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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27
votes
14answers
35k views

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 ...
10
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4answers
405 views

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 $|t_1-t_2|$. ...
0
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1answer
810 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 ...
1
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0answers
25 views

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}{\...
3
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2answers
201 views

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 \...
0
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0answers
27 views

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 ...
0
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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 ...
8
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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 ...
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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\...
0
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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....
0
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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, ...
1
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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 ...
1
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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, ...
4
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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 ...
0
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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: $$ g(x) = \frac{ax}{b+x} + cx = f(x) + cx $$ ...
0
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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.
2
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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 ...
0
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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,...
2
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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 ...
1
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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 ...
1
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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) ...
1
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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 ...
-1
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1answer
43 views

Orthogonal trajectories of unit circles centered on x-axis [closed]

Find ( $c$ is an arbitrary constant ) orthogonal trajectories of circles: $$ (x-c)^2+ y^2= 1 $$
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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 ...
0
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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?
2
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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 $P(x, y) = Q(x)$ ...
0
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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). ...
4
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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 ...
2
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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 ...
1
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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 ...
0
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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 ...
1
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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 $\alpha$ ...
0
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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 ...
0
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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 ...
0
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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 ...
7
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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 ...
1
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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 ...
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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 ...
2
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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)
3
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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 ...
8
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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 ...
5
votes
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 ...
0
votes
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{...
16
votes
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 ...
1
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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-...
2
votes
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 ...
1
vote
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 ...
0
votes
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{...
0
votes
1answer
116 views

The area of a region around a curve

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 ...
2
votes
1answer
84 views

Curvature inequality involving a Curve within a disk

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 ...