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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regular curves and ordinary cusps

Let $\Gamma$ be the subset of $\mathbb{R}^{2}$ given by $$ \Gamma = \{(t^2,t^3) : t\in \mathbb{R}\} $$ Does there exist a regular curve of class $C^{3}$ (3−times continuously differentiable), say $\...
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Is it true that re-parametrization of a curve does not change its length?

I'm trying to understand what is happening with the arc length parametrization, but it seems that something is missing. I considered the curve $a(t)=\left(e^{t}\cdot \sin(t), e^{t}\cdot \cos(t)\...
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31 views

Showing that a circle is an osculating circle of a unit-speed curve

Let $\alpha : I\to\mathbb{R}^2$ be a smooth plane curve parametrized by arc length, and assume that $0\in I$. A circle with radius $r$ centred at $p$ is called the osculating circle of $\alpha$ at $0$ ...
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Equilong (Path-Length-Preserving) Map Between Square and Circle?

Does there exist a map between the (half-open) square and the (half-open) circle which preserves path lengths? Such a map would be called an equilong map. Specifically I am searching for a map $f:[0,...
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68 views

Is it always possible to draw an Euler diagram with $2^n$ distinct regions?

Assume we have a set $L$ of $n$ Jordan curves which intersect each other and divide a plane into regions. Given a region $r$, we characterize it by $C(r)=\{l \in L: \text{$r$ lies inside $l$}\}$. Let ...
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386 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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Stereographic Projection of circle of Complex plane to Sphere

Source : Complex Variables by Ablowitz and Fokas Q : Show that a circle in the z- plane corresponds to a circle on the sphere. Sphere : $X^2+Y^2+(Z-1)^2=1$ Attempt : x,y are in z-plane (complex) ...
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Moving a point along an ellipse given initial point $P$ and arc-length $d$

Fix a reference frame in $\mathbb{R}^2$. Suppose you have an ellipse $\mathcal{E}$, a point $P \in \mathcal{E}$ and a real number $d$, where it is implicitly understood that $d>0$ means movement ...
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682 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
38 views

Formula for the osculating conic of a plane curve

A follow up to this question. Presumably similar curves have similar osculating conics, which in turn have identical eccentricities. Thus, the 'local eccentricity' of a plane curve at a point is the ...
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Finding the tangent line(s) to a curve in 3D parallel to a plane

Given the curve $$r(t) = (1 − 2t)i + (t^2)j + (t^3/2)k, ~~ t > 0~.$$ Find a point on the curve at which the tangent line is parallel to the plane $$5x + y + z − 3 = 0~.$$ I have done everything ...
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Derivation of the formula for the vertex of a parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the ...
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A curve is a circle or a line

Let $\gamma(t):\mathbb{R}\to\mathbb{R}^2$ be a continuous curve in the plane such that for every $t_1,t_2\in\mathbb{R}$ the euclidean distance $d(\gamma(t_1),\gamma(t_2))$ depends only on $|t_1-t_2|$. ...
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835 views

Fourier transform for a 2D curve

I am stuck on the following problem about a Fourier transform of a 2D curve: I have to calculate the Fourier transform (using 1D complex FT) (and the opposite of it) for a 2D curve z(t). The curve is ...
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Computing integral when the derivative factors (separates)

I have a specific smooth function $F(x,y,z) = 0$ which implicitly defines a function $\widehat{z}(x,y)$. In my analysis of this function's properties, the term $$M(x,y) \equiv -\frac{\widehat{z}_y}{\...
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Are winding number and index of a not smooth closed curve the same?

Let $\gamma:[0,1] \longrightarrow \mathbf{C} \backslash \{0\}$ be a closed curve (continuous and of bounded variation). We call $$\operatorname{Ind}_\gamma(0) \overset{\mathrm{def}}{=} \frac{1}{2 \...
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Relationship between affine plane curves and projective plane curves.

Let $\mathbb K$ be a field and $\mathbb A^2_{\mathbb K}:=\mathbb K^2$. An affine plane curve is a set of the form $$C_f(\mathbb K)=\{(a, b)\in\mathbb A_{\mathbb K}^2: f(a, b)=0\}$$ where $f\in \mathbb ...
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$[u:v]\mapsto[x(u^2v^3):y(u^2v^3):z(u^2v^3)]$ gives a rational curve

Given the map $\mathbb P^1\to\mathbb P^2$ with $[u:v]\mapsto[x(u^2v^3):y(u^2v^3):z(u^2v^3)]$, s.t. $[x:y:z]$ is on $C$(curve) how shall I deduce that the curve is rational ? I think I must show the ...
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Can a portion of a hypocycloid be a regular polygon?

Hypocycloids are curves that generally don't include straight lines. A significant exception is a hypocycloid with 2 cusps, generated by rolling one circle inside another having twice the radius of ...
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Help with parametrization of a Surface and finding tangent plane

So i have Surface defined as: $$(x^2+y^2+z^2)^3= (x^2−y^2)^2$$ Where $|x|\leq y$ So I was thinking Spherical Coordinates as base, so something like: $$x=\cos\theta \cos\phi$$ $$y=\cos\theta \sin\...
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963 views

Log power rule problem

According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian plane....
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Space-Filling Jordan Curve

My question is about a simple closed curve that is also a space-filling curve. The figure shows 6 iterations of the formation of a Hilbert curve (limit), whose trace is a solid square. I think we may, ...
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1answer
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prove an inequality between the curvature of these curves

The exercise is: Let $\alpha$ a plane curve such that $|\alpha'(s)|=1$ with curvature $k(s)$. Let $\beta(s)=\alpha(s) + k(s)N(s)$ such that $\beta'(s)\ne 0$ $\forall s$. ($N$ is the normal vector ...
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693 views

Point normal equation of plane

Im doing an homeassignment in linear algebra! This is the question I'm having problems with! The plane M contains two lines; l1 : (x, y, z) = t(2, −1, 0), t ∈ R, l2 : (x, y, z) = t(0, 1, ...
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When are cone geodesics planar

I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section. One asked whether if you 'unroll the cone' the conic section becomes a straight line ...
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If there any plane curve whose critical points' curvature are invariant by linear transformation?

I was studying if the curvature of $f$: $$ f(x) = \frac{ax}{b+x} $$ can have the critical points located at the same vertical than this other $g$ curve: $$ g(x) = \frac{ax}{b+x} + cx = f(x) + cx $$ ...
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hexagon with plane symmetries with maths and technology. [duplicate]

Find all plane symmetries (rotations and reflections) of a regular pentagon and of the regular hexagon.
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Finding Line tangent to surface and parallel to plane.

I had an exam today and I wanted to know if I solved this question correct. It asked me to, given a surface curve, like $z=x^2+y^2+10$, find the line tangent to it at point $(2,2,1)$ and parallel ...
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Intersection multiplicity and contact order of plane curves

A standard fact about intersection multiplicities of algebraic plane curves is that, for curves $X=V(F(x,y))$ and $Y=V(G(x,y))$, if $p\in X\cap Y$ is a non-singular point of $X$ and $Y$ then $$ I_p(X,...
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Total curvature of a closed polygonal curve

Let $\tau$ be a closed polygonal curve in the plane, that is, $\tau$ consists of $n$ piecewise-linear segments between the vertices $v_1, \ldots, v_n, v_1$. The total curvature of $\tau$ is defined as ...
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1answer
43 views

Total curvature of a parametrized-by-arc-length curve

Suppose we have the following smooth curve $\sigma:]0,2\pi[\leftarrow\mathbb{R}^2, \sigma(t) = (t, \sin t)$. I want to find the total curvature $\kappa := \int_0^{2\pi}||\sigma''(t)||dt$, but first I ...
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1answer
33 views

Does the system $xy = ab, G(x)+G(y)=G(a)+G(b)$ always have exactly two solutions if $G$ is continuous and injective?

If $f$ and $g$ are commutative operations $$\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R},$$ then for any constants $a,b \in \mathbb{R}$, the system of equations $$f(x,y) = f(a,b), \qquad g(x,y) ...
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Formulations of Cauchy's theorem that don't seem consistent

So I am reading through the book Complex Analysis by Lars Ahlfors, and there is one point that is causing some confusion for me: In chapter 4.2 - Cauchy's integral formula, we first encounter the ...
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Orthogonal trajectories of unit circles centered on x-axis [closed]

Find ( $c$ is an arbitrary constant ) orthogonal trajectories of circles: $$ (x-c)^2+ y^2= 1 $$
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313 views

Chess Board Coloring of a Paper using an Arbitrary Curve

Pick a piece of paper and a pen. Put the pen on a starting point and begin to draw an arbitrary curve and don't withdraw your hand until you reached the starting point. You can meet your curve during ...
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2k 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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254 views

Show that $P(x,y)=0$ is an ellipse if $b^2-4ac<0$.

I tried the following: I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ ...
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1answer
25 views

Is it possible to construct a closed, simple curve in $\mathbb{R}^2$ that has a segment with zero curvature and is differentiable everywhere?

The question title says it all. Presume I am interested in creating a closed, simple curve in $\mathbb{R}^2$ that contains a segment with zero curvature (that is, part of it is a straight line). ...
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896 views

Do line integrals along non-piecewise-smooth curves exist?

This article at Wolfram Mathworld has the following theorem on conservative vector fields: Theorem. The following conditions are equivalent for a conservative vector field $ \mathbf{F} $ defined on ...
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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 does one find a parameter representation for bounded region?

I need help with this question. I have been stuck at it for a few days. My main problem is how I use the curve $K_r$ to find the parametric representation. I have a curve $K_r$ in the $(x,y)$-plane ...
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What kind of Planar Quartic Curve might this be?

I'm trying to smoke out the parameters for a family of curves showing up in a particular optimization problem. I have convinced myself that the solutions always lie on a quartic curve, which is ...
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4answers
81 views

Confusion about a tangent line approaching an asymptote

I'm working from do Carmo's Differential Geometry of Curves and Surfaces, 2ed. He tends to use language like"the curve $\alpha$ and its tangent line approach [some line] $L$" or "the curve $\alpha$ ...
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35 views

Strong/weak tangents and limit positions, with rigor

As I'm working from do Carmo's Differential Geometry of Curves and Surfaces, I have found some of his imprecise language regarding strong and weak tangents to be most irksome. I've seen similar posts ...
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1answer
70 views

How many straight lines and circles can be drawn in the plane so that they are equidistant from all four points?

Four points on a plane are given which are not collinear or all on one circle. How many straight lines and circles can be drawn in the plane so that they are equidistant from all four points? If not ...
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Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
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Prove that the curve $\alpha(t)$ is tangent to the $x$ axis.

I have the curve $\alpha(t):(-1,\infty) \rightarrow R^2$ given by $\alpha(t)= ((\frac{nat}{1+t^3}), (\frac{nat^2}{1+t^3}))$ with $n$ a natural an $a$ a constant both of them fixed. I need to prove ...
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3answers
45 views

Can an arbitrary curve in $\Bbb R^2$ be a graph of a certain equation?

Can any curve in $\Bbb R^2$ (which doesn't intersect itself) be a graph of a certain equation? In other words, if given an arbitrary curve in $\Bbb R^2$ (which doesn't intersect itself), is there a ...
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What is the general formula for NURBS curves?

Give me the general mathematical formula for NURBS curves, with special cases (B-spline and Bézier curves)
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688 views

Formula to create a Reuleaux polygon

The Wikipedia articles for Reuleaux triangle and curve of constant width do a good job of describing the properties of a Reuleaux polygon, but they don't give a straightforward formula for computing ...