Questions tagged [curves]
For questions about or involving curves.
3,461
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Path created by a line wrapping a circle [closed]
I have a circle and a line with the length of its circumference. One end of the line is attached to a point on the circle circumference, forming the shape of the lowercase 'b'.
Now, the line begins to ...
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Does every curve have an equation that can represent it perfectly? If no, what curves have no perfect equation? [closed]
I consider a curve to be a series of connected points, and i want to know if all curves have perfect (not approximate) equations that represent them. If no, what curves have no equation?
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Confusion about coordinate representation and their relations and curve fitting
Can someone review a problem I am having related to coordinates and curve fitting
Part 1
I have a curve in the coordinate frame XY. This curve is represented by a list of $(x,y)$ coordinates
I also ...
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Fitting a modified Bézier curve
I am connecting various vectors with a Bézier curve, using the De Casteljau algorithm. These vectors have a variety of lengths and directions, and when they are equal and orthogonal the curve (purple) ...
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Hopf Umlaufsatz (circulation/rotation index/rotation angle theorem) on a Riemannian 2-manifold
The book I am studying (Loring W. Tu's Differential geometry, Connections, Curvature and Characteristic Classes) makes use of the following theorem:
Theorem 17.4 (Hopf Umlaufsatz). Let $(U,e_1,e_2)$ ...
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Finding an equation for a curve along circle between two points
I am trying to find an equation for a curve along circle with radius $5$ and centre at $(-2,3)$ and between points $(3,3)$ and $(-2,-2)$ on circle
Now I know that equation for this circle is $$(x+2)^2 ...
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If a closed curve has no antipodal points it must be contained in some open hemisphere
Whilst doing an exercise, on bounds on total absolute curvature I encountered the following obstacle:
Let $ \mathit c$ be a closed curve, whose trace is contained in $\mathbb S^2$, if c has no ...
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Why do we find the parameters for a catenary curve the way we do?
I have been searching for an explanation and walkthrough about finding the parameters for a catenary curve. I understand that I have to take the length of the curve squared and subtract the vertical ...
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Line of Best-Fit Between Two Polymetric Curves
I have been using basic regression in excel to find the correlation between two variables.
I have since been able to produce two curves in an attempt to model the variable against each other.
Two ...
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Does the mean area of triangles with equal perimeter $p$ and circumradius $R$ have a local minima at $p=4R$?
Definition: Isoperimetric triangles are triangles which have the same perimeter and the same circumradius.
Isoperimetric area curve: The largest perimeter of a triangle that can be inscribed in a ...
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Does a curve intersect itself always finitely many times at a given point?
I thought of this question in Differential geometry class. Probably when topic is about self intersecting curves, professors draws (if) a curve coming from below, making a loop and going towards below ...
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On the definition of singular points for algebraic curves
For simplicity, let us consider curves in the Euclidean plane ${\mathbb{R}}^2$. They may be defined as the set of points $(x_1, x_2)$ satisfying an equation of the form $P(x_1,x_2)=0$ where $P(x_1, ...
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Determining the graph a parametric function.
I know that the graph of $\overline{r}(t) = \langle \cos(t), t, \sin(t)\rangle$ is a cylinder aligned along the $y$-axis, however how can I determine from $\overline{r}(t) = \langle \cos(t), -\cos(t), ...
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Complex variable integration
Question: Consider $\gamma_1 = [z_0 - r, z_0 + r]$ and $\gamma_2: [0, \pi] \rightarrow \mathbb{C}$, $\gamma_2(t) = z_0 + re^{it}$. Let $z \notin \gamma_1^* \cup \gamma_2^*$.
Evaluate
$
\frac{1}{2\pi i}...
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Is it possible to define a single parametric curve that equals three other related curves for specific inputs?
The Curves
I have worked out parametric equations for the curves of involute gear profiles using the seven gear parameters given in the "Background" section below. However, currently the ...
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Dragon curves, fractional dimensions and whatever could be deduced
I put out a question about a thesis on Dragon Curves which was deleted. I did not know it is against the guidelines to advise on academia. Please don't consider my question as seeking advice on that ...
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A line perpendicular to a tangent curved at both ends
I made a guess two years ago. I have a strong feeling that there is a proof using the fixed point theorem with geometric visualization, but I couldn't do the proof.
If you have a simple closed convex ...
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How to calculate the area of the region enclosed by $x^3+y^3=3axy$？ [duplicate]
Let $y=tx,$ after calculation, I get
$$x=\frac{3at}{1+t^3}, y=\frac{3at^2}{1+t^3}.$$
Use the Green Formula, the area is equal to
$$\frac{1}{2}\oint_{\Gamma}xdy-ydx.$$
My question is: how to determine ...
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Does Hartshorne assume curves are irreducible when proving that pullback along a finite morphism of nonsingular curves preserves linear equivalence?
The question is how it can be proved the statement in the title:
Let $f: X\to Y$ be a finite morphism of nonsingular curves.
Then the pullback morphism $f^*:\operatorname{Div} Y \to \operatorname{Div} ...
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Analytic parameterization of a residue disk
Let $C/\mathbb{Z}_p$ be a non-singular, projective, curve, with good reduction. Now, let $P\in C(\mathbb{F}_p)$ be a point of its special fibre, and denote by $\mathcal{D}$ the residue disk of $C$, ...
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Curves And Surfaces Exercise (6) of Chapter 1
I am currently reading the book "CURVES AND SURFACES" [ SECOND EDITION 2009 ] by Ros and Montiel.
I am having trouble to solve and understand Exercise (6) of Chapter 1.
It says the following ...
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equation of 3d curve passing through three points
Let $P=(x_1,y_1,0)$ and $Q=(x_2,y_2,0)$ be two points in the $xy$ plane.
Find an equation for a curve (could be a half-ellipse) passing through $P$, $Q$, and a third point $R=(\frac{x_1+x_2}{2},\frac{...
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Find $\epsilon$ such that $\varphi^{\epsilon} = \sigma + \epsilon(\cos t \mathbf{n} + \sin t \mathbf{b})$ is injective ($\sigma$ biregular curve) .
I’ve been struggling for an entire day trying to approach the third point of this exercise (Ex. 4.23 from ‘Curves and Surfaces’ by M. Abate and F. Tovena).
Let $\sigma:I \to \mathbb{R}^3$ be a ...
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Can the boundary of any closed curve be represented as a series of circles of the same diameter (or any other geometrical shape)
The boundary of a simple closed curve has each point on it, touching two adjacent points one on each side. Can the entire boundary be represented as a series of circles of the same diameter "d&...
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The area of closed curve made by sine functions that are rotated on some angle $\theta$ [closed]
I constructed interesting figure and want to know its function on Cartesian plane and the area enclosed by this figure.
Suppose you have the sine function (explicitly, $m \cdot \sin(kx)$) from 0 to $\...
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Exponential Power Fit
I have 3 data points for a material property (permeability of a gas through an solid elastomer) as a function of temperature. My goal is to extrapolate the property at a lower temperature.
As a ...
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Estimating mean curvature of a surface by two perpendicular curves of the surface
Let S be an oriented smooth surface containing a circle of radius 1 and a straight line, which intersect perpendicularly at a point $p\in S$. Show that if the Gauss curvature K of
S satisfies
K(p)=0, ...
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Parametrising a piecewise circular curve in 3d
The curve in question is a closed curve $C: = \overline{abcda}$ made of three circular arcs $C_1=ab, C_2 =bc$, $C_3 = cd$ and a straight line segment $ C_4 = da$. The arc $C_1$ lies on a plane $P_1$ ...
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Parametric Surfaces Appearing completely different in graphing software than in original paper
I tried to plot the parametric surfaces from this paper to try to play around with the variables and potentially adapt it for other purposes. However I attempted to plot the surfaces in both Geogebra ...
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The length of a curve that has Riemann integrable derivative [Mathematical anaylsis]
Let $-∞<a<b<+∞$ and $k∈ℕ$. If a function $γ:[a,b]→ℝ^k$ has bounded derivative on $[a,b]$,
(a) prove that $\gamma$ is rectifiable.
(b) If $\gamma'$ is Riemann integrable on $[a,b]$, determine ...
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Does $\kappa = |T \times \frac{dT}{ds}|$?
If $T$ is the unit tangent vector and $\kappa$ the curvature, is it true that $$\kappa = |T \times \frac{dT}{ds}|$$?
I believe this is true (proof below), but am surprised that I cannot find it ...
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How to write clear proofs involving curves and surfaces, when switching between different frames?
I'm having trouble wtih the exposition to problems such as the one below, and I'd therefore like to request suggestions on how to improve it. The problem, my solution, and my struggles are below.
...
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Properties of Mannheim pairs
Two regular curves $\alpha$ and $\beta$ are considered a Mannheim pair when $N_{\beta} = \pm B_{\alpha}$ and there is a differentiable function $\lambda: I \rightarrow \mathbb{R}$ in order that $$\...
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Intrinsic vs extrinsic quantity of curves
I have recently started learning the differential geometry of curves and surfaces.
I am not sure what intrinsic/extrinsic means.
My current understanding is this: an intrinsic quantity on a curve is ...
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Consider the curves $C_1: x^2y^2-a^2(x^2+y^2)=0, C_2: xy=a^2, C_3: x^2+y^2=a^2(a≠0).$ Count the number of points & tangents common to all three curves
The following question is taken from JEE practice set :
Consider the curves
$C_1: x^2y^2-a^2(x^2+y^2)=0$ ,
$C_2: xy=a^2$ ,
$C_3: x^2+y^2=a^2(a≠0).$
Let number of points common to all three curves is $...
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NURB curves with interior knot multiplicy higher than curve degree
I have the following question. If I have a NURB curve where one of the interior knots has multiplicity higher than the degree of curve (I did not chose to have such curves, I am writing a code for ...
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Conceptually Understanding the Mathematical Definition for an Envelope of Family of Curves
Assuming we have a one-parameter, two-dimensional, family of curves, given by $f(x, y, p) = 0$, there are two requirements for the envelope (see https://en.wikipedia.org/wiki/Envelope_(mathematics)#) ...
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Is the parameterization by arc length unique?
I have the following problem which I don't know if it's true:
Let $\varphi: I\rightarrow \mathbb{R}^n$ and $\psi: J\rightarrow \mathbb{R}^n$ be parameterizations of a curve $\Gamma$ such that $\varphi ...
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Dualizing sheaf - normalization of nodal curve
Given a normalization of a nodal curve $\alpha : \tilde{C} \to C$ over an algebraically closed field. Assume for simplicity we only have one node $p$ with $\alpha^{-1}\left(p\right)=\left(q_1,q_2 \...
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Connected components of image of non-degenerate boundary component
This is a follow up to my previous question, asked and answered here: Connected components of conformal image of boundary
I omitted this by accident from the last question, so I have created this ...
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How to get enough data to draw an arc or a curve from 3 points?
Note: Although I want to accomplish this in java, I think the question is more suitable for this site since it is mostly mathematical.
I am in the following scenario. I want to draw a curve and I have ...
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Convexity of spherical curves
Let $\gamma$ be a smooth curve in $\mathbb{R}^{3}$. We say that $\gamma$ is convex if it lies on the boundary of its convex hull. Recall that the convex hull of $\gamma$ is the set of all convex ...
user242708
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Curve which have no tangent
I am currently reading "Science and Hypothesis" written by Poincaré.
In chapter 2, "Mathematical Magnitudes and Experiments", I found the following sentence.
"We can show that ...
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$\mathbf{r}(t):=(t^3,t^7)$. I want to say $(1,0)$ is the unit tangent vector to the curve defined by $\mathbf{r}$ at $\mathbf{0}$. Why bad?
The vector $\mathbf{r}'(t)$ is called the tangent vector to the curve defined by $\mathbf{r}$ at the point $\mathbf{r}(t)$, provided that $\mathbf{r}'(t)$ exists and $\mathbf{r}'(t)\neq\mathbf{0}$.
...
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An identity theorem due to Koebe
There is likely a misprint on pages 31,32 and 33 of the book "Geometric Theory of Functions of a Complex Variable" by G.M. Goluzin, given below (see Lemma 1 and its proof ):
This is because ...
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Is a non-fractal, continuous curve made of tiny line segments?
EDIT: This question was in-part made unclear by my misconception of the hyperreals not being dense, since I thought there were no numbers between zero and consecutive infinitesimals
($0, 1/\infty, 2/\...
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Convex hull of convex space curve
Let $\gamma$ be a smooth closed curve $\mathbb{R}^{3}$. We say that $\gamma$ is convex if it lies on the boundary of its convex hull, which we denote by $\mathrm{conv}(\gamma)$.
I know that the convex ...
user242708
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Consider the curve $C=\bigl\{(t,|t|): t \in \mathbb R\bigr\}$.
Consider the curve $C=\bigl\{(t,|t|): t \in \mathbb R\bigr\}$.Show that if $\alpha:\bigr]a,b\bigr[ \to \mathbb R$ is a differentiable curve whose trace lies in C and $t_0 \in \bigr]a,b\bigr[$ is such ...
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Trying to fit a quasi-periodic sequence
I am trying to fit dataset with a function, but with little success, as illustrated below. The blue line is the data I wish to fit, and the red line is my attempt.
Below is the function I used:
$$ g(...
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Measuring the smoothness of a curve
In deep learning, accuracy curves are crucial for evaluating a model's performance. Typically, an accuracy curve resembles a logarithmic function, although the reasons for this are beyond the scope of ...
