# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...