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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Cycloid arc lenght question

I've parametrized the cycloid with the function $$\gamma(t)=(\cos(\frac{3}{2}\pi-t)+t,\sin(\frac{3}{2}\pi-t)+1) ; \space t \in [0,+\infty)$$ I am asked to find the arc lenght of the curve which ...
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63 views

Problem with calculating winding number in sum of curves

Let $$ \begin{align} \gamma= \gamma_1 +\gamma_2+\gamma_3,\\ \gamma_1(t)=e^{it}, t\in[0,2\pi] \\ \gamma_2(t)=-1+2e^{-2it}, t\in [0,2\pi]\\ \gamma_3(t)=1-i+e^{it},t\in [\frac{\pi}{2},\frac{9\pi}{2}] ...
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1answer
40 views

winding number of $\gamma$ and point exterior to $\gamma$

$$ \begin{align} n(\gamma,z_0)=\frac{1}{2\pi i}\int_\gamma\frac{1}{z-z_0}dz . \end{align} $$ Is it safe to say that $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$
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2answers
289 views

Finding the equation for a (inverted) cycloid given two points

If I have two points on a Cartesian plane, and I know that they are connected by a cycloid, then how do I find the equation for that cycloid? For background information, I have been playing around ...
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1answer
28 views

Problem in showing that contours $\gamma_2$ is equivalent to $ \gamma $

Let $\gamma_2(t)= e^{-it^2}, t\in[0,\sqrt{2\pi}]$ and $\gamma(t)=e^{2\pi it}, t\in[0,1]$ Show that $\gamma_2 \sim \gamma $. I think that for the latter to be true $\gamma_2$ should be ...
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1answer
43 views

Does $\gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}]$ represent the unit circle?

$$ \begin{align} \gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}] \end{align} $$ Is my thinking correct that $\gamma$ represents the unit circle correct?
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1answer
48 views

Image of a parametrised curve and its geometrical meaning

$\gamma_1(t)= t^2 + i\, t^4 , t\in [ -1, 1]$ which is the Image of this curve and / or what is the geometrical description of the set of points $ z : z=\gamma_1(t)$ I am helping a friend study ...
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1answer
61 views

There is at least one straight line that can bisect both the perimeter and the area of the curve. [duplicate]

Consider a closed curve of finite length. There is at least one straight line that can bisect both the perimeter and the area of the curve. Why is this statment true?
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1answer
39 views

The motion of the particle satisfies $\textbf{v} = \textbf{c}\times \textbf{r}$

Why is the path is contained in a circle that lies in a plane perpendicular to $\textbf{c}$ with centre on a line through the origin in the direction of $\textbf{c}$
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1answer
50 views

How can we find the arc length of the curve? [closed]

How can I find the length of the curve $$\left(\frac{t^3}{3} - t\right)\mathbf{i}+ t^2 \mathbf{j}, \quad 0≤t≤1?$$
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7answers
499 views

Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
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2answers
973 views

Using equations to draw out complex objects

How do people come up with equations of curves to draw out complex objects? Some popular examples would include: batman curve & PSY curve. This stackexchange link explains the rationale for the ...
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1answer
110 views

How is equation art created [duplicate]

A friend of mine sent me a Wofram Alpha link to a parametric curve that creates a detailed drawing of a game character. The parametric equation that draws it is about ~9 screens tall on my ...
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0answers
48 views

How to derive Jordan curve theorem for three arcs from the two arc version?

The Jordan curve theorem can be stated as follows: Let $p,q\in\mathbb R^2$, $p\ne q$ and $a,b$ be arcs between $p$ and $q$ intersecting only in the endpoints. Then $a\cup b$ divides $\mathbb R^2$ ...
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2answers
428 views

What is a French curve, as mentioned by Feynman?

I'm reading "Surely You're Joking, Mr. Feynman!", he says: I often liked to play tricks on people when I was at MIT. One time, in mechanical drawing class, some joker picked up a French curve (a ...
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0answers
71 views

A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
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2answers
162 views

Retrieve the initial cubic Bézier curve subdivided in two Bézier curves

I have a cubic Bezier curve subdivided to two cubic Bezier: Assuming that "t_cut" is the t value where this initial Bezier is cut: example of function subdivision(BezierCurve initialCurve, ...
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4answers
5k views

How to find the parametric equation of a cycloid?

"A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia In many calculus books I have, the cycloid, in parametric form, is ...
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3answers
5k views

Polar to Parametric Equation?

I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right. Curve C has polar equation ...
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0answers
112 views

Maximum area enclosed by a string attached at fixed points

Two fixed points A and B have a string of length L attached between them. Supposing that the string does not intersect the line segment AB, then the string and AB will form a closed figure. What ...
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1answer
74 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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4answers
147 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
81 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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127 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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228 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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2answers
44 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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1answer
36 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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0answers
14 views

Difference between containing point and pass through point?

I do not understand this, What is the difference between the equation of the plane containing the points and the equation of the plane through the point? Is it the same thing or are they different?
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1answer
61 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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3answers
395 views

How to find the length of a curved path.

We have to find a continuous model for a curved path which you then solve. A woman is running in the positive y-direction starting at x=50 (50,0) which is orthogonal to the x axis. At this point a dog ...
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1answer
69 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
416 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
105 views

convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
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1answer
95 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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1answer
45 views

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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1answer
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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1answer
134 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
123 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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0answers
26 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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5answers
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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 ...
2
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0answers
200 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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0answers
101 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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127 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
629 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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2answers
30 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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3answers
994 views

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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1answer
56 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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2answers
3k views

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
70 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 ...