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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Convex Curve Parametrization

How can I parametrize a convex plane curve using the angle $\theta$ between the tangent line and the $x$-axis?
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Circumference parametrization

Let $C=\{(x,y)\in \Bbb R^2: (x-x_0)^2+(y-y_0)^2=r^2\}$ and let $\varphi :[0,2\pi]\to \mathbb{R}^2$, $\theta \mapsto (x_o+r\cos \theta, y_0+r\sin \theta)$, with $r>0$. I'm trying to prove that ...
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201 views

Symbolic integration of vector norm

I'd like to symbolically integrate the expression $\int_0^1{\|r'\left(t\right)\|_2\,dt}$ where $r$ is a function $\mathbb{R} \rightarrow \mathbb{R}^2$ (so the expression is the arc length of the curve ...
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789 views

Find area of a simple, smooth, closed curve lying in a plane

I was given this question in class and I assume it is a spin off of Green's theorem for finding the area of a closed curve $\lambda$ in 2D but expanded to 3D I believe. Anyways I am pretty confused ...
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How to calculate the area between 2 polar curves: $r=\frac{4}{2}-\sin\theta$ and $r=3\sin\theta$?

How to calculate the area between 2 polar curves: $r=2-\sin\theta$ and $r=3\sin\theta$? I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ...
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703 views

How can I find the points of intersection between the curves $r=1+\sin\theta$ and $r=1-\sin\theta$?

Find the points of intersection for the curve $r=a(1+\sin\theta)$ and $r=a(1-\sin\theta)$ My book says the answer is $(0,0),(a,0),(a,\pi)$. However I calculated $ (a,0),(a,\pi),(a,2\pi)$.
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Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
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46 views

Plan curve with zero area has at least two points of zero curvature

Let $\alpha=(x,y)$ be a smooth closed plan curve defined on $[a,b]\subset \mathbb{R}$. We can define the oriented area of $\alpha$ by $A=\int_{a}^{b} x(s)y'(s)ds$. So, if A=0 then there exists $t_1 ...
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A circle rolls along a parabola

I'm thinking about a circle rolling along a parabola. Would this be a parametric representation? $(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$ A gives us the radius of the circle, B changes the frequency ...
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139 views

Find the point where equations $x=t^2-t$ and $y= t^3 -3t-1$ cross itself.

Find the point where equations $x=t^2-t$ and $y= t^3 -3t-1$ cross itself. This's the first time I meet this kind of problem, can someone give me some idea? Thank you.
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41 views

Curve is approximatable by function

I want to show that for $\gamma: [a,b] \subseteq \mathbb{R} \rightarrow V$ continously differentiable where V is a bounded subset of $\mathbb{R}^2$. There is always a sequence of functions ...
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37 views

Approximation theorem

I am looking for some theorem that gives me that each curve $x(t)=(x_1(t),x_2(t))$ that is continously differentiable and has $\dot{x_1}(t)\ge 0$ can be approximated by continously differentiable ...
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1answer
83 views

How To Write The equation for a line given a set of co-ordinates

I'm trying to learn how can I write equation for a line given all the points that belongs in the line. I'm looking to find equation for a curve. An example Set of points is: { (24,11) (25,11) ...
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153 views

Is it possible for a Jordan curve in the plane to enclose a set with area zero?

I read about the Isoperimetric Inequality the other day. It says that for any Jordan curve, $$ \frac{4 \pi A}{L^{2}} \leq 1, $$ where $ L $ is the length of the curve and $ A $ is the area of the ...
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2answers
77 views

Graph of a curve

Today in my test, there was a question which had contour C: $|z+\dfrac{1}{z}| = 2$. What does the curve represent? Is it a discrete set of points or really a curve?
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166 views

A problem on Residue Theorem

Today I had a problem in my test which said Calculate $\int_C \dfrac{z}{z^2 + 1}$ where C is circle $|z+\dfrac{1}{z}|= 2$. Now, clearly this was a misprint since C is not a circle. I tried to find ...
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162 views

Is there a $C^1$ curve dense in the plane?

Is there a curve $\gamma : \mathbb{R} \to \mathbb{R}^2$ injective and $\mathcal{C}^1$ whose range is dense in $\mathbb{R}^2$?
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1answer
88 views

Radius function from length of curve

I have the following function definition for the length of a curve: $$ l(\theta) = {K_0 \times \sin(\theta) \over \cos(\theta) + K_1} \\ 0 \le \theta \lt \frac \pi 2 \\ K_0, K_1 \ge 0 $$ I would like ...
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1answer
91 views

Name for this problem regarding chords, and the area between two closed convex curves?

I want to read more about the amazing result that, when given a closed, convex curve in the plane that can be traversed internally by a chord of length $p$+$q$, and on that chord lying at p, a point, ...
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Characterize a large class of shapes using a finite number of parameters

I am doing some numerical computations searching for an optimal shape for a certain functional. In my particular case, the shape $\Omega$ is a 2 dimensional star shaped domain by the origin, which ...
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595 views

Fireworks under inverse-cube gravity

What is the path of a projectile under an inverse-cube gravity law? Imagine that the law of gravity was changed overnight from $F(r) = G m_1 m_2 / r^2$ to $F(r) = G' m_1 m_2 / r^3$. To be ...
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1answer
338 views

What does the definition of curvature mean?

First question, am I right in saying that curvature measure how quickly the direction of a cruve changes? Also, we have been given the "definition": $$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$ where ...
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Change parabolic equation to canonical form

I have equation $y = -x^2 + 2x + 7$. How can I change it to canonical form, which looks like $y^2 = 2px$ ? ($p$ will be parameter) What i ve tried so far: $$\begin{align} y &= -x^2 + 2x + 7\\ y ...
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Normalisation of an algebraic curve.

I need to compute explicitly the normalisation of a singular algebraic curve $C$ which is given by an explicit equation in $\mathbb{A}^2$. This task is mostly reduced to finding the integral closure ...
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Parameterized curve describing trajectory of thrown object

We describe the trajectory of a thrown object (neglecting friction and similiar effects) with the curve $$k(t) = \left(v_0\cos(\beta)t,\,v_0\sin(\beta)t-\frac{g}{2}t^2\right)$$ with ...
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1answer
145 views

For any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.

Find the minimum possible value of $A$ such that for any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.
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Functionals defined on curves

I am looking for classical (real) functionals defined on rectifiable curves (neither necessarily simple nor closed) in either $\mathbb{R}^2$ or $\mathbb{R}^3$. There's length, turning number, total ...
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1answer
110 views

Could you help me to find a model for this curve?

I am very bad in mathematics and I'm not able to find by myself the model corresponding to this kind of curve. I wish to have a quick growth at the beginning, then it should increase slowly for a ...
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2answers
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Finding a general equation for a quadratic curve passing through three points.

I have three points (250, 0), (500,500) and (750, 0). To find a curve passing through these points all I have to do is plug-in these values into the general quadratic equation: f(x) = ax^2 + bx + c ...
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What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid?

What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid? If possible, show some reference please?
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Addition law on moduli space of curves

Dislaimer: I know very little about this, so if parts of my question don't make sense, please feel free to edit in a way that does, or ask me to clarify. Let $\mathcal{M}_{1,2}$ be the moduli space ...
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616 views

Reconstructing space curves from its curvature and torsion

I have to write a program which gets two functions (curvature and torsion) and 3 vectors of the Frenet-Serret "frame" at the starting point - and I have to reconstruct the space curve from this given ...
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198 views

Plane curve: orhogonal projection and closest points

I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28: "Let two curves $C_1$ and $C_2$ have a regular point $P$ ...
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1answer
92 views

Defining the movement of an object on a 2-dimensional plane

I am trying to define the movement of an object for a danmaku game I am making. Here is a link to some example gameplay (not my game, but a popular series in this genre made by Zun). Basically, I was ...
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1answer
100 views

Representing a curve as a plane curve in different ways

Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$. I know that $X$ has a plane model. More ...
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How to define a perspective circle in xy?

You can see a perspective view of a square(FCED) and a circle in 2D screen. O is center of the circle. How can I define the perspective circle equation that shown as red in the picture? Thanks a ...
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1answer
125 views

Real Curves/Circles

Unfortunately I am very ignorant when it comes to mathematics. Please understand and forgive me if this question reflects that. Thank you! I have an observation: Every curve and every circle in the ...
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1answer
97 views

Detecting the type of singularity with the Jacobian

Say we have a plane curve $\mathcal{C} = V(f(x,y)) \subset \mathbb{A}^2_{\mathbb{C}}$. The partial derivatives tell us about the singularities: if they all vanish at a point $p \in\mathcal{C}$ then ...
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1answer
211 views

An almost straight curve with infinite curvature?

I played around with computing the curvature of some curves, and found this weird example that is driving me nuts. Consider the following (Bézier) curve (on a plane, the first point is $[-1,0]$): ...
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160 views

Assigning alternate crossings to closed curves

This is a minor curiosity that I've been wondering about. Suppose that we draw a closed curve in the plane and that this curve intersects itself several times, but never twice in one spot. We can knot ...
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1answer
204 views

Need to find the equation of a curve having only the direction of it at a given point

Temperature T of a plate lying in xy plane is defined T(x,y)=50-(x^2)-(2y^2). An ant, which is initially at (2,1) moves along a curve ensuring the temperature is decreasing as rapidly as possible. I ...
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311 views

A question about curves in $\mathbb{R}^2$

I need to show this result: Let $\alpha :I\rightarrow \mathbb{R}^2$ a smooth curve, where $I$ is a compact interval of the real line. If $\lVert \alpha (s) - \alpha (t) \rVert$ depends only on ...
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1answer
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Determine the path that created by a sector of circle

ABC circle sector turns on ground (x axis) as shown in the figure. A is the center of the circle. $\angle{OAB}=\angle{OAC}=\alpha $ $|AB|=r$ $\cfrac{|AP|}{|PC|}=k$ The corners meet on point $H$ ...
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Locus of points generates several very different curves. Closed form?

Consider, for the sake of simplicity, a circle $C$ centered at he origin with radius $a$. Let $F=(h,k)$ be a point not necessarily inside the circle. Let $M=(a\cos\theta,a\sin\theta)$ be a point in ...
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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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1answer
185 views

Regular parametrization of a curve

Let $\gamma : \left\{ \begin{array}{ccc} \mathbb{R} & \to & \mathbb{R} \\ t & \mapsto & (t^2,t^3) \end{array} \right.$ and $\Gamma= \gamma(\mathbb{R})$. Because of the singularity at ...
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1answer
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find length of curve of intersection

I have come to a dead end on a problem and I need someone to tell me either if I did it correctly, or how to fix it if I did not. This is Stewart Calculus 7th edition, problem 13.3.12. Here is the ...
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2answers
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Find the length of the curve $y(x) = \int_1^x\sqrt{t^3 - 1} \, dt$, $1 \leq x \leq 4$

Find the length of the curve $y(x) = \int_1^x\sqrt{t^3 - 1} \, dt$, $1 \leq x \leq 4$. Not sure if I should be using first principle theorem (having some trouble with that) or if there is a ...
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66 views

practical question about developable surface

there's my question: Given 2 regular plane curves (let's say $\mathcal{C}^1$) in the 3D space, is there always a developable surface which contains both curves ? Thanks, anders
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367 views

Ways to define a curve

I'm trying to give shapes in my physics engine roundness/ curvature. I am aware of various methods for mathematically defining curvature such as bezier-curves, ellipses, etc; but I'm not sure which ...