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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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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58 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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451 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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0answers
56 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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149 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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47 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
97 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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1answer
68 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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23 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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69 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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80 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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1answer
274 views

The Trace-Determinant Plane

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
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1answer
46 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
167 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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2answers
104 views

Curves with “constant speed”?

I am new to the concept of curves. Let us a assume we have a simple function such as $f:\mathbb{R}^+\rightarrow \mathbb R^+\quad f(x) := \sqrt{x}$. (Or $f(x)=\exp(x)$ or a polynomial etc.). We can ...
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5answers
267 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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3answers
196 views

Scalar Equation of a Plane

Determine the value of $k$ such that the line with parametric equations $x = 2 + 3t, y = -2 + 5t, z = k$ is parallel to the plane with equation $4x + 3y – 3z -12 = 0.$
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49 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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46 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 ...
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1answer
225 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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71 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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122 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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33 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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46 views

Osculating circles intersecting a given point

Well, the problem is a question in Montiel's book. How to prove that a planar curve $\alpha$ such that all osculating circles intersects a given point is actually a circle (or a part of it)? I've ...
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48 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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94 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
217 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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42 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 —> ...
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2answers
181 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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2answers
54 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
148 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
102 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
241 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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Areas and Volumes using integrals [closed]

I am practising for an engineering calculus exam from past year papers. My main problem topic is Integration. My exam is in one day and I need help on how to find areas and volumes of graphs using ...
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2answers
312 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
115 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
74 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
319 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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2answers
287 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 ...
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1answer
60 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 ...
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2answers
52 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 ...
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2answers
930 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
133 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 ...
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1answer
131 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 ...
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46 views

Approximately space-filling parametric curves from unit line to n-cube

I am trying to find the solution of TSP in 2D Euclidean space using parametric curves of two interpolation polynomials constructed so that at times $t_k$ the curve passes through point $p_t$. See my ...
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114 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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46 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 ...
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129 views

What is the parametric equations for the following closed curves?

First case Second case For the sake of simplicity, let the circle of radius $R$ be at the origin, the rectangle width and height be $w$ and $h$, respectively.
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93 views

Constructing shortest interpolation curve from points in $\mathbb R^2$ with parametric equations.

Assume we are given a set of $n$ points from $\mathbb R^2$, $(x_1,y_1),(x_2,y_2)\dots(x_n,y_n)$. We want to construct a path connecting all these points using a pair of parametric equations ...
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52 views

Evaluate $\|f\|$ on complex domain

Given a function $ f\colon \mathbb{C} \rightarrow \mathbb{C}$, how do I find $||f||_{\operatorname{Trace}(\gamma)}$, where $\gamma$ is a curve in the complex plane? In particular, I need to do this ...