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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1answer
103 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 ...
1
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1answer
38 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 ...
5
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2answers
96 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 ...
1
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2answers
55 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 ...
0
votes
0answers
90 views

Finding the tangent vector to the vertical trace curves

Why does z have to be the derivative of f(x,y) in respect to x? I get that I am finding the tangent line with y constant. But I need help visualizing this. From ...
2
votes
2answers
81 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
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1answer
59 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. ...
0
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0answers
40 views

Approach to store result of intersecting $n$ planes

The equation of the plane is ax + by + cz + d = 0, where (a,b,c) is the plane's normal, and d is the distance to the origin. This means that every point (x,y,z) ...
2
votes
2answers
110 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 ...
11
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4answers
431 views

When is the moment of inertia of a smooth plane curve is maximum?

Given a smooth plane curve $(x(s),y(s))$, parameterized in arc length $s$, of fixed finite length $L$, its moment of inertia about its center of mass (axis perpendicular to the plane) is given as $$MI ...
1
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1answer
46 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 $E$ ...
1
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1answer
43 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 ...
6
votes
0answers
81 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 $S$. ...
0
votes
2answers
373 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 ...
1
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2answers
631 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 $e'(s) = \kappa(s)v(s)n(s)$ $n'(s) = -\kappa(s)v(s)e(s) + \tau(s)v(s)b(s)$ ...
0
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2answers
63 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
votes
1answer
38 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 ...
0
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2answers
609 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 ...
2
votes
2answers
271 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
71 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
101 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
113 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 ...
1
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1answer
58 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 ...
17
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4answers
477 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 ...
1
vote
1answer
159 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 ...
2
votes
0answers
60 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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0answers
161 views

Convex curve in R^2 pass through two point with fixed normal direction

Given two distinct point in $\mathbb{R}^2$ and two distinct normal directions what are some explicit convex $C^2$ curve (e.g. a map $[0,1] \to \mathbb{R}^2$ which satisfy many equivalent property) ...
0
votes
1answer
25 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
votes
1answer
116 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 ...
1
vote
2answers
302 views

Question on the catenary

The catenary minimizes the potential energy of a cable and has equation $y - y_0 = A \cosh (\frac{x-x_0}{A})$. It is physically intuitive that the catenary is unique, but is there a mathematical ...
3
votes
0answers
138 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 ...
9
votes
5answers
305 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
87 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
297 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
52 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
54 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 ...
3
votes
1answer
85 views

Genus of Edwards curve

Let us work over a field $\Bbbk$ of characteristic not equal to two. Let $d\in\Bbbk\setminus\{0,1\}$. It is said in the wikipedia article about Edwards curves that the plane quartic defined by the ...
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0answers
209 views

Definition of multiplicity of a point (in a plane curve)

In the book "Basic Agebraic Geometry I (third edition, 2013)" at page 14 Shafarevich says, about plane curves, what it follows: If $P=(0,0)$ and the leading terms (note:by leading terms I suppose ...
4
votes
0answers
121 views

In how few points can a continuous curve meet all lines?

Inspired by this (currently closed) question, I'm wondering about a related topic: given that we have a continuous parametric curve (that is, a curve of the form $(x=x(t), y=y(t))$ for ...
6
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1answer
268 views

Does every continuous everywhere but differentiable nowhere curve have an infinite length?

Given a curve $\!\,\gamma : [a, b] \rightarrow ℝ^2$ that is continuous everywhere but differentiable nowhere (or almost nowhere), is its length: $$\text{length} (\gamma)=\sup \left\{ \sum_{i=1}^n ...
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votes
2answers
261 views

What is a the intuition behind a parametric equation?

I have always used equations for the line (y=a + bx) in R2. Recently I came upon this thing called parametric equations. I cannot grasp the difference between them and the equations for lines that I ...
7
votes
6answers
3k views

A Math function that draws water droplet shape?

I just need a quick reference. What is the function for this kind of shape? Thanks.
2
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2answers
69 views

Find lattice points on a planar curve

I have the following curve in the plane: $$y = \frac{c-x}{6x+1}$$ Given a constant value $c \in \Bbb N$; is there a technique(s) I can apply to find lattice points on this curve?
3
votes
2answers
158 views

why $x^2 = y^3$ is not smooth?

I read a definition of a smooth curve on the plane: A smooth curve is a map from $[a,b] \to \mathbb R^2: t\mapsto ( f(t),g(t) )$, where $f$ and $g$ are infinitely differentiable functions. According ...
1
vote
1answer
116 views

Straightness measure for smooth 2-d plane curves of a given fixed length

Consider a smooth, 2-d plane curve of given fixed length $d$. Any straight line of length $d$, is also a curve of this type. What i am interested in is, How straight a curve of a fixed length, is? In ...
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votes
2answers
690 views

Conditions for a smooth parametric curve

A curve defined by $x=f(t), y=g(t)$ is smooth if $f'(x)$ and $g'(x)$ are continuous and not simultaneously zero. Why do we have the second condition(simultaneously zero)?
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2answers
146 views

Finding arc length and binormal vector for a given curve

Can somebody show me the arc length of a curve formula, and the binormal vector formula. The curve C with equation $r(t)=(\sqrt{3}\cos t,t,\sqrt{3}\sin t)$ How do you find the arc length from $t=0$ ...
2
votes
6answers
88 views

Why $\vec{r}$ is commonly use for vector equation?

I'm wondering why $\vec{r}$ is commonly use in mathematics (vector calculus, line integrals) and physics for denote the vector equation. Edit/Added clarification: I'm wondering why the letter $r$ is ...
3
votes
1answer
390 views

Minimize the sum of distances between two point and a circle

Let's $A$,$B$ and $O$ be random point in a plane, such that they are not colinear. Let's $c$ be a circle centered on $O$, such that points $A$ and $B$ are outside of it. Find a point $X$ that lies on ...
1
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1answer
53 views

References for implicit [intrinsic - I remembered wrongly] co-ordinates

When I was at school I learned about "implicit coordinates" for curves in a plane. Essentially these mapped the path in terms of arc length $s$ from a fixed point, and direction of travel $\psi$ ...