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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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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157 views

Proving $\dfrac{dN}{ds}=-\kappa T+\tau B$

Currently revising for a differential geometry exam. The question I am working on is one of those types where the next part of the question follows from the last. I've gotten to the point where I have ...
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2answers
106 views

Suppose $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ lie on a plane through the origin.

What determinant is zero? What equation does this give for the plane? I need some help here, am pretty stuck
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130 views

Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
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248 views

Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
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49 views

Lattice Points in x-y plane

What are Lattice Points? Which points in x-y planes are Lattice Points? Is (m,n) a lattice point where m,n are any integers?
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38 views

Two definitions for a smooth curve equal.

I've encountered these two definitions: 1. $\gamma\colon [a,b]\longrightarrow\mathbb{R^3}$ is smooth if all three derivatives exist and $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$ ...
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64 views

injective curve inside curve

I am struggling to prove the following intuitive result: Take $\phi:[a,b]\rightarrow \mathbb{R}^{n}$ a continuous mapping with $\phi(a)\neq\phi(b)$. Then there is a continuous injective mapping ...
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75 views

Paths followed by Morley triangle vertices as apex moves parallel to base

Let the vertices of a triangle $T$ be $(A,B,C)$, and $(a,b,c)$ the vertices of its Morley triangle $M$. Designate vertex $C$ as the apex of $T$. Now move apex $C$ parallel to $AB$, all the while ...
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1answer
450 views

Understanding the Spiro Spline

My name's Wray. This is my first time here. Firstly, I like curves. I've been keeping a pet project for a long time that would implement a delightful new curve-interpolation algorithm named the Spiro ...
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1answer
125 views

How to estimate curve length from random points along a (possibly not connected) curve

I'm working with a closed, smooth, planar curve. I don't know the curve exactly, but I do have an effectively-random collection of points along the curve. What's a good way to estimate the curve ...
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Why is $\sum^{n}_{i=1} \int_{t_{i-1}}^{t_i} = \int_{a}^{b} $?

This is a part of my proof: $$\begin{align} \left| \sum^{n}_{i=1} V(r(\tau_i)) \cdot \int_{t_{i-1}}^{t_i} r'(t) dt - \int_{a}^{b} V(r(t)) \cdot r'(t) dt \right| &\leq \sum^{n}_{i=1} ...
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35 views

Intersection of planes

A line perpendicular to the plane $ 3x-5y+4z-11=0 $ passes through the origin. At what point does this normal intersects the plane?
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26 views

finding the line of intersection

Find the line of intersection between two planes x+y+z=1 and x-2y+3z=1 ? I found r1,r2,n1 and n2 but I don't know what are the other steps
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141 views

Is this variant of the Jordan Curve Theorem true?

This feels as though it should be falsifiable, but it's not immediately obvious to me. The informal version of the statement is 'for every non-intersecting curve between two opposite corners of a ...
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1answer
126 views

How do I turn a “broken” plot into a smooth curve

I developed and solved a differential equation that predicts fluid temperature along the length of a long pipe with time. Analytical solution is such that it is causing a "discontinuity" in the ...
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28 views

What is the meaning of “slope of ca”?

I'm reading a paper, when this article refers to the function: $$\beta(v)=\frac{(\frac{v}{I})^k}{1+(\frac{v}{I})^k}$$ It say that "around the $I$, $\beta$ is approximately linear in $I$, and has a ...
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Ambiguous Curve: can you follow the bicycle?

Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
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228 views

Equation for the Maximum-Area Rotor in a Wankel Engine

I am attempting to construct a specialized Wankel Engine in Autodesk Inventor. For my particular project, the rotor must take up the maximum area, in order to separate the air spaces of the top and ...
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104 views

Harris Exercise 5.13 (points in general position)

I have a question about the second part of Exercise 5.13 in Harris' Algebraic Geometry: A First Course: Given $n \leq 2d + 1$ points in $\mathbb{P}^2$, characterize the subset of $(\mathbb{P}^2)^n$ ...
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142 views

How to estimate/determine surface normals and tangent planes at points of a depth image (point cloud)?

I have depth image, that I've generated using 3D CAD data. This depth image can also be taken from a depth imaging sensor such as Kinect or a stereo camera. So basically it is depth map of points ...
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1answer
704 views

arc length parameterization of planar curve in Matlab

Let $\gamma (t)$ be a planar curve parameterized by time $t$. For fun, let it be a limacon. In Matlab $\gamma (t)$ looks like this. ...
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31 views

Lengths of Plane Curves - Calculus 2: $\sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2}$

$$ \sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2} $$ I am having problems setting this up. Taking the derivative of $\sqrt{1-x^2}$. Leaves me with: $$ \frac{1}{2}\left(1-x^2 ...
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A hyperbola as a constant difference of distances

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 $2a$, the distance between the two ...
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74 views

Curvature of the boundary curve of convex set

I've got a very simple question, but I can't get a rigorous proof. Suppose that $X\subset\mathbb{R}^2$ is a convex set, and suppose further that $\gamma$ is a closed regular curve with image $\partial ...
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Find the area of the parallelogram with vertices (4,1), (6, 6), (7, 7), and (9, 12).

I am trying to find the area of the parallelogram with vertices (4,1), (6, 6), (7, 7), and (9, 12). So I believe the way to solve this problem is through the cross product and then taking the ...
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2answers
73 views

Finding singularities of a projective curve

For $w \in \mathbb{C}$ we define the projective curve $$p(x,y,z):= x^3+y^3+z^3+wxyz.$$ Now I have to find all $w \in \mathbb{C}$ for which the projective curve $p(x,y,z)$ is singular and show that ...
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69 views

How can I find a curve based on its tangent lines?

Let's say for some curve its tangent lines at every point have a property that the length of a segment within the first quarter $[0;+\infty)^2$ is exactly $C>0$. How can such a curve be defined ...
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38 views

$r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$

What does the following equation represent? $r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$ My approach: I factorized the equation and it became $(a+r\cos\theta)(a-r)=0$ I feel that ...
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69 views

Green's Theorem and Divergence (2D)

I am reading the book Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson. In Chapter 1 he talks about the Possion Equation, and to prove that FEM ...
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56 views

Determing if a parametric curve is smooth

I have to determine whether the following curves are smooth or not and I'm having trouble with the following two functions: Consider $f(t) = (t^{2}-1,t^{2}+1)^{T}$ The solution states: $f'(t) = ...
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227 views

How can a simple closed curve not look locally like the rotated graph of a continuous function?

A simple closed curve is a continuous closed curve without self-intersections. The question of whether you can inscribe a square in every simple closed curve is currently an open problem, but this ...
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33 views

Manipulating An Equation into A Workable Form

The question asks me to find the arc length of $$y= (x-x^2)^{1/2} + \sin^{-1}(x^{1/2})$$ I know I need to take the derivative: $$\frac{1-2x}{2(x-x^2)^{1/2}} + \frac{1}{(1-x)^{1/2}}$$ I've tried ...
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132 views

GNU Octave draw figure of 2 planes

How can I draw two planes in same figure in GNU Octave? $$ x + y + z = 1\\ 2x - y + 3z = 4$$ Thanks!
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253 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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1answer
59 views

The area of a region around a curve

If we are given a simple closed curve with length $L$ in the plane, and we have a fixed number $r$ such that for each point $x$ on the curve there is a related disc $D(x,r)$ with radius $r$, how can ...
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275 views

arithmetic and geometric genus for a reducible plane curve

If $C$ is an irreducible plane curve we have the well known formula relating the airthmetic (obtained via the degree-genus formula) and the geometric genus $$\frac{(d-1)(d-2)}{2} - \sum ...
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2answers
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Strange property of parametrization of a class of plane curves

My studies lead me to the following parametrization of perhaps a new class of plane curves ( which are similar in shape to the classical sinusoidal spirals but not identical ). If the curves are not ...
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83 views

Creating a $C^2$ smooth convex curve which joins circular arcs

Suppose that I have many points $p_1$,..., $p_n$, all very close to origin 0. For R very, very large, I have circular arcs of radius R, $\alpha_i$, about $p_i$. I do not speak about their ...
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26 views

Limit of an Expression on a Plane Curve

Say we have a curve $\mathcal{C}$ like this: $$ x^5 + y^5 = 5xy.$$ Say then that we want to find (and prove) the limit of the quantity $x/y$ (if it exists) as $x \to + \infty$ on the curve ...
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1answer
23 views

Limit of an expression as $x$ tends to a particular quantity on a curve

Say we have a curve like this: $$ xy + y^2x^2 = x.$$ Let's call it the curve $\mathcal{C}$. Say then that we want to find the limit of the quantity $x^2/y$ (if it exists) as $x \to - \infty$ on the ...
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95 views

Degree of ramification divisor and number of fixed points under a group action.

Let $C$ be a projective plane nonsingular complex curve and a finite group $G$ acts on $C$. Consider the quotient $f: C\rightarrow C/G=:C'$. Then, by Riemann–Hurwitz $2g_C-2=|G|(2g_{C'}-2)+\deg R,$ ...
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Parameterization of simple closed curve

A curve $Z$ in a two dimensional space is parametrized by $0\leq t < 1$ , and satisfies $Z(t) = Z(t+1)$. If it is sufficiently well behaved, it can be represented using a Fourier series with a ...
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Find the distance of the point $( 1,2,3 )$ to ,,,,,,,,,,,,,,

Find the distance of the point $( 1,2,3 )$ of the plane $3x-2y+5z+17=0$ .
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Curve parametrization

When plotting following implicit function $x^4 + y^4 = 2xy$, what is the best way to parametrize it? I tried $y=xt$, but then I get $x=\sqrt{2t\over {1+t^4}}$ and $y=\sqrt{2t^3\over {1+t^4}}$, which ...
3
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2answers
313 views

find maximum area

Consider a problem here : There is a wall in your backyard. It is so long that you can’t see its endpoints. You want to build a fence of length L such that the area enclosed between the wall and the ...
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1answer
47 views

Real valued function associated with the Diophantine equation $a^2(2^a-a^3)+1=7^b$

The parent question that maybe still remains to be answered at this moment is:Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$ . As far as the parent question is concerned, when generalizing to ...
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39 views

Domain and Range problem(plane)

Consider the function $z = \ln{(y + 1)}+\sqrt{x-3}$. Find the domain and range, and sketch the domain in the plane.
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show that $\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' (t)\rangle$

Let $\gamma,\eta:[a,b]\to \mathbb R^n$ be continuous, differentiable, curves. show that $$\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' ...
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PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...