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Questions tagged [curves]

For questions about or involving curves.

8
votes
0answers
198 views

Variational gradient descent on the space of curves

I want to find a curve $\gamma : [0,T] \to \mathbb{R}^n$ that minimizes the following functional $$J[\gamma]=\int\limits_0^T F(\gamma(t),\gamma'(t))\,dt$$ Suppose I know the endpoints, $a, b \in \...
7
votes
0answers
81 views

Can any curve in 3D space be described by an intersection of two surfaces?

Can any curve in 3D space be described by an intersection of two surfaces? If not, what assumptions I need to let it be true? If this is too general, what if I restrict the scenarios to twice ...
7
votes
0answers
106 views

Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
5
votes
0answers
264 views

Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves ...
4
votes
0answers
94 views

There are no non-constant morphisms from $\mathbb{A}^1$ to a cubic curve

Consider the cubic curve $X$ defined, over an arbitrary field $K$, by the equation $y^2 =x(x−1)(x−\lambda)$ in $\mathbb{A}^2$, where $\lambda \neq 0,1$. Show that there are no non-constant morphisms $...
4
votes
0answers
61 views

Consider homotopy of closed curves. Show equality of winding numbers.

a) Let $H: [0,K] \times [0,1] \rightarrow \mathbb{R}^2 $ be a homotopy of closed curves, so $H$ is continuous and for every $\sigma \in [0,1]$ it holds that $ c_{\sigma}:[0,K] \rightarrow \mathbb{R}...
4
votes
0answers
83 views

Show that the following space curve $c$ is $C^{\infty}$ and Compute binormal.

Consider the space curve $c : (-\frac{1}{2},\frac{1}{2}) \rightarrow \mathbb{R^3}$ , $$c(x)= \begin{cases} (x,e^{-\frac{1}{x^2}},0) &\ x<0\\ \ \\ (0,0,0)&\ x=0 \\ \ \\ (x,0,e^{-\frac{1}{x^...
4
votes
0answers
321 views

(first) cohomology $H^1$ of sheaves on curves

Say we have a reduced, irreducible, complete curve $X$ over a field $k$ and a sheaf $\mathcal F$ on $X$. I am interested in understanding the cohomology groups of $\mathcal F$. By flat base change, ...
4
votes
0answers
100 views

Does roots of L-polynomial of curve being roots of unity imply the curve is supersingular?

If $C$ is a supersingular curve over $\mathbb F_q$, then $\frac1{\sqrt q}$ of roots of $L$ function are roots of unity. What about converse? If $\frac{1}{\sqrt q}$ of roots of $L$ polynomial are ...
4
votes
0answers
44 views

Effect of adding a constant to the torsion of a 3D curve

Let $\gamma$ be an arc-length parametrized curve in $\mathbb{R}^3$. Let say I add a constant to the torsion of $\gamma$ and let $\widetilde{\gamma}$ be the curve associated to the curvature of $\gamma$...
4
votes
0answers
125 views

What's an algebraic curve's polar line for?

I can take an algebraic curve, and I can draw its first polar. By this, I mean that I can take an arbitrary point not on the algebraic curve, and then I can identify all the points on that algebraic ...
4
votes
0answers
135 views

Detecting the respiratory rate of a breating lung.

I am currently working with some data-sets that represents the movements of a beating heart and breathing lungs. The data-sets are represented as a collection of floats that range from 47 to 51. We ...
3
votes
0answers
86 views

Intersection of two curves only in two points.

Is it always true that for any given curve and for any two points on that given curve (no matter how close these points are), it is possible to construct some curve that intersect that given curve ...
3
votes
0answers
45 views

Elliptic curves (Tate normal form?)

I basically have two question, the other question can be found below. Let $E/k$ be an elliptic curve with $P\in E(k)$ a point of order $\geq 4$. Show that $E$ can be described by \begin{equation*} y^...
3
votes
0answers
65 views

Symmetries of a curve on the unit sphere

Let $\mathbb{S}^2 \subseteq \mathbb{R}^3$ be the unit sphere. Let $\gamma : [0, l] \to \mathbb{S}^2$ be a smooth embedded closed curve. Let $\vec{\nu}$ be one of the two continuous unit normal vector ...
3
votes
0answers
55 views

“Successive extension of invertible $\mathcal{O}_S$-modules”

I am currently reading Takeshi Saito's book "Fermat's Last Theorem: The Proof". In the proof of Proposition 8.12, there is a part that I do not understand clearly. For my problem in particular, I am ...
3
votes
0answers
48 views

Contact order of a space curve with one of it's tangent lines

Definition $1$: Let $\alpha: I \mapsto \mathbb{R^3}$ and $\beta: \overline{I} \mapsto \mathbb{R^3}$ be two regular curves such that $\alpha(t_0) = \beta(t_0)$, where $t_0 \in I \cap \overline{I}$. $\...
3
votes
0answers
161 views

Show the equivalence of arc length definitions

Definition 1: Let $r: [a,b] \to \Bbb R^d$ be a continuous differentiable function. Then the arc length is given by $$L(r) = \int_a^b || r'(t) || \, dt$$ Definition 2: Let $r: [a,b] \to \Bbb R^d$...
3
votes
0answers
111 views

Calculating singular points of a quintic curve in the projective plane?

I have the following question for an assignment. "An irreducible quintic curve in the real projective plane $P^2(R)$ is defined by $F: (X^2-Z^2)^2Y-(Y^2-Z^2)^2X=0$ Verify that the quartic curve ...
3
votes
0answers
740 views

Definition of smooth curve - why nonzero derivative?

Above is the definition of smooth parameterized curve given in Stein and Shakarchi's Complex Analysis. (i) Why do we require $z'(t) \not= 0$? I have not seen this condition used in any proofs. (ii) ...
3
votes
0answers
163 views

Line integrals along space filling curves

Let ${C_n}$ be a family of curves (on the unit square $M$) such that $C_\infty$ is a space filling curve and $f(x,y)$ a function of two variables. Does the following identity hold? $$lim_{n\to\infty}\...
3
votes
0answers
1k views

What is the significance of Brachistochrone curve in the real world?

I want to know how does the brachistochrone curve is significant in any real world object or effect. Are there any machines or devices which are based upon the principle of shortest time?
3
votes
0answers
171 views

What's the name of the curves obtained by intersecting a n-holed torus and a plane?

Cassini ovals are a well-known family of quartic curves which include the lemniscate as a special case. They can be thought as the vertical cross sections of a torus, that is, the intersections of a ...
3
votes
0answers
193 views

B-splines and Catmull-Clark subdives. What are the similarities between them?

I've a special question between mathematics and 3d. I struggle for two days understand relation between B-splines and Catmull-Clark subdivs. Everywhere wrote the Catmul-Clark subdivs is based on ...
3
votes
0answers
202 views

Does every conceivable curve have a possible equation?

We know that every equation has a graphical representation by a curve, but does every curve have an equation? If I scribble something crazy on a coordinate plane, do we know if there's an equation ...
3
votes
0answers
199 views

Approximate equation for tapered cycloid offset curve without cusps

Is it possible to create parametric equations to approximate a tapered cycloid offset curve without cusps, that does not require manual adjustment of values when the primary curve parameters are ...
3
votes
0answers
51 views

Find a differentiable curve on the paraboloid $\,z= 2x^2 + y^2$ with minimum curvature

Let $S$ be the graph of the function $\,f(x,y) = 2x^2+ y^2$ in $\,\mathbb{R}^3$ (a paraboloid with vertex at the origin). It is clear that $S$ is a regular surface which can be parametrized by the ...
3
votes
0answers
318 views

What difference between analytic curve, and analytic function? .

What is the difference between an analytic curve, and an analytic function?
3
votes
0answers
375 views

Is it known whether the boundary of the Mandelbrot set is not continuous?

I might be missing something obvious here, but my understanding is that nobody currently knows whether the boundary of the Mandelbrot set is a Jordan curve because otherwise we would know that the ...
3
votes
0answers
76 views

Show that $|\alpha'(t)|^2-1=0$ for any arbitrary parameter $t$

There is this supposed to be not so hard question but concept of re-parametrisation by arc length bothers me a bit. Let $\alpha:I\to R^2$ be a regular paramatrised plane curve (arbitrary parameter) ...
3
votes
0answers
83 views

Intersection of translated simple curves in $\mathbb{R}^2$

I'm trying to prove the following conjecture: Let $\gamma$ be a simple (i.e. non self-intersecting) plane curve going from point $p$ to $q$. Let $T(\gamma)$ be a translated copy of $\gamma$ going ...
3
votes
0answers
123 views

Curvatures of 4-Dimensional Parallel Curve

Let $\vec{\alpha}: I \rightarrow \mathbb{R}^4$ be an arc-length parametrized curve in $\mathbb{R}^4$ with curvatures $k_1, k_2, k_3$. The principal normal unit vector is $\vec{n} = \frac{\vec{\alpha}'...
2
votes
0answers
38 views

Reparameterisation of Curve as a Regular Curve (Topology)

There is a result that a curve or topological path can be reparameterized as a regular curve contained in the paper "Reparametrizations of continuous paths - Ulrich Fahrenberg and Martin Raussen" ...
2
votes
0answers
30 views

Orthogonal monomials on a curve

Let $\Gamma \subset \mathbb{C}$ be a smooth curve such that monomials are orthogonal on it, i.e. with $n,m \in \mathbb{N} \cup\{0\}$ $$\int_{\Gamma} z^n \overline{z^m} |dz| = 0, \qquad \qquad \forall ...
2
votes
0answers
33 views

What is the definition of ''geometrically irreducible closed curve''?

In the algebraice geometry, one says about "geometrically irreducible closed curve" over field $k$. For example, the theorem 5.4.5 (pp. 147) of ''Heights in Diophantine Geometry'' of E. Bombieri wrote ...
2
votes
0answers
45 views

How to prove that Frenet frame is independent of the choices of parameters?

When I am reading ''A course in differential geometry'' of Klingenberg, I cannot be sure the Frenet frame defined in this book is independent of the choice of parameter of a curve. As a result, the ...
2
votes
0answers
28 views

Inscribed Rectangle Proof (Basic Question)

Here's a video on the problem that I am referencing (it's well worth the watch either way) https://youtu.be/AmgkSdhK4K8 The problem asks whether you can find an inscribed rectangle on any Jordan ...
2
votes
0answers
67 views

Osculating circle at curvature minimum point of simple closed curve encloses the curve

The story begins with seemingly unrelating situation. I was trying to find out an elementary solution of the following problem. A circle is called a separator for a set of five points in a plane if ...
2
votes
0answers
43 views

Glueing two affine curves along a map

Let $C = V(y^2 - x^4 - 1) \subset \Bbb A^2$ and set $C_0 = C \times \{0\}, C_1 = C \times \{1\} \subset \Bbb A^3$. Consider the space $X$ obtained by quotienting $C_0 \sqcup C_1$ but the relation $((...
2
votes
0answers
30 views

Approximating the intersection of a line and the iPhone X screen as well as its normals

I am creating a simulation where little, fast moving, particles need to intersect with the edges of the iPhone X screen. Previously I have had no difficulty with my collisions. The particles have ...
2
votes
0answers
28 views

Computing the geodesic curvature of curves in $\mathbb{H}^1 \times \mathbb{R} \subset \mathbb{L}^{3}$

Here $\mathbb{L}^3$ is $\mathbb{R}^3$ with the metric $\langle u, v\rangle_\mathbb{L} = -u_1v_1 + u_2v_2 + u_3v_3$, where $u = (u_1, u_2, u_3)$ and $v = (v_1, v_2, v_3)$ and $\mathbb{H}^1 \times \...
2
votes
0answers
47 views

Closed curve $\gamma$ such that least area of a surface with boundary $2\gamma$

What does the following statement mean? Any geometric intuition would be very helpful. L.C. Young constructed a closed curve $\gamma$ in the Euclidean space $\mathbb{R}^4$ such that the least area ...
2
votes
0answers
25 views

parameterisation of a set $M$. $ M = $ rotation of the image of a curve $c$ around x-axis.

Let $ c : I \rightarrow \mathbb{R}^3 $ be a space curve with $c(t) = (f(t),g(t),0)$. if we rotate the image of $c$ around the x-axis, we get the set $R$. First task: Find a parameterisation of $...
2
votes
0answers
55 views

In Calculus, why must the gradient of a specific point of a curve be given by a line that locally touches the curve at one and only one point?

What I don't understand is why the gradient of a specific point of a curve, the tangent line, must only locally touch the curve at one and only one point. What is the logic behind that? Why does the ...
2
votes
0answers
62 views

Good references to start studying the curve shortening flow

I want to study the curve shortening flow on curves in $\mathbb{R^2}$. I have knowledge of the local theory of plane and space curves (obviously that comes with linear algebra and calculus as well), ...
2
votes
0answers
232 views

Whats the relation between the Darboux Frame and the Frenet-Serret on a oriented surface?

i'm studying diferential geomtry and i'm in the part of geodesics, my professor always define a curve that can define the tangent field but for calculating the geodeiscs and normal curvatures at each ...
2
votes
0answers
141 views

Curve fitting N points with n(fixed) quadratic curves

I essentially have a constrained curve fitting problem that I need to solve efficiently. The following problem arises when performing practical calibration of RSSI (signal strength), providing ...
2
votes
0answers
89 views

Dynamic programming approach of Hermite interpolation

Let's define the $k$-th degree Hermite interpolating spline as a curve which interpolates between points $P_0$ and $P_1$ and also between their derivatives $v_0^j$ and $v_1^j$ up to order $j=(k-1)/2$. ...
2
votes
0answers
145 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 ...
2
votes
0answers
58 views

Inequalities and logarithm

If $f(x) = k^3x + k^3 – 2$ cuts the curve $g(x) = \frac{1}{2}ln(x)^2$ at exactly one point then What interval would 'k' belong to ? I basically tried to equate both the equations together but got ...