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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Reparametrization of curve

I found this in a book: Let $\alpha(t) = (g(t),h(t),0)$ be a regular curve in the $xy$-plane. If $g'(t) \neq 0$ for all $t$, then $g$ is strictly increasing. Thus, it is one-to one and has an inverse ...
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2answers
20 views

derivative of closed parametric curve

Suppose, a parametric function $\beta:[0,1]\mapsto\mathbb{R}^2$ is a closed curve, that is $\beta(0)=\beta(1)$. For example $\beta(t)=(\sin 2\pi t,\cos 2\pi t)'$. Then my question: Is the derivative ...
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18 views

Definition of roulette (curve)

I want to give a formal mathematical definition of a roulette, a curve described by a point attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. I ...
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2answers
70 views

Prove using an example that there is no plane on $\mathbb{R}^3$ that contains every group of 4 points

Well, this is a homewrok question (which I know I should not be asking, but I cannot find an answer to this anywhere): The exercise is as follows: i) Find the equation of the plane of $\mathbb{R}^3$ ...
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3answers
28 views

determine unit outward normal vector on a curve

It is necessary for me to find unit outward normal vector for the curve: $$\gamma=(x(t),y(t))$$ where $$x(t)=(0.6)\cos(t)-(0.3)\cos(3t)$$ and $$y(t)=(0.7)\sin(t)+(0.07)\sin(7t)+(0.1)\sin(3t)$$ I ...
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1answer
24 views

Roulette (curve) parameterization

I was wondering about the parameterization of a roulette on Wikipedia. A roulette is a curve formed by a point associated to one curve as it rolls upon another fixed curve. Wikipedia says, if $f$ is ...
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3answers
285 views

Need help with Curves and parameterizations

I'm having some trouble solving a couple of problems: I know this one must be pretty easy but can't find the way to solve it. I need to find the arc length of a curve described by $ r=1- ...
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17 views

Delaunay surfaces - plane as surface of revolution

According to Wikipedia (and other sources) "Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These ...
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1answer
40 views

Intersection numbers of plane curves - constructing a counterexample.

There is the theorem that if $P$ is a simple point on a plane curve $F$, then for any plane curves $G,H$ we have $I(P,F\cap(G+H))\geq\min(I(P,F\cap G),I(P,F\cap H))$. I need to find a counter-example ...
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1answer
16 views

Gradient as Normal Vector Problem

Here's my problem: Use the normal gradient vector to determine the equation of the line/plane tangent to the given curve/surface at point P. $x^4 + xy + y^2 = 19$, P(2,-3) I know how to use the ...
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1answer
351 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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19 views

Gradient Vectors as Normal Vectors: How to Find D in Form Ax + By + Cz = D?

I've got a question that looks like this: Use the normal gradient vector to determine the equation of the line/plane tangent to the given curve/surface at point P. $x^4 + xy + y^2 = 19$, P(2,-3) I ...
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1answer
31 views

Finding the uniformising parameter of a DVR [duplicate]

I am looking to find the intersection number of the affine plane curves $F=Y^2-X^3+X$ and $G=(X^2+Y^2)^3-4X^2Y^2$, and I need to do it with the order function of the local ring of $F$ at the origin. ...
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2answers
314 views

It is physically intuitive that the catenary is unique?

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 ...
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16 views

Parameterized Curvature Problem

Massively deviating from the given answer on this one, have no idea what's going on. Find the curvature of the following function: x = 5cos(t) ; y = 4sin(t) ; t = $pi$/4 Formula for curvature is k ...
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14 views

Definition of an “arc” and possible error in proof of length of projection of regular $1$-set in $\mathbb{R}^2$.

Here is an extract from Falconer's The Geometry of Fractal Sets. I cannot see how an "arc" is defined and was wondering whether someone could help me with the definition. Also if $ \begin{align} ...
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1answer
291 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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1answer
28 views

Perpendicular line to plane

Consider the plane $\large2x+y-5z=7$ and the line with parametric equation $\large r=r_0+tu$ a) Give a value of u which makes the line perpendicular to the plane. So I'm confused on what to do. I ...
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23 views

Determine nodes inside a curve

Let $$x=0.5\cos(t)-0.3\cos(3t)$$ $$y=1.2+0.6\sin(t)-0.07\sin(3t)+0.2\sin(7t)$$ How could I know an arbitrary point is inside or outside of this curve? Also, another ...
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3answers
20 views

Functions that go up two and then down two

I'm trying to make a function that goes up two and then down two (kind of like sin(x) but without the curves). I keep drawing a blank on what I can do to even create this functions as I haven't done ...
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14 views

Curve fitting using a graph extracted from an article, is it possible?

I want to curve fit a graph from an article which I can only extract from the pdf file as a screenshot. Therefore, I do not have the coordinates of the data points explicitly, yet I know that the ...
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44 views

How to find the tangent vector to a curve at a special point

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. The line segments $AC$ and $BC$ make equal angle $\alpha$ with the horizontal plane through $C$. The angle ...
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1answer
44 views

$dx$-notation in analysis

In the context of integrals and differential equations, often the symbol $df$ or $dy$ appears, where in some previous steps $f$ and $y$ were functions. What do these symbols mean $df$ and $dy$? ...
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47 views

Showing that a specific curve is regular.

Define the curve by $c(t):=(sin(pt)+r)(cos(qt),sin(qt))$ for $p,q \in \mathbb{Q}$ and $r\in \mathbb{R}$. Determine for which $p,q$ is the curve regular, i.e. $c'(t) \neq (0,0)$ for any $t\in ...
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50 views

Total curvature of a piecewise smooth planar curve

I am trying to show that the total curvature of a piecewise smooth planar curve $\alpha$ has the property that $\displaystyle \int_{C}\kappa $ $\text{d}s + \displaystyle \sum_{j=1}^{l} \epsilon_{j} ...
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24 views

Relative motion between line/circle

By rolling a (line,circle) on a fixed (circle, line) in $\mathbb{R}^2$, the locus of a point on the former is an (involute, cycloid). By what procedure can we inter-convert their parametric ...
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23 views

Is there a unique solution?

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a given continuous function and $t_0\in (a,b)$ a fixed point. Is it true that the following problem has a unique continuous solution ...
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1answer
100 views

Rolfsen exercise, chord theorem

Here's a problem from Rolfsen's Knots and Links that has me scratching my head: Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to ...
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1answer
32 views

Focus of a rolling parabola traces a catenary - geometric explanation

It is known that the focus of a rolling parabola along the x-axis traces a catenary. I'm interested in a geometric explanation. I found this derivation in The surfaces of Delaunay by J. Eells in The ...
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31 views

Another type of primitive

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $\Vert\mathbf{v}(t)\Vert=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}\colon (a,b)\to\mathbb{R}^2$ so ...
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1answer
26 views

Another type of derivative, another type of differential equation

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Is it possible to find a continuous function $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so ...
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0answers
21 views

A curve is contained in a circle [duplicate]

I need to prove that if $\alpha: I\rightarrow \mathbb{R}^{2}$ is regular with curvature $\kappa$, then $\alpha$ is on a circle with radius $r>0$ if and only if $|\kappa(t)|=\dfrac{1}{r}$.
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46 views

Tangent vector for a curve defined by a discrete set of points

I have a curve defined by a discrete set of points (x,y). How can I approximate the tangent vector at a point for such a curve?
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3answers
40 views

Chess Board Coloring of a Paper using an Arbitrary Curve

Pick a piece of paper and a pen. Put the pen on a starting point and begin to draw an arbitrary curve and don't withdraw your hand until you reached the starting point. You can meet your curve during ...
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1answer
94 views

Differential geometry: Conformal map

Let $f:\mathbb{R}_{>0} \times (0,2\pi) \rightarrow \mathbb{R}^3$ $$f(t,\xi) := (r(t) \cos( \xi) , r(t) \sin(\xi),z(t))$$ be a surface of revolution, where we assume that $r>0$ and ...
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1answer
30 views

How can I get a smooth distortion on a circle with a function g(x,y)

Let's say, $$f(x,y)=x^2+y^2=1$$ gives the unit circle. Now I would like to get a smooth distortion on the circle with a function $g(x,y)$. my guess is to consider the perimeter as one dimension, so ...
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0answers
43 views

Cusp singularity

What is the most general way of defining cusps singularities for a continuous curve $\mathbf{r}:(a,b)\to\mathbb{R}^2$ and how can we define them? What are the minimum requirements for the curve ...
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1answer
13 views

Image of a locus via stereographic projections

Yesterday evening I was playing around in my head with stereographic projections and I've come up with this idea. Let $\gamma(t)=(x(t),y(t))$ be a certain curve on a plane. Define a new curve ...
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3answers
2k 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 ...
0
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1answer
29 views

Maximum of the length of a curve and its curvature

I'm trying to solve this problem but I can't find any way to do it. Some hints or helps will be very useful and I'll be very thankful. The problem says: Let $\alpha : (a,b) \rightarrow \mathbb{R}^2$ ...
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1answer
20 views

troublesome area problem

Calculate the area of the region of the graph bounded by: $$\begin{eqnarray} y &=& x \\ y &=& x^2 + 1 \\ y &=& 2 \\ x &=& 0 \end{eqnarray}$$ My final result is ...
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0answers
10 views

Equation for a Vinyl curve

This video seems to show an explicit map from the torus to $\mathbb{R}^2$. Does it factorize through the projection $\mathbb{R}^3 \to \mathbb{R}^2$? What is the equation of the curve?
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10answers
357k views

Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
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23 views

Show that if $v_1$ and $v_2$ are any two vectors in this plane, then for any two scalars, $c_1v_1 + c_2v_2$ is also a vector in the plane

Let $a,\,b$ and $c$ be constants (not all zero) and consider the equation $ax + by + cz = 0$, which has a graph that is a plane that passes through the origin in $\mathbb{R}^3$. Show that if $v_1$ and ...
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1answer
18 views

Distance between two skew lines

I have 2 skew lines $L_A$ and $L_B$ and 2 parallel planes $H_A$ and $H_B$. The line $L_A$ lies in $H_A$ and $L_B$ in $H_B$. If the equations of $H_A$ and $H_B$ are given like this: $x+y+z = 0$ (for ...
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1answer
45 views

Torsion vs Curvature of a curve

I was wondering if there is a simple explanation of the torsion and curvature in $\mathbb{R}^3$ of a curve. The curvature measure somehow the acceleration perpendicular to the tangent vector, but ...
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1answer
39 views

Does every plane curve contain a rational point?

Does every plane curve contain a rational point? I think the answer is yes, but I can not prove this. Please help. However, if it is possible to build a pathological curve - without rational points, ...
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1answer
18 views

Question on a function defined on some plane curve.

Let $C$ be the plane curve defined by $y^d =f(x)$, where $f(x) \in \mathbb{C}[x]$ is a polynomial. Let $g(x,y)$ and $h(x,y)$ are polynomials relatively prime with each other. We consider $r(x,y) ...
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1answer
42 views

Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$

Let $x,y,z$ be elements of $\mathbb{R}^2$ Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$ d is usual euclidean metric.
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453 views

Reconstructing a curve from its curvature

I'm with a problem in an exercise form Do Carmo's Differential Geometry of Curves and Surfaces; it is number 9 section 1.6. I have a differentiable real function $k(s)$, $s\in I$, and I need to show ...