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Questions tagged [curves]

For questions about or involving curves.

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25 views

What curve is this and where is it used?

While playing around with the normal distribution, or more precisely, the probability density function: $$\displaystyle f(x | \mu,\sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{(x - \mu)^2}{2 ...
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1answer
28 views

Does there exist a constant curvature curve on paraboloid surface?

I'm learning differential geometry,and need help with the following question: Does there exist a constant curvature curve on paraboloid surface $ z = x^2 + y^2$ different from plane curve, i.e $z$ ...
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24 views

How important is it to have a deep understanding of the graphs of common differentiable functions? [on hold]

Recently I have been trying to refresh my understanding of calculus in an in-depth way and I have been going through a calculus text book, doing every other problem. I am tempted to skip over the ...
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31 views

Find the orthogonal to a clothoid spiral from an offset coordinate

At the moment, I'm implementing an algorithm which defines a spiral using the method outlined in the following paper: https://www.ams.org/journals/mcom/1992-59-199/S0025-5718-1992-1134736-8/S0025-...
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3answers
39 views

Proving a line is tangent to a curve

1) Imagine a point T(g, f) on a Cartesian plane, where g, f ≠ 0. Deduce, in terms of g and f, the equation of the line, in which its x-intercept and y-intercept have midpoint T, which means that when ...
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24 views

Penrose Singularity Theorem Proof

I have problems understanding the first part of the proof for the Penrose singularity theorem in the book "Leonor Godinho José Natário An Introduction to Riemannian Geometry": I know that $\langle\...
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1answer
24 views

condition of curvature continuity - Is direction of the bending important?

What is the condition of curvature continuity? Do we only consider the value of the curvature $\kappa$, or do we have to also take account of the direction of the curvature vector $\mathbf{k}$ as ...
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2answers
40 views

How to calculate the derivative of this curve?

How do I go about calculating the derivative of this curve from $\mathbb{R}$ to $\mathbb{R}^3$? $$γ(t) = (t\cdot\cos(2t), t\cdot\sin(2t), t)$$ I have tried simply taking the derivative of the three ...
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20 views

Curves and angles between them

How do you define-: (a) Angle between curves (b) Angle between straight line and a curve (c) Angle between tangent and a curve
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3answers
44 views

Find projectile speed given maximum height and range.

I want to simulate a catapult throwing a rock in my computer game, but by design, I want all my units to shoot from a certain height, reach a maximum height and also hit a target that can be meters or ...
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27 views

Confusion on the formula of Rational Bezier curve in Duncan's book

Duncan Marsh's book Applied Geometry for Computer Graphics and CAD stated I'm really confused about this formula. First of all, why do we replace $w_i\mathbf{b_i}$ with $\mathbf{b_i}$ when $w_i=0$? ...
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1answer
31 views

Proof that the tautochrone is a cycloid

In the Wikipedia article about the tautochrone curve, there is a proof of the fact that the tautochrone curve must be a cycloid. The proof starts with the following statement: One way the curve can ...
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2answers
59 views

Curves $y=a^{x}$ and $y=x^2$.

Let us assume two curves $y=a^{x}$ and $y=x^2$. Let us assume that a>0. Values range of a for which curves have one solution, two solution and three solution. Is there a way to check the values. I ...
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9 views

shortest path in surface $P$

Let $P$ be a surface in $\mathbb{R}^3$ given by $x = \frac{6-y^2}{2}$. Find the shortest path in $P$ which contains the points $A_1=(3,0,0)$ and $A_2=(1,2,1)$. I started this exercise with the aim to ...
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1answer
17 views

Strict inequality of the H1 seminorm of a curve emerging from a convex set and its convex projection

Let $\gamma(t) \in W^{1,2}([0,1]; \mathbb{R}^d)$ such that $\gamma(0) \in C$, with $C \subset \mathbb{R}^d$ some closed convex set, and $ \gamma(t) \not \in C, \ \forall t\in (0,1]$. Furthermore we ...
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2answers
32 views

Maximum ( or minimum ) of two functions on arbitrarily small interval.

Is there an example of two continuous functions say f and g ( they are continuous on whatever was the interval in hand ) , such that you can't find an interval ( no matter how small the interval was ) ...
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1answer
15 views

Dimensionality reduction for high dimensional curves?

I have a continuous curve in high dimensional space, and I'd like to visualize it in lower dimensional (2D or 3D) space to get an intuition of what it looks like. I'm familiar with PCA and t-SNE, but ...
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1answer
18 views

Question on definition of time-orientability and future-directed curves

I can't properly understand the definitions for a future directed tangent vector. Now I know the following definitions: A spacetime $(M,G)$ is called time-orientable, if there exists a vector field $...
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2answers
61 views

How to shrink a circle to a point that is not its center

Let's say we have a circle $C$ with a fixed radius $R$ on $\mathbb{R}^2$. An obvious parametrization to the circle would be: $$C(t)=(R\cos(t),R\sin(t)) \\ t\in[0,2\pi]$$ I'm interested in shrinking ...
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87 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 ...
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Equation of Peano Curve

In 'Space-Filling Curves' by Hans Sagan is presented equation defining the Peano curve. Peano, defined a map $f_p$ from $I$ to $Q$ in terms of the operator $kt_j = 2-t_j(t_j = 0, 1, 2)$ as ...
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3answers
34 views

Using one integer point to find other integer points on a hyperbola

Is it possible to use an integer point on a hyperbola of the form $x^2-y^2 = a$ to find other integer points on the hyperbola? For example if we have the parabola $x^2-y^2 = 221$ and we know that $(\...
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What is the mathematical relationship between the Hype cycle and Adoption Curve against Normal Distribution?

What is the mathematical relationship between: 1) The Hype cycle against Normal Distribution? 2) The Adoption Curve against Normal Distribution? This is not a homework assignment nor for academic ...
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1answer
20 views

Path for fastest end velocity while accounting for friction

How would one calculate the path for the fastest velocity for a rolling object while accounting for frictions? Because ideally in the theoretical world, the path would not matter as long as there was ...
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2answers
34 views

Can a curve be defined by the function of the angle between the tangent line and an axis?

The answer to this question is something I should know...but without regular practice working mathematics the answer is falling into some shadow of my understanding. I have a curve which lies ...
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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" ...
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2answers
27 views

Surface of Revolution (parametrization)

Question For a curve $\gamma(t)=(x(t),y(t),z(t))$ what would be the parametrization of the surface that we get by the revolution of $\gamma$ in the zz' axis? What about a random line as an axis? ...
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35 views

How to use Hilbert Curve Formula?

What is the Hilbert curve's equation?! In the question linked, there is a formula for Hilbert Curve. It involves $e_{kj}$ and $d_j$ terms. It says that $e_{kj}$ denotes number of k's preceding $...
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2answers
36 views

Max Velocity on a curved ramp and ideal ramp for highest Velocity

Is it possible to find the max exiting velocity of an object roll down a curved ramp? Since the brachistochrone curve has the path of the fastest time, does this mean it also has the highest exiting ...
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46 views

Shortest path inside the rectangle but not in the obstacle between two points

Given a convex closed curve, a rectangle and two points $A$ and $B$ inside the rectangle but not in the closed curve, I'd like to calculate the shortest path between $A$ and $B$ such that the whole ...
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1answer
39 views

Generate a S-curve between two points

I am working to generate a path for an autonomous vehicle. I have to generate a S-curve between two points A(11 20.88) and B(0.80 27.5). The vehicle orientation at point A being 0 degrees and at end ...
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Calculating an elliptical arc/curve from imperfect data points?

I am trying to do figure out how I can apply math to do some construction work. I have to calculate a an ellipse curve of a roof in order to cut and fit a piece of wood along the curve of it. Problem ...
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3answers
32 views

Find the parametric equation of a circumference in a non traditional reference frame

I know that the parametric equation of a circumference center in (0,0) is: $y=\sin(\alpha)*r, x = \cos(\alpha)*r$. I try to write the equation of the same circumference with respect to the ...
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1answer
25 views

Understanding of the definition of a Riemannian metric

I have have an exercise where it says: Let $M=\{x \in \mathbb{R}^2 \vert x_2>0\}$ with metric $g=\dfrac{1}{x_2^2}((dx_1)^2+(dx_2)^2)$. Now I have trouble understanding how this metric works, let ...
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1answer
49 views

Does an immersed curve have to be “curve-like” somewhere?

Let $C$ be a smooth immersed curve, i.e. an image of a smooth immersion $\varphi\colon \mathbb{R} \to \mathbb{R}^n$. We will say that $C$ is curve-like at a point $p \in C$ if, for every two smooth ...
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2answers
56 views

Reparametrised curve is geodesic

I do not know how to continue with my exercise: I have a pseudo-Riemannian manifold $(M,g)$, $I,J \subset \mathbb{R}$ open intervals and $\gamma:I \rightarrow M$ a smooth curve, $\gamma' \neq 0$. ...
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Definition of curves with two variable

I have a question regarding a diagram that I have made. It reflects measurement results. For different duty cycles, a gain factor was measured. The time reflects the turn-on time of transistors. I ...
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33 views

derivative of composition curve

a quick question: Let $(M,g), (N,h) $ be pseudo-Riemannian manifolds, $\gamma:I \rightarrow M$ a curve. $\gamma^{'}(t_0):= d \gamma \dfrac{\partial}{\partial t} |_{t_0}$ Let $F:M \rightarrow M$ be ...
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1answer
36 views

Understanding of differential of a curve as a vector field

I have trouble understanding the definition for a geodesic curve: Let $\gamma:I \rightarrow M$ be a curve, $M$ pseudo.Riemannian manifold. Then $\gamma$ is geodesic, if $\nabla_{\gamma^{'}} \gamma^{'...
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88 views

Why (possibly) are all of these integrals either $\frac \pi2$ or $\pi$?

I have encountered with a rather interesting vector field while studying Green's Theorem: $$\vec{F}(x,y)=-\frac{y}{x^2+y^2}\hat{i}+\frac{x}{x^2+y^2}\hat{j}$$ This field turned out to be quite special ...
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2answers
70 views

How to solve a differential equation when the right-hand side of the equation is a non-parametric curve?

I just asked another question (see here: How to calculate the shape of a curve given y coordinates and slope?) and was advised by the user who answered my question to ask a new question. I would ...
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1answer
37 views

How to calculate the shape of a curve given y coordinates and slope?

I apologise in advance if my description of the problem does not use the correct terminology but I'm still learning! Let me know if something is ambiguous or not clear and I'll try to rephrase it. ...
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33 views

How to draw astroid given by Parametric equation $a(t)=(\cos^3(t),\sin^3(t) )$

I wanted to draw Asteroid with parametric equation $a(t)=(\cos^3(t),\sin^3(t) )$ I know that $x^{2/3}+y^{2/3}=1$ is equation . Also $x^2+y^2=1$ is equation of circle. By using online app I had ...
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48 views

Determine value for which a curve is geodesic

Let $(N,g_N)$ be Riemannian manifold, $I \subset \mathbb{R}$ open with coordinate $r$ and $M:= I \times N$ with metric $g=dr^2+f^2 g_N$. Now let $\gamma:J \rightarrow N$ be a geodesic curve. I need ...
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1answer
22 views

Using Green's theorem to compute integral on curve

Prove: $$\ \int_C (\sin x - y^2)dx +(x-y \tan^{-1}(y^2))dy = 2.4 $$ where $\ C $ is the curve from $\ (1,2) $ to $\ (-1,2) $ on $\ y = x^2 + 1 $ Using green's theorem $$\ \int \int_D (Q_x - P_y)dx ...
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1answer
27 views

Find the points of the trajectory in which the velocity is 9

I have a parametric curve given by $M:[0,2]\rightarrow\mathbb{R}; x(t) = 3t^2, y(t) = t^3,z(t)=6t$. I need to find the points of its trajectory in which the velocity is 9. I also need to represent ...
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2answers
39 views

Weird Notation in Differential Geometry (Variational Vector Field)

I can't seem to understand what my professor wants to say with his notation. So I hope that some of you can help me. We're looking at some variational curve $\gamma_t : I \to \mathbb{R}^2, \ t \in (-\...
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1answer
27 views

Regular homotopy between a circle and an oval

How do I build a regular homotopy between an oval $$\frac{x^4}{a^4}+\frac{y^4}{b^4}=1$$ and a circle $$\ (x-x_0)^2+(y-y_0)^2=R^2$$? I know I need to find parametrizations for both curves and there ...
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1answer
23 views

How do you render the union of a simple closed curve and its interior?

In this question there is confusion about the answer because it appears (to me, anyway) that the origin was supposed to be rendered as not on or inside a simple closed curve; but the question is ...
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2answers
52 views

Extremum of ln(x)/x = C (analytical)

I want to find the maximum value of y of the following equation: $A\cdot \sqrt{x^2 +y^2} = e^{\frac{\sqrt{x^2 + y^2}-x}{B}}~(1)$ I tried to use polar coordinates with $x = r \cos(\phi)$ and $y=r \...