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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336 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
486 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 $e'(s) = \kappa(s)v(s)n(s)$ $n'(s) = -\kappa(s)v(s)e(s) + \tau(s)v(s)b(s)$ ...
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2answers
60 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 $\phi(0)=\phi(1)$ ...
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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 ...
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2answers
521 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
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2answers
212 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 ...
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1answer
66 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
89 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 ...
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1answer
86 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
45 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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461 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
121 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
58 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
152 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) ...
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0answers
65 views

Inverse of a pythagorean hodograph curve.

I'm trying to solve this question about pythagorean hodograph curves: If $\alpha$ is a Pythagorean hodograph curve, and $\alpha(t)\neq(0,0)\;\forall t$, then the curve $\frac{1}{\alpha(t)}$ is also a ...
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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 ...
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1answer
100 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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2answers
243 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
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0answers
103 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
284 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 ...
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1answer
69 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 ...
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1answer
222 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 ...
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1answer
50 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: ...
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1answer
50 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
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1answer
82 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
154 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 ...
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0answers
35 views

Singular point of a plane curve: the geometrical meaning

Consider a plane curve $C\subset\mathbb C^2$ where $$C=\{(z,w)\in\mathbb C^2\,:\, P(z,w)=0\}$$ A singular point of $C$ is a point $(z_0,w_0)$ such that $\frac{\partial P}{\partial ...
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0answers
50 views

Coordinate rings of different curves, and what they're isomorphic to

I'm trying to teach myself about coordinate rings, and algebraic geometry in general, through examples, but I'm struggling a bit. Apparently the coordinate ring of $(t, t^2, t^3) \in \mathbb{A}^3$ ...
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105 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 ...
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1answer
243 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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0answers
43 views

A simple exercice about curves. [duplicate]

Somebody can to give me a hint about this exercise? I don't know how proceed. I try to show that $\alpha$ has curvature zero, but I have no successfully. Prove that a curve $\alpha : I —> ...
4
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2answers
225 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 ...
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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.
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2answers
60 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?
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2answers
152 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 ...
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1answer
111 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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2answers
416 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
131 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 $t=0$ ...
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6answers
75 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
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1answer
338 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 ...
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1answer
52 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 $\psi$ ...
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3answers
99 views

Non-linear function on $\mathbb{R}^2$ preserving the origin and maps lines onto lines?

Is there an $f:\mathbb{R}^2 \to \mathbb{R}^2$ such that: $(0,0)\mapsto (0,0)$; and for any $a,b,c$ with $a^2 + b^2 >0$, the set $A=\{(x,y):ax+by=c\}$ is mapped onto ...
3
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2answers
339 views

Do simply connected open sets in $\Bbb R^2$ always have continuous boundaries?

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 ...
2
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1answer
61 views

Separation of variables (ODEs)

Here is the question I am currently stuck on: Here is what I have done so far: My apologies as I understand this post seems fairly lengthy. However I cannot seem to get the final answer ...
4
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2answers
56 views

Rotations around the origin

The problem is to find the number of rotations around the origin for the function $$f(z)=z^{2013}+2z+1 $$ when $z$ moves through $\left\{|z|=1\right\}$. I tried to solve it with the help of argument ...
2
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1answer
165 views

Support function of a convex domain

Let $Q$ be a compact, convex domain in the plane, with smooth boundary $\partial Q$. We further assume that the origin is contained in $Q$. For a concrete example, let's take an ellipse ...
2
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1answer
175 views

Equation for a distorted circle

When you view a circle posted on a wall at a distance and at a glancing angle, the circle elongates. However, I don't think it is just an ellipse because it will also become asymmetric. It has more of ...
2
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2answers
126 views

Self-intersection of vector valued function

A vector valued function $r(t)$ has the following coordinates: $$x = 4\cos\left(\frac12t\right)+2\cos(2t)+\cos(4t)\\ y = 4\sin\left(\frac12t\right)+2\sin(2t)+\sin(4t)$$ I have to find the $t$-values ...
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129 views

curve which crosses itself at every point

Does there exist a continuous curve $C$ on $\mathbb{R}^2$ that crosses itself at every point on $C$? For two pieces of curve to cross and not just touch, each must have some length either side ...
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1answer
48 views

Minimum/Maximum Extremas in a Plane

Explain one way in which extrema of a function of two variables f(x,y) defined on a closed, bounded region of S of plane differ from extrema of a function of one variable f(x) defined on a closed ...