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Questions tagged [plane-curves]

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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Function of curve length s in term of x

I am looking into a certain problem and decide to formulate it in a way that use the equation s=F(x) to describe a planar curve. Normally, we express the equation of a planar curve in the form y=f(x). ...
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Parametrizing $x^2(x^2+y^2)=4(x-y)^2$

I need to find $x=x(t)$ and $y=y(t)$ so that the implicitly defined curve on $\mathbb R^2$ $$x^2(x^2+y^2)=4(x-y)^2$$ is converted into an explicit function of the parameter $t$ that can be analysed ...
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298 views

If a curve lies in a circle of radius $r$, show there is a point at which the curvature $|k(s)|\geq 1/r$

I am self studying curves and came across this problem: Let $\alpha: I \to R^2$ be a simple smooth closed plane curve. i) If the curve lies inside a circle of radius $r$, show there is a point ...
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Calculating the area of cardioid with trisectrix with green's theorem: Will the area of the loop be added twice? See picture inside

I have a cardioid with a trisectrix, making a loop inside. This is what the cardioid looks like: . My question is the following, if we use Green's theorem to calculate the area of C, which is the ...
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66 views

Parametrizing $(2x+y)^2(x+y)=x$

I need to find $x=x(t)$ and $y=y(t)$ so that the implicitly defined curve on $\mathbb R^2$ $$(2x+y)^2(x+y)=x$$ is converted into an explicit function of the parameter $t$ that can be analysed using ...
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142 views

Plane curves of bounded curvature

Here is a "conjecture" that should be known (but I have not found any good reference to it): consider the class $\cal F$ of all $C^2([0,1])$ functions $f$ with the property $f(0)=f'(0)=f''(0)=0$ and ...
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398 views

If a plane curve has curvature bounded from below, is it contained in a disk?

Let $\gamma: (0,1) \to \mathbb{R}^2$ be a $C^\infty$ regular plane curve, and suppose that its curvature is at least $k_0$ everywhere. Is the image of $\gamma$ contained in disk of radius $\frac{1}{...
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61 views

Tangents of a curve whose trace remains in a half-plane.

I need to prove the following result: If a curve's trace is contained in a half-plane then the tangents at the intersection of the curve with the line defining the half-plane $R$ coincide with $...
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2answers
171 views

Spiral tangents form constant angle with polar lines.

Given the logarithmic spiral $$\alpha(t) = e^{-t}(\cos(t),\sin(t))$$ I take a ray from the origin given by $\lambda(\cos \theta, \sin \theta)$ and I have to prove that in $\alpha(\mathbb{R}) \cap R_{\...
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56 views

Logarithmic spiral is an embedding

Consider the logarithmic spiral $\alpha(t) = e^{-t} (cos(t),sin(t))$. When $\alpha:\mathbb{R} \to \alpha(\mathbb{R})$, I have shown that this is a bijective continuous mapping. I would like to ...
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75 views

Condition for a plane curve to intersect its osculating circle

The osculating circle of a curve $\alpha$ at the point $p \in \alpha$ is the circle $\mathbb{S}^1$ which is tangent to $\alpha$ at $p$ and has radius $\frac{1}{k(p)}$. Show that, if $k'(p) \neq 0$, ...
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Good references to start studying the curve shortening flow

I want to study the curve shortening flow on curves in $\mathbb{R^2}$. I have knowledge of the local theory of plane and space curves (obviously that comes with linear algebra and calculus as well), ...
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49 views

Can a twisted or 3 dimensional curve have an asymptote?

Also, I recently encountered a definition for 'asymptotes' in an old engineering mathematics book that says,"An asymptote is a straight line which cuts a curve in two points at an infinite distance ...
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38 views

On points with same $x$-co-ordinate on certain cubic curve

Consider the cubic curve $y=x^3+ax^2y+bxy^2+cy^3$. If $(t,r)$ and $(t,s)$ are two distinct points on the curve, then is it necessarily true that $t=0$ ?
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37 views

How does Raph Levien's Spiro choose angles for the ends of a path?

I've read Raph Levien's paper on splines (http://www.levien.com/phd/phd.html), and think I mostly understand chapter 8, the nuts and bolts of fitting a piecewise polynomial spiral to a sequence of ...
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1answer
30 views

Question on the span of a tangent plane

I was reading this answer regarding the span of a tangent plane here. The answer says the graph of $f$ is also the graph of the map $F(x,y) = (x,y,f(x,y))$. The tangent plane is spanned by $(1,0,f_x)...
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Could anyone suggest me some good references on interpolation that include other mathematical structures than just single variable functions?

I have been studying interpolation methods because I find then fascinating and they give me a greater sense of freedom for mathematically defining curves that have a given shape, but when I began ...
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Find the equation of the plane which bisects the pair of planes $2x-3y+6z+2=0$ and $2x+y-2z=4$ at acute angles.

Find the equation of the plane which bisects the pair of planes $2x-3y+6z+2=0$ and $2x+y-2z=4$ at acute angles. My try: The normal to the plane $2x-3y+6z+2=0$ has direction ratios $2,-3,6$ and the ...
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Let $F(X,Y,Z)=5X^2+3Y^2+8Z^2+6(YZ+ZX+XY)$. Find $(a,b,c) \in \mathbb{Z}^3$ not all divisible by $13$, such that $F(a,b,c)\equiv 0 \pmod{13^2}$.

Let $F(X,Y,Z)=5X^2+3Y^2+8Z^2+6(YZ+ZX+XY)$. Find $(a,b,c) \in \mathbb{Z}^3$ not all divisible by $13$, such that $F(a,b,c)\equiv 0 \pmod{13^2}$. Although it looks like a homework problem it is not. It ...
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93 views

Show this curve is a closed set in $R^2$ by using the definition

Let $S = \{<x,y>\,:\,xy=1\}$, prove S is closed in $R^2$. A similar exercise is proving a line is closed. This is requested to prove by using the definition that says that the open sets are the ...
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209 views

Tangent plane when gradient is zero

I'm attempting to find the tangent plane of the function $f(x,y,z)=xz+2y^2z^2$ at $(x,y,z)=(-1,1,0).$ The partial derivatives are $z, 4yz^2$, and $4zy^2$ when done in respect to $x, y$ and $z$ ...
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65 views

Subdivision of a simple polygon into triangles? [duplicate]

Suppose that we have a simple and closed plane polygon. How to show (inductively) that every such polygon can be subdivided into triangles, see the following picture? Are there good reference books? ...
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162 views

Interpolation method that gives the least arc lenght of the curve.

this question may be a bit sloppy on my part but I will make it anyway, I recently have been fascinated by the idea that the surface of a soap bubble film restricted to a boundary will be so as to ...
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3answers
58 views

Suggest parametric equations for a given curve

Can anyone suggest a pair of parametric equations involving trigonometric functions that could give a curve with the following shape?
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174 views

How to use homogeneous coordinates and the projective plane to study the intersection of two lines

How to use homogeneous coordinates and the projective plane to study the intersection of two lines, and how to compare this to a standard homogeneous coordinates. *The equation of a line in ...
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0answers
63 views

Finding a quartic with some prescribed multiplicities

Let $F(X_0,X_1,X_2)=-X_1^4+X_1^3X_2+X_0^3X_2$ and $C:=\{F=0\}\subset\mathbb{C}\mathbb{P}^2$. I am asked to find a quartic $D\subset\mathbb{C}\mathbb{P}^2$ subject to the following conditions (here $I(...
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Proving regularity and orthogonality of two curves

Let $\alpha(s) = (x(s)),y(s))$ be a regular plane curve that is parameterized by arclength, and let $n(s)$ be the normal vector and $k(s)$ be the curvature of $\alpha$. Consider the family of curves: ...
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2answers
110 views

How do I prove this curve has the shape of a logarithmic spiral?

I am working with $\alpha(t) = \cfrac{e^{at}}{a^2+1}(\sin(t) + a\cos(t),a \sin(t) - \cos(t))$, a reparametrization of $\alpha(s) = \left(\cfrac{(as+b)\left(\sin\left(\cfrac{\log(as+b)}{a}\right) + a\...
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214 views

Boundary of a general plane domain?

Is it trivially true that the boundary $\partial U$ of a domain $U\subset \mathbb{R^2}$ is a finite (or countable?) union of disjoint Jordan curves?
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Getting back the original curve having only the curvature

More specifically, I'm trying to prove that all plane curves with curvature $k(s) = \cfrac{1}{as + b}$ are logarithmic spirals and trying to describe all the ones such that $k(s) = \cfrac{1}{\cosh(s)}$...
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52 views

Regular plane curves such that the line they determined is mutually orthogonal to both

Let $\alpha(t)$ and $β(t)$ be two regular plane curves such that the line determined by $\alpha$ and $β$ is mutually orthogonal to $\alpha$ and $β$. Is it really this simple to prove that the segment ...
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1answer
55 views

On normal lines and centers of curvature at points where the curvature is maximum/minimum

Let $\alpha(s)$ be a plane curve. I need to check that the normal line to $\alpha$ at $s$ is orthogonal to the curve that is determined by the centers of curvature on the points in which the ...
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94 views

Proper parametrization of a closed curve

Let $\gamma:I\to\mathbb{R}^2$ be a closed plane curve, for simplicity, a unit circle. Therefore, we have $$\gamma(\varphi) = (\cos \varphi, \sin \varphi).$$ What is the proper domain of $\varphi$? ...
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47 views

On determining the plane curves on which the length of the segment of the normal lines between the curve and the x axis are constant

So, if $\alpha(t)$ is such a curve, then $g(r) = \alpha(t) + rn(t)$ is the normal line and $w(s) = \alpha(t) + s\alpha'(t)$ is the tangent line, where $n(t)$ is the orthogonal vector to $\alpha'(t)$. ...
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44 views

Find implicit equation of the following plane parametrized curves

I have the following parametrized plane curve: $C_1 =$ Image of $c_1$, where $c_1(t)=(2\cos t,3\sin t)$, and I need to find implicit equation. I don't know exactly what are implicit equation. This ...
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1answer
36 views

Parameterizing the pedal curve of an arbitrary plane curve

Let $\alpha(t) = (f(t), g(t)), t \in \mathbb{R}$ be a regular curve and $P(x_0, y_0)$ be a fixed point in the cartesian plane. The set of points $X$ such that the line $PX$ is perpendicular to the ...
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41 views

The equation of a curve

I need to write the equation of a curve which exactly goes throw point points: $$\left(0,\frac{\pi ^2}{6}-1\right),\left(\frac{1}{2},\frac{2}{3}\right),\left(\frac{3}{4},\frac{2\pi}{3}-\frac{88}{63}\...
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1answer
104 views

image of parametric quadratic curve with three components contained in a plane

I am studying Differential geometry I tried to prove this by taking all three components as quadratic with $t$ as a parameter but could not be successful. If all three component functions of a ...
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1answer
223 views

Simple Proof of the Isoperimetric Theorem in the plane

I am wondering whether there is a ‘simple’ proof of the Isoperimetric Theorem in the plane, i.e. that any simple closed curve in $\mathbb{R}^2$ with length $L$ and enclosed area $A$ fulfils $$4\pi A \...
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1answer
28 views

Algebra Linear Transformations

Hi I have a question which I don't really understand and wondered whether I could get some help on it. Question: Write the standard matrices representing the following linear transformations in the ...
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145 views

Describing the plane curve $α(θ)$ that has the following property: the area of the triangle given by $cQT$ is constant (details below)

A plane curve, $α(θ)$, has the following property: if $c(θ)$ is the center of curvature of $α$ in $θ$, $Q(θ)$ is the projection of $α(θ)$ on the x axis and $T(θ)$ is the intersection point of the ...
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How to prove smooth plane quintic curves are not trigonal? [closed]

It's Hartshorne iv.5.6. I really do not know how to use the condition "plane" to make a contradictions.... Emm..... Then I find an exercise in ACGH’s Geo. of Alg. Curves says every smooth plane ...
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Proving that strictly monotonic curvature implies no self intersections (more specifically, using the following inequalities)

Let $a(s)$ be a regular curve that is parametrized by arclength. Prove that, if the curvature $k(s)$ is a strictly monotonic function, then $a(s)$ has no self intersections. Suggestions: a) [will be ...
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54 views

Homotopy of Jordan curves

I'm struggling with the following question: Let $\gamma$ be a (not necessarily smooth) closed Jordan curve and let $\operatorname{int} \gamma, \operatorname{ext}\gamma$ denote the interior and the ...
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1answer
46 views

Show that $ \int_\Omega f \, dA = \frac{3\pi}{2} \, f(0) + \frac{\pi}{2} \, f'(0)$

Let $\Omega = \{ z + z^2/2 : |z| < 1\}$ be the interior of a shape in the complex plane, and let $f(z)$ be an analytic function. Show the following formula: $$ \int_\Omega f \, dA = \frac{3\pi}{...
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46 views

Find the equations of all planes that pass through the points (1,1,1) and (2,0,1) and also are tangent to the surface $x^2 + y^2 + z^2 = 1$ [closed]

Find the equations of all planes that pass through the points (1,1,1) and (2,0,1) and also are tangent to the surface $x^2 + y^2 + z^2 = 1$
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311 views

Transforming quadratic parametric curve to implicit form

I have a algebraic parametric curve $$ \mathbf{p}(t) = (x(t), y(t)) $$ where $x$ and $y$ are both polynomials of degree $\leq p$. Now, I want to find the implicit form $f(x, y) = 0$. A document I'm ...
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47 views

How to calculate the length of the portion of curve from given conditions.

Let $l$ be the length of the portion of the curve $x=x(y)$ between the lines $y=1$ and $y=2$ where $x(y)$ staisfy $\sqrt \frac {1+y^2+y^4}{y} \ , x(1)=0$. Then find $l$ . The main thing I didn't get ...
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1answer
321 views

How to calculate an area enclosed by two parametric curves?

I know the area under the curve given by parametric equations$x=f(t),y=g(t),\alpha\leq t\leq \beta$ is given by$$A=\int_{\alpha}^{\beta}g(t)f'(t)dt$$ That is in $\int_{a}^{b}ydx$ we have substituted $...
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1answer
395 views

Derivatives of the curvature of a plane curve

I would like to know what is the name of the derivative of the curvature of a plane curve. It should be called "sharpness", but I cannot find a reference. There should also be a connection with the ...