# 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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### 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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### 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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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 $f\in \mathbb ... 0answers 27 views ###$[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 ... 1answer 234 views ### 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 ... 2answers 25 views ### 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: $$x=\cos\theta \cos\phi$$ $$y=\cos\theta \sin\... 1answer 938 views ### Log power rule problem According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian plane.... 0answers 23 views ### Space-Filling Jordan Curve My question is about a simple closed curve that is also a space-filling curve. The figure shows 6 iterations of the formation of a Hilbert curve (limit), whose trace is a solid square. I think we may, ... 1answer 24 views ### prove an inequality between the curvature of these curves The exercise is: Let \alpha a plane curve such that |\alpha'(s)|=1 with curvature k(s). Let \beta(s)=\alpha(s) + k(s)N(s) such that \beta'(s)\ne 0 \forall s. (N is the normal vector ... 2answers 688 views ### Point normal equation of plane Im doing an homeassignment in linear algebra! This is the question I'm having problems with! The plane M contains two lines; l1 : (x, y, z) = t(2, −1, 0), t ∈ R, l2 : (x, y, z) = t(0, 1, ... 1answer 52 views ### When are cone geodesics planar I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section. One asked whether if you 'unroll the cone' the conic section becomes a straight line ... 0answers 22 views ### If there any plane curve whose critical points' curvature are invariant by linear transformation? I was studying if the curvature of f:$$ 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 $$... 0answers 16 views ### hexagon with plane symmetries with maths and technology. [duplicate] Find all plane symmetries (rotations and reflections) of a regular pentagon and of the regular hexagon. 0answers 29 views ### Finding Line tangent to surface and parallel to plane. I had an exam today and I wanted to know if I solved this question correct. It asked me to, given a surface curve, like z=x^2+y^2+10, find the line tangent to it at point (2,2,1) and parallel ... 0answers 34 views ### Intersection multiplicity and contact order of plane curves A standard fact about intersection multiplicities of algebraic plane curves is that, for curves X=V(F(x,y)) and Y=V(G(x,y)), if p\in X\cap Y is a non-singular point of X and Y then$$ I_p(X,... 2answers 68 views ### 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 ... 1answer 41 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 ... 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 $$f(x,y) = f(a,b), \qquad g(x,y) ... 0answers 44 views ### Formulations of Cauchy's theorem that don't seem consistent So I am reading through the book Complex Analysis by Lars Ahlfors, and there is one point that is causing some confusion for me: In chapter 4.2 - Cauchy's integral formula, we first encounter the ... 1answer 43 views ### Orthogonal trajectories of unit circles centered on x-axis [closed] Find ( c is an arbitrary constant ) orthogonal trajectories of circles:$$ (x-c)^2+ y^2= 1 $$4answers 310 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 ... 2answers 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? 1answer 253 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) ... 1answer 24 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). ... 3answers 881 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 ... 0answers 146 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 ... 0answers 35 views ### 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 ... 0answers 13 views ### 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 ... 4answers 79 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 ... 0answers 32 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 ... 1answer 66 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 ... 0answers 15 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 ... 0answers 122 views ### 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 ... 0answers 29 views ### 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 ... 3answers 43 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 ... 1answer 3k views ### 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) 3answers 670 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 ... 2answers 141 views ### If \gamma\colon[a,b]\to\mathbb{C} is continuous and \gamma(b)=-\gamma(a), must the curves \gamma and e^{ic}\gamma intersect for all real c? If [a, b] is a compact interval of \mathbb{R} and \gamma: [a, b] \to \mathbb{C} is continuous, denote the connected, compact set \gamma([a, b]) by [\gamma]. If h is a complex number of ... 0answers 54 views ### Must a curve \eta \colon [a, b] \to \mathbb{R}^2 intersect the curves \eta + \frac{\eta(b) - \eta(a)}{n} (n \geqslant 1)? Must a curve \eta \colon [a, b] \to \mathbb{R}^2 intersect the curves \eta + \frac{\eta(b) - \eta(a)}{n} (n \geqslant 1)? This is the fourth - and with any luck, the last! - in a series of ... 0answers 66 views ### Parametrization of special family of tori knots Finding the parametric equations of an (a-c)tori knot knowing that one turn has the following parametric equation:$$\alpha(t)=\begin{pmatrix} x=\Big(R_1+R_2\cos(t)\Big)\cos\biggr(c\arctan\Big(\dfrac{... 4answers 23k views ### How to find the parametric equation of a cycloid? "A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia In many calculus books I have, the cycloid, in parametric form, is used ... 1answer 40 views ###$\tilde \phi:\Gamma (W)\rightarrow \Gamma(V)$may not extends to all$k(W)$A problem from Fulton's Algebraic Curves:-- Let$\phi:V\rightarrow W$be a polynomial map between two affine varieties and$\tilde \phi:\Gamma (W)\rightarrow \Gamma(V)$be the induced map between co-... 1answer 61 views ### If two closed plane curves are outside each other, can there be a point inside both of them? I think this recent question (also here) has a quick answer if the conjecture below is true. It looks "obviously" true, but I've learned to distrust my judgement in such matters. It also looks as if ... 1answer 25 views ### Plane clipping by cubic limits I have a plane equation given by a point and a normal vector, for example. This plane has to lay between$xyz$limits,$300<x<2700$,$150<y<1350$,$130<z<1370$. I want to know the ... 0answers 33 views ### Matrix powers and hyperbola (We're in$\mathbb{R}^2$) How to find hyperbola equation, that has symmetry axis crossing (0,0) point and for$n=1,2,\ldots$points${\begin{pmatrix} 4 & 3 \\ 1 & 1 \end{pmatrix}}^n \begin{...
If we are given a simple closed curve with length $L$ in the plane, and we have a fixed number $r$ such that for each point $x$ on the curve there is a related disc $D(x,r)$ with radius $r$, how can ...
If a closed plane curve $C$ is contained inside a disk of radius $r$, prove that there exists a point $p \in C$ such that the curvature k of C at p satisfies $\lvert k\rvert \ge$ $1/r$. I understand ...