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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84 views

Prove $\int_{[a,b]}f=\int_{[a,c]}f+\int_{[c,b]}f$

Let $a,b\in\mathbb C$ and $c\in[a,b]$. Let $f$ be continuous on $[a,b]$. Use the definition to show that \begin{equation} \int_{[a,b]}f=\int_{[a,c]}f+\int_{[c,b]}f \end{equation} Note: You should ...
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1answer
22 views

Multivariable Calculus: Intersection of surface and planes

I have this math problem: Let $S$ be the surface that consists of all points $(x,y,z)$ that satisfy the equation $x^2+y^2=z^2$. 1) What are the intersections of $S$ with horizontal planes ...
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1answer
41 views

Do Carmo DG Q. 1.7.2 finding the arc with a given length which bounds the largest area

I'm struggling with the following question for long. I tried to apply isoperimetric inequality $4\pi A\leq L^2$, but my attempt has been unsuccessful. Could anyone give me a hint? Let $AB$ be a ...
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1answer
38 views

Alebgraic curve and Riemann surfaces

How do we prove that any smooth complex algebraic curve $C\subset\mathbb{P}^2$ is a Riemann surface? Does there exist a complex version of the implicit function theorem?
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1answer
15 views

Deriving tangent plane equation from scalar equation of plane

There's one step in the derivation that I don't understand. I'll explain some of the derivation and then explain which step I don't understand. Begin with scalar equation of plane: A(x-x0) + ...
2
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1answer
41 views

Arc length of curve (regular condition)

I have a question regarding the defintion of arc length of a curve in $\mathbb{R}^n$. If $\gamma$ is a regular curve, the define arc length as $S(t)=\int_{t_0}^t|\gamma'(t)|dt$. Since $\gamma$ is ...
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1answer
37 views

(hyper) elliptic curve in characteristic two and the Jacobian criterion

Let $k$ be a field of characteristic two and let $E$ be a curve given by $$ y^2=x*(x+1)*(x^2+x+1)*(x^3+x+1)\quad\text{or}\quad y^2=f(x) $$ Now we have $dy^2/dy=2y=0$ and consider the Jacobian ...
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2answers
40 views

how to prove plane $ax+by+cz = d$ has normal vector $(a,b,c)$

given a plane function $ax+by+cz=d.$ How to show that unit normal vector is $n = \pm(ai+bj+ck)/(a^2+b^2+c^2)^{0.5}$
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2answers
154 views

Do line integrals along non-piecewise-smooth curves exist?

This article at Wolfram Mathworld has the following theorem on conservative vector fields: Theorem. The following conditions are equivalent for a conservative vector field $ \mathbf{F} $ defined ...
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1answer
40 views

Pull-back of regular map and rational function field

I don't understand what I'm missing in this example. Let $X=V(X_1^2+X_2^2-X_0^2)$ the circle in $\mathbf{P}^2_k$, being $k$ an algebraically closed field. Let be also $f:X\longrightarrow ...
3
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1answer
92 views

Is simple closed curve homeomorphic to a circle?

For sure every curve that is homeomorphic to a circle is a simple closed curve, but is every simple closed curve homeomorphic to a circle? Is there a proof for that, or is there some topological ...
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0answers
18 views

Determining the curvature of $y=Asin(bx)$

I am asked to determine the curvature of $y=Asin(bx)$. Unfortunately, I don't think I am on the right track. So the curvature of a curve is: $\kappa = \frac{1}{\lvert \vec{V}\rvert}\lvert ...
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0answers
76 views

Embedding a $deg \geq 4$ curve in $\mathbb{P}^2$

Reading the paper by Kollar "The structure of algebraic threefolds", among some examples he does talking about intrinsic and extrinsic geometry over $\mathbb{C}$, he mentions that "an irreducible ...
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1answer
501 views

Calculating equidistant points around an ellipse arc

As an extension to this question on equiangular fisheye distortion, how can I calculate equidistant points around an ellipse (or 1/4 segment of) given it's aspect ratio? When it's circular, I can use ...
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1answer
31 views

For which functions $r$ is the curve $r(t)(\cos t,\sin t)$ regular or unit speed?

Decide conditions on smooth function $r$ such that $\nu$ is a regular curve and decide functions $r$ for which $\nu$ has speed $1$. Let $r: \mathbb R \rightarrow \mathbb R$ be a smooth function ...
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1answer
26 views

Isomorphism from $\mathbb{R}^2$ to $\mathbb{D}$ such that lines become circular arcs

I'm currently working on a hobbyist math project that require taking lines on an infinite plane, and projecting them onto a finite (euclidean) surface such that intersections are preserved. Does ...
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0answers
23 views

Laplacian discretization for parametric curves

I know how to compute the discrete Laplacian of a graph and of a mesh (the Laplace-Beltrami operator). Is there an analogous definition for the computation of the Laplacian of a parametric curve ? ...
2
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2answers
57 views

How to explain why a curve is on a cylindrical surface?

The question may be a bit general but I'm unsure about how to define it. I have a curve: $\vec r(t) = (2\cos(t),2\sin(t), 2t)$, for $0\le t \le 2\pi$, The problem I'm trying to pose : "Show that ...
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1answer
42 views

critical point for the curvature does not correspond to a local maximum/minimum.

Draw an example where a critical point for the curvature does not correspond to a local maximum/minimum Does the curve for infinity sign satisfy this? I am having trouble seeing why it's true, if of ...
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0answers
15 views

The Cusp $w^2 + p(z,w)=0$ is desingularizable in the origin $O \in \mathbb{C}$

I have just studied a method in projective geometry over complex numbers on how to desingularize a curve in a point but i'm a little bit confused. I don't know the name of this classical method in ...
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1answer
61 views

Parameterizing the path of a point on a circle rolling on another circle

Problem: A wheel of radius $a$ rolls on the outside of a circle with radius $b$ (see figure). Find the parameterization for the curve a point on the wheel follows. You may choose freely how you ...
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1answer
41 views

The Point of Tangency Between a Sphere and a Tangent Plane

Find the equation of the sphere centered at (2,0,-3) that is tangent to the plane x=y. What is the point of tangency? Describe the interior of the sphere with an inequality. What I have thus far: ...
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19 views

Points at infinity correspond to asymptotic slopes

Let $ P^2\mathbb{C} = \{ [a, b, c] | a,b,c \in \mathbb{C}^* \} $ the complex projective plane. So $ [a,b,c] \sim [x,y,z] $ iff $ \exists \lambda \in \mathbb{C}^* \colon \lambda(a,b,c) = (x,y,z) $. In ...
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1answer
184 views

Fourier transform for a 2D curve

I am stuck on the following problem about a Fourier transform of a 2D curve: I have to calculate the Fourier transform (using 1D complex FT) (and the opposite of it) for a 2D curve z(t). The curve is ...
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2answers
76 views

Equation of a vertical plane given $2$ points

A vertical plane passes through points $(1, -1, 1)$ and $(2, 1, 1)$. With three points on a normal plane, I just found two vectors and found the normal by cross-product of the vectors, but I'm not ...
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1answer
671 views

Circumference of a superellipse?

Could someone help me formulate the circumference of a superellipse? $$\frac{x^n}{a^n} + \frac{y^n}{b^n} = 1$$ If it makes things easier, I'm considering only the cases $n>2$, and ...
2
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1answer
56 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. But I don't get why $\cos \angle PFK = \frac{dx}{ds}$. Can someone ...
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1answer
42 views

Degree of min distance function between two algebraic curves

Suppose I have two algebraic curves $C_1$ and $C_2$ in the plane. I would like to find the minimum distance between the two curves. If the two curves have degrees $n_1$ and $n_2$, what is ...
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1answer
24 views

How can I accurately depreciate a set of elements?

I have the following function to determine the value $v$ of an element: $$a =\space\text{age in days} \\ v = a\left(\frac{-1}{730}\right)^\frac{5}{8} + 1$$ My intention is for each element, ...
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3answers
347 views

Arc Length Formulas

Use the arc length formula to find the arc length of the upper half of the circle with center at $(0,0)$ and radius $3$. Also, find the arc length of the curve in the first question by using ...
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1answer
87 views

Detection of self intersection point of curve

What numerical procedure is be adopted to detect self-intersecting parametrized points $ [x(t), y(t) ] $ in $ \mathbb R^2 $ ? Observation : @ roots ( t= 2, t=-1 ) parabola has double value with ...
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5answers
483 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 ...
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1answer
299 views

s-shaped reverse logistic curve

Is there any curve that grows very slow at the beginning then growth picks up exponentially before hitting the wall. I need sort of reverse behavior of the logistic curve ...
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1answer
17 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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9 views

Rotations at point of concurrency

$ L_1 = 0, L_2 = 0, L_1 + \lambda \cdot L_2 = 0 $ are equations of concurrent curves where $\lambda $ is an arbitrary constant. What is the geometric significance of $\lambda$? Does a curve ...
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2answers
37 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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2answers
74 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
249 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
63 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
301 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- ...
0
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1answer
62 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
30 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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0answers
24 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
41 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
358 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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0answers
20 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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0answers
25 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
26 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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0answers
17 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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0answers
59 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 ...