is:Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$ . As far as the parent question is concerned, when generalizing to ...
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1answer
38 views

Domain and Range problem(plane)

Consider the function $z = \ln{(y + 1)}+\sqrt{x-3}$. Find the domain and range, and sketch the domain in the plane.
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1answer
28 views

show that $\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' (t)\rangle$

Let $\gamma,\eta:[a,b]\to \mathbb R^n$ be continuous, differentiable, curves. show that $$\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' ...
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0answers
40 views

PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...
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4answers
248 views

how to find the equation of one curve in XY plan which satisfies such functions?

I want to find the equation of one curve in $X-Y$ plan which satisfies the functions as follows: 1) $A(x_1,y_1)$, $B(x_2,y_2)$ are two known points and $f(x,y)$ (or $y(x)$) is the equation of the ...
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0answers
35 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
3
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1answer
131 views

Degree of the dual curve to $XY^2 - Z^3$

I have a question about the dual curve to the curve $C$ cut out by the equation $F(X,Y,Z) = XY^2 - Z^3 = 0$ in $\mathbb{P}^2$. (Assume that everything is over an algebraically closed field of ...
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1answer
96 views

Express parametric curve as graph of a function

I have a parametric curve in $\mathbb{R}^2$ given by $$ t\mapsto f(t)\left(\begin{array}{c}1\\1\end{array}\right)+\sqrt{-f'(t)}\left(\begin{array}{c}1\\-1\end{array}\right),\quad ...
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7answers
8k views

Drawing heart in mathematica

It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer ...
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2answers
199 views

Line of intersection/quadric surfaces

Let $C$ be the curve of intersection of the cylinder $\frac{x^2}{25}$ + $\frac{y^2}{9}$ = $1$ with the plane $3z = 4y$. Let $L$ be the line tangent to $C$ at the point $(0,-3,-4)$. What is the ...
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2answers
130 views

Plane intersection

The planes $5x + 3y + 2z = 0$ and $ 2x + 8y - 5z = 0$ intersect. Find the equation of the intersecting line. I get the parametric equation: $x = t$ y = $\frac{29}{34}t$ z = $\frac{-121}{170}t$ ...
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1answer
203 views

Relations between curvature and area of simple closed plane curves.

Let $\gamma$ be a simple closed plane curve. We know that a curve with constant curvature $\kappa$ will trace a circle in the plane. The radius of this circle is the inverse of its curvature. Now, ...
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4answers
47 views

Finding the arc length of $r(t)=ti+j+(\frac{1}{6}t^3+\frac{1}{2}t^{-1})$

How would I find the arc length of the following curve from $t=0$ to $t=2$ $r(t)=ti+j+(\frac{1}{6}t^3+\frac{1}{2}t^{-1})$ I took the first derivative and got ...
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1answer
39 views

To prove a relation for a smooth, asymptotic plane curve, in arc length parametrization.

Given a smooth plane curve, parametrized in arc length as $\alpha(s) \equiv (x(s),y(s))$ and given that $$\lim_{s \to \infty} \frac{y(s)}{s} = k,$$ $k$ a constant, and $$\lim_{s \to \infty}x(s) = 0,$$ ...
2
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1answer
116 views

Line tangent to curve in complex plane

For what pairs $a,b\in\mathbb{\mathbb{C}}$ is the line $L(x,y)=ax+by+1=0$ tangent to the curve $C(x,y)=x^4+y^4+1=0$? By definition of "tangent", if I have a point $(c,d)\in C$, and a line ...
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1answer
41 views

How to determine a point is outside or inside

How could I determine a point is outside or inside of a domain with variable raduis. like this: $$x(t)=(0.3+0.2(\sin3t))\cos t$$ $$y(t)=(0.3+0.2(\sin3t))\sin t$$ where 0$\leq t< 2\pi$. I tried ...
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2answers
101 views

What is the slope of the tangent at $(0,0)$ on the curve $x^2 y^2 = 4 x^5 + y^3$

This question is arising from the answer to another one: How find this equation integer solution: $x^2y^2=4x^5+y^3$ . For $x < 27$ and $y > -243$ , the basic equation $x^2 y^2 = 4 x^5 + y^3$ is ...
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2answers
75 views

Find a parametrization of the intersection curve between surfaces

Find a parametrization of the intersection curve between the surfaces $−3x^2+2z=10$ and $4x^2+10y^2=5$. You should parametrize such that $y=k\sin(t)$ for some constant k. The answer should be in ...
2
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2answers
113 views

When does this parametric curve cross itself?

Find the points where the curve given parametrically by$$\mathbf{r}(t)=\left(2+\cos\frac{3}{2}t\right)\left(\begin{matrix}\cos t\\\sin t\end{matrix}\right)$$crosses itself. So, I understand that ...
2
votes
1answer
67 views

There exists a constant arc length parametrization

I heard that for any curve in the plane that can be given parametrically by $\vec{r}(t)=\langle x(t),y(t)\rangle$ for $a\leq t\leq b$ that there exists a constant arc length parametrization, i.e. ...
2
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2answers
130 views

When is a curve smooth at points where $dy/dx$ does not exist?

The curve $y=x^{1/3}$ is smooth everywhere even though $dy/dx$ does not exist at $x=0$. Why? In general; Wherever $dy/dx$ does not exist on a curve, how can I show that it could still be smooth at ...
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1answer
49 views

Minimum distance from curve

I was thinking about the following problem: Let $\gamma \subset \mathbb R ^2$ be a curve that admits a $C ^{\infty}$ regular parametrization. Is it always possible to choose an open set $E$ ...
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1answer
48 views

Finding the orthogonal family of curves to a given family of curves. I am missing some.

Given the family of curves: $F(x,y,x_0)=0$ where $F(x,y,x_0) = (x-x_0)^2 + y^2 - R^2\ ,\ x_0 \in \mathbb{R}$ find the orthogonal family. This is my attempt: I first get the differential equation ...
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0answers
87 views

Shortest closed loop containing all extreme points of a convex set

Suppose $S\subset \mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to\mathbb{R}^2$ is a continuous map with $\Gamma(0)=\Gamma(1)$. Suppose $\Gamma$ passes through all extreme points of $S$. ...
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2answers
433 views

Moment of inertia about center of mass of a curve that is the arc of a circle.

Let $(x(s),y(s))$ be a smooth 2-d plane curve which is an arc of a circle of a certain radius $r$. Assume it is represented by an inelastic string $S$ of finite length, lying in a 2-d plane. Let there ...
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2answers
71 views

Curve is a submanifold

In our class today we said today that an 1-times continuously differentiable immersion $\phi:[0,1] \rightarrow \mathbb{R}^n$ is a submanifold of dimension 1 if it is closed such that $\phi(0)=\phi(1)$ ...
2
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1answer
41 views

Equation of a plane (not sure if I got this)

When I was doing my calculus midterm, I came across a question that I didn't really know how to solve, I think I skipped over these problems in my studies. The question is: Find an equation of the ...
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2answers
691 views

Finding vector and parametric equations provided only one point.

Normally to answer these questions I have a point and one or two vectors. However, for this one I only have a point. How can I concoct these equations provided there is limited information? Find ...
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2answers
322 views

Can an involute gear profile be modeled with a Bézier curve?

In the context of a game, I want to draw gears. The most common curves available on the platforms I'm using are third degree Bézier curves. Is there an exact representation of the involute gear ...
0
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1answer
74 views

$d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$

If $f: S_1\to S_2$ is a smooth map betweentwo regular surface $S_1$ and $S_2$, then $d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$ I cannot prove this statement. I think that ...
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0answers
134 views

How to see and proof that the hyperbola as a constant difference of distances holds for $\frac{1}{x}$?

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, which is the distance ...
0
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1answer
135 views

Hausdorff dimensions of smooth but non-rectifiable curves

Smooth curves of finite length have a Hausdorff dimension of 1. How about smooth but non-rectifiable (i.e. infinite-length) curves? Are they also of Hausdorff dimension 1, or does it depend on the ...
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1answer
83 views

Equation for this graph

I am trying to find an equation for the graph that crosses the Y axis at (0,100), X axis at (100, 0), is a curve with adjustable degree of "bending" and has an axis of symmetry y(x) = x. Here are few ...
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4answers
498 views

For which $L^p$ is $\pi=3.2$?

This is a silly question that came to mind after watching numberphile video on How Pi was nearly changed to 3.2. For which $p\in(1,+\infty)$ is the ratio of the perimeter of the $L^p$ disc in $\Bbb ...
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1answer
257 views

Show the curve $\alpha$ is differentiable and regular

Consider the map: $$ \alpha(t) = \left\{ \begin{array}{ll} (t,0,e^{-1/t^2}) & t>0 \\ (t,e^{-1/t^2},0) & t<0 \\ (0,0,0) & t=0 \end{array} \right. $$ a. Prove that ...
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0answers
69 views

Should a “good” equation divide the plane?

At the question Is there any equation for triangle? (MSE) the answer given by Henning Makholm received the most upvotes. Therefore let's define the following triangle function $H$ with (a,b,c) the ...
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1answer
27 views

r(t) curve vector

Here's the problem: Integrate $f$ over the given curve. $$ f(x,y) = \frac{x+y^2}{\sqrt{1+x^2}}\qquad C: y=\frac{x^2}{2} \text{ from } (1,1/2) \text{ to } (0,0) $$ In the solutions manual it says ...
0
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1answer
168 views

Why is this not a space-filling curve?

From Wikipedia, a space-filling curve is a curve (i.e. a continuous function whose domain is the unit interval $[0,1]$) whose range contains the entire 2-dimensional unit square. Many examples of ...
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0answers
385 views

Guilloché security printing — can it be cracked?

Money uses Security printing, and often uses Guilloché patterns. These curves are inscribed by wheels on wheels on wheels, ten wheels deep in some cases. For example, the back of the US $1 bill has ...
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5answers
313 views

Parametric equation of $x^y=y^x$ curve

What is the easiest/most natural way to parametrize the following curve? $$\{(x,y)\in\Bbb R^2\mid x^y=y^x, x\neq y\}\cup\{(e,e)\}$$ The best I could do was taking it apart, and for $x>y$ use ...
1
vote
1answer
93 views

Curve described by a point inside an ellipse

It's well known that a point inside a circle rotating on a line describes a trochoid of parametric equation: $$x=c_0\phi-c_1\sin(\phi)$$ $$y=c_2-c_3\cos(\phi)$$ in which the constant ...
2
votes
1answer
369 views

multiple tangent lines to a plane curve

Suppose that $C=V(F)$, with $F\in k[X,Y]$ ($k$ algebraically closed), is an irreducible plane curve such that $P=(0,0)$ is a nodal singular point (or an ordinary multiple point according to Fulton's ...
4
votes
1answer
54 views

Computing a quotient of rings

Let $R=k[x,y]/(y^2-x^2-x^3)$ and $I=(x,y)\cdot R \subset R$. I would like to show that $$ \bigoplus_{i=0}^{\infty} I^i\,/\,I^{i+1} \cong \,k[x,y]\,/\,(x^2-y^2). $$ Could you please help me? Remark: ...
2
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1answer
56 views

Is this union of tangent spaces a known object in Algebraic Geometry?

Let $C$ be a family of smooth (all but one curve is smooth, I'll explain later) cubic curves in $\mathbb{P}_{\mathbb{C}}^2$ with the following property: given any two elements $a,b \in C$, the curves ...