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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20 views

Differential geometrie : geometric interpretation of a scalar product.

I am trying to solve an exercise regarding parametric curves. So I will give you what was given : let s be a parametric of a curve plane over the interval I. For any real number $\theta$, we define N($...
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Fulton problem 8.32: a projective variety of curves

I am trying to understand the following problem of Fulton's "Algebraic Curves": Notation: let $d \in \mathbb{N}$, $p_1, \cdots p_m \in \mathbb{P}^2$ distinct points and $r_1, \cdots, r_m \in ...
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Isolated Point Singularities on Curves

I know that a curve $C: f(x, y) = 0$ is singular provided there is a point $(x_0, y_0) \in C$ for which $$\dfrac{\partial f}{\partial x}(x_0, y_0)=\dfrac{\partial f}{\partial y}(x_0, y_0)=0.$$ However,...
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How to determine mathematically that an expression with one independent variable forms a closed area or not

Is there any mathematical method {other than sketching the curve from given expression (or equation)} to determine whether a given expession (or equation) forms a closed area or not? For example- ...
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Show if $\gamma$ has an ordinary cusp at a point p then so does any reparameterisation of it

(iii) Let $\tilde\gamma(\tilde t)$ be a reparametrization of $\gamma(t)$, and suppose $\gamma$ has an ordinary cusp at $t=t_0$. Then at $t=t_0$, $$d\tilde\gamma/d\tilde t=(d\gamma/dt)(dt/d\tilde t)=0,\...
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How to minimize exposed surface for half a pie?

I bought a large pie (of radius $R$). I cut off a half and gave it to my friend. This exposed an area or $2Rh$ -- where $h$ is the pie's thickness -- to air. I watched one Numberphile video too many,...
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Ordinary cusp on a plane curve

I calculate the 3 derivatives to be (I don’t know the line break) $\dot{\gamma} = (m t^{m-1}, n t^{n-1}) $ $\ddot{\gamma} = (m(m-1) t^{m-2}, n(n-1) t^{n-2})$ $\dddot{\gamma} = (m(m-1)(m-2) t^{m-3}, n(...
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Plane curve - conditions of regularity

For the the attached question I have proceeded as far as shown $\left \| \dot{\gamma}\right \| = \sqrt{\dot{r}^{2}+r^{2}}$ Not sure how to proceed...
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Modern Reference for Arc Accessibility?

The only reference I'm familiar with that deals with arc accessibility in the plane is Wilder's Topology of Manifolds, which is getting close to 100 years old. There is a book on Cluster Sets which ...
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Proving that two curves are orthogonal

Let $α(s)=(x(s)),y(s))$ be a regular plane curve that is parameterized by arc length, and let $n(s)$ be its normal vector. Consider the family of curves: $β(s,r)=α(s)+rn(s),−ϵ≤r≤ϵ$ I need to prove ...
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Finding planar curve $\beta(s)$ given curvature $k(s) = \frac{1}{s}$, $\beta(1) = (0, 0)$, $\beta'(1) = (1, 0)$

Find the curve $\beta(s) = (x(s),y(s))$ with $|\beta'(s)|= 1$ for all $s >0$ such that $\beta(1) = (0,0)$, $\beta'(1) = (1,0)$ , and curvature $k(s) =\frac{1}{s}$ for all $s >0$. I am guessing ...
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Does $\mathbb{R}^2$ Contain Uncountably Many Disjoint Copies of the Warsaw Circle?

The Warsaw Circle is defined as the closed topologist's sine curve, with an additional arc attached at its free end point and one of the end points of the critical line: Since we don't have an ...
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Converting this parametric curve to a level curve

I want to convert the parametrized curve $\gamma(t) = (\cos^{3}(t), \sin^{3}(t)), \ t \in \mathbb{R}$ to a level curve. Let $C = \{(x,y) \in \mathbb{R^{2}}: x^{2/3} + y^{2/3} = 1\}$. I claim that $x^{...
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General equation of a roulette

Currently I am trying to derive the equations x and y shown in the attached picture. I am struggling to find the horizontal and vertical components of lines u and v with angle phi and phi one. Could ...
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A Catalog of Special Plane Curves Book - Dennis

Can anyone recommend a book similar to above? The idea is to sketch out the locus of the various curves and derive their parametric forms which I enjoy and in the process learn about their properties ...
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Volume of body bounded with surfaces

I need to find volume bounded with $x=2y-y^2-z^2$ and $2y-x-1=0$. So first one is paraboloid that is located on x-axis and its "maximum" is in $(0,2,0)$ and the second is plane paralele to z-...
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Tangent plane at a point of ellipsoid and sphere.

The given ellipsoid and sphere, respectively, are $$\psi(x, y, z)=2x^2+10y^2+z^2-2=0,$$ $$\xi(x, y, z)=x^2+y^2+z^2-x-4y-2z+8=0.$$Given that above two are tangent to each other at the point $P=(r, s, t)...
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Boundaries in 2D space

Find the volume of figure that is bounded with the three following surfaces:$z=y^2,y=x^2,z=4$. I don't think this has anything to do with polar coordinates etc. So I tried to put this problem in $OXY$ ...
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How to find the genus of curve $x^2-\cos(y)=c$?

$x^2-\cos(y)=c$ is not an algebraic curve, and its genus may depend on the constant $c$, how to find them? In general, do we have any algorithms for transcendental curves?
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Asymmetry between definitions of curvature in 2-space and 3-space

The definition of curvature for a curve in 2D space I know of is the following: Given the velocity versor $T(s)=dP/ds,$ the accelleration $dT/ds$ is orthogonal to it, hence it must be parallel to the ...
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What are $A,B,C,D$ in the proof of Chasles' theorem?

I read A Treatise on Algebraic Plane Curves by Julian Lowell Coolidge and I have a question about the proof of Chasles' theorem: I do not really understand where $A,B,C,D$ are first defined? What are ...
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evolute of an astroid is another astroid

I want to prove that the evolute of the astroid, $(a\cos^3{t},a\sin^3{t})$, $t \in [0,2\pi)$, is another astroid turned by a $\frac{\pi}{4}$ angle. I have tried a variable change from the parametric ...
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Points of inflexion at circular points at infinity.

According to the Wikipedia article on Cassini Ovals, a Cassini oval has double-points, which are also inflexion points, at circular points I and J at infinity. I don't understand how to show that I ...
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Cuspidal cubic in characteristic 3 with no inflection points?

Fix an algebraically closed field $k$ of characteristic $3$. Consider a curve in $\mathbb{P}^2_{k}$ given by $$xy^2+yz^2+zx^2=0.$$ It seems to be a cubic with a cusp at $[1:1:1]$, whose Hessian $(x+y+...
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Thoughts on Bézier curve intersection and reliability of the test

Suppose we have two planar parametric curves f(t) and g(t), which are guaranteed to be second- or third-degree Bézier curves, which means they can be placed in "polynomial" form: $$ \textbf ...
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Midpoint of Circle tangent to Parabola [closed]

I am wondering what the function describing the locus of the positions of the midpoint of a circle with a given radius rolling on the inside of a parabola with a given focal length looks like.
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Is the logarithmic spiral the only (up to dilations and rotations) self similar curve?

A plane curve $\cal C$ (not a circle) given in polar coordinates $r=r(\theta)$, say with $r(0)=1$, has the property that for every $t>0$ there exists a function $\alpha(t)$ such that $t\cdot r(\...
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Find the surface area of the region defined by the intersection of $z=2y$ and $z=x^2+y^2$

I know A(s) = $\iint\sqrt{1+\frac {dz}{dx}^2+\frac {dz}{dy}^2}dA$ and solving for the domain D I can get to $x^2+(y-1)^2=1$. So logically, my answer should be $\int_0^2\int_{-\sqrt{1-{(y-1)^2}}}^{\...
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Tangent planes for the function $f(x,y) = 1 - x^2 - y^2$

I have to solve the following statements, finding the tangent planes to $f$ and its points of tangency: that contains the line in $\mathbb{R}^3$ that passes through the points $(3,0,3)$ and $(0,-3,3)$...
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Resolution of an ordinary multiple point on an irreducible plane algebraic curve

Let $C \subset \mathbb{A}^2$ be an irreducible plane algebraic curve and $P \in C$ be its ordinary point of multiplicity $m$, i.e., there are exactly $m$ tangents of $C$ at $P$ and they are pairwise ...
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Number of connected components of complement of plane curve

Let $$P(x,y)=\sum_{k=0}^m\sum_{l=0}^n a_{k,l}x^k y^l$$ be a polynomial over $\mathbb R$ in $x$ and $y$ of degree $m+n$. The zero locus $$\mathcal C=\{(x,y)\in \mathbb R^2\,|\, P(x,y)=0\}$$ is a plane ...
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What is the shape of the shadow of the rim of my lamp shade?

My lamp has a cylindrical shade and a light bulb sits in the center of it. It projectes the pictured shadow on the wall. That U-shaped curve is the shadow of the rim(s). What shape is it? A parabola? ...
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Permutations of n pairs of elements accounting for relabelling symbols and a number of other symmetries

I am doing some work on gauss codes in higher genus surfaces (good resource on the planar case here if you are unfamiliar) and am trying to figure out how many distinct gauss codes there are for a ...
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Spiral function with constantly increasing separation between its loops

Spirals following the polar equation $r=a+bθ$ are known to have constant separation between each of its loops, I show this by transforming the polar equation into Cartesian equation by using the ...
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Is the closure of the region bounded by a piecewise $C^1$ Jordan curve $C^1$-diffeomorphic to a polygon?

Given a piecewise $C^1$ Jordan curve $\sigma$, there is a region $A$ bounded by $\sigma$. Is it true that $\overline A$ is $C^1$-diffeomorphic to a closed polygon? (I believe this is the same as ...
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Jacobian Determinant of frenet transformation

If anybody can help with this. Given a point $\boldsymbol x = (x,y)$, it can be represented as $\boldsymbol x=\boldsymbol p(s)+r \boldsymbol u(s) $. p is a curve parametrized with arc length s. $r$ is ...
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Transform algebraic curves

For $i=1,2,3$ let $P_i$ be real polynomials such that $P_3$ is positive. Define the mapping \begin{align*} f(x,y) = \left(x+\frac{P_1(x,y)}{1+\sqrt{P_3(x,y)}},y+\frac{P_2(x,y)}{1+\sqrt{P_3(x,y)}}\...
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Existence of a plane curve with a given curvature $\kappa(s)$ that passes through two given points $P_1, P_2$ and arc length $L$.

As the title, says, is it possible to find a curve (i.e. prove its existence) whose curvature is equal to a given function $\kappa(t)$, begins in $P_1$, ends in $P_2$, and has arclength $L$? The ...
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Check if a curve is contained in a plane

Let $$\gamma :\mathbb{R}\rightarrow\mathbb{R}^{3},$$ $$\gamma(t)=(t,t^{2},\ln(1+t^{2})).$$ be a parametrization of a curve. Establish if $\gamma$ is contained in a plane. I have a problem here. First ...
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41 views

Arc length when the curve intersect itself

I came a cross a theorem saying that the arc length of a smooth non self-intersecting parametric curve is given by $ L= \int_{0}^{2\pi} \sqrt(y’^2+x’^2)dx$ Why we specify that the curve should be ...
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Differential Geometry, plane rotating around coordiante line

I am self studying Differntial Geometry with Do Carmo and I am attempting problem 9 of section 2-4. Specifically I am confused about the rotation statement. After not understanding that part I had to ...
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60 views

How to identify a specific type of lemniscate

I have the following equation: \begin{equation} f(x,y)\equiv -1296 x^4 \left(4 y^2+9\right)+9 x^2 \left(-128 y^4+864 y^2+2187\right)-y^2 \left(8 y^2+81\right)^2=0 \end{equation} Which represents a ...
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Approximately finding self intersection points of closed plane curves?

Let $0 = t_0 < t_1 < t_2 < \cdots < t_N = 2 \pi$. I compute the points on the closed curve $F(t)$ in the complex plane , (Set $\epsilon = +\infty$) : $$P_i = F(t_i)$$ $$P_{i+1} = F(t_{i+1})...
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Tangent at two different points of a simple closed convex curve

Let $C$ be a simple closed convex plane curve and let $p$ be line tangent at two points of curve $C$, $t_0 $ and $t_1, t_1\ne t_2$. Show that $C$ contains every point from $[t_1,t_2]$ on $p$, i.e. $p$ ...
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In how many components/regions does this curve divide the complex plane and how can I color them (with an algorithm)?

I am reading a book (Creating symmetry, by Frank A. Farris) on how to construct "symmetric images" or wallpaper and came across the following curve: $$\gamma(t) = \exp(it)+\frac{1}{2}\exp(...
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38 views

Find the differential equation that solves a certain problem

I'm at the beggining of a differential equation's course, and I need to solve the following problem: Find every curve in the plane $XY$ so that the middle point of the normal's segment between each ...
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help integrating

I'm really struggling to integrate this. $$\int_0^t \sqrt{\cos^2\left(1\right)\sinh^2\left(u\right)+4\sin^2\left(\dfrac{1}{2}\right)\cosh^2\left(\dfrac{u}{2}\right)} \,\mathrm{d}u$$ This is to find ...
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50 views

Inequality used to bound curvature terms

I've been poring over the article: Gage and Hamilton's The Heat Equation Shrinking Convex Plane Curves (here). In Lemma 4.4.2 , it's supposed to find bounds for the higher derivatives of k.In the part ...
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Find a unit speed parametrization of the semicubical parabola $t\to\left(t^{2},t^{3}\right)$, valid for $t>0$.

This is Chapter 1.1, Problem 7 of Modern Differential Geometry of Curves and Surfaces with Mathematica, by Alfred Gray, et al. Find a unit speed parametrization of the semicubical parabola $t\to\left(...
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A small calculating problem in the article by M. Gage and R. S. Hamilton

In the article " The heat equation shrinking convex plane curves " by M. Gage and R. S. Hamilton, I didn't finish the calculation in 4.3.4:$$\frac{\partial }{{\partial t}}\int\limits_0^{2\pi ...

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