Questions tagged [curves]

For questions about or involving curves.

1,840 questions
25 views

While playing around with the normal distribution, or more precisely, the probability density function: $$\displaystyle f(x | \mu,\sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{(x - \mu)^2}{2 ... 1answer 28 views Does there exist a constant curvature curve on paraboloid surface? I'm learning differential geometry,and need help with the following question: Does there exist a constant curvature curve on paraboloid surface  z = x^2 + y^2 different from plane curve, i.e z ... 0answers 24 views How important is it to have a deep understanding of the graphs of common differentiable functions? [on hold] Recently I have been trying to refresh my understanding of calculus in an in-depth way and I have been going through a calculus text book, doing every other problem. I am tempted to skip over the ... 0answers 31 views Find the orthogonal to a clothoid spiral from an offset coordinate At the moment, I'm implementing an algorithm which defines a spiral using the method outlined in the following paper: https://www.ams.org/journals/mcom/1992-59-199/S0025-5718-1992-1134736-8/S0025-... 3answers 39 views Proving a line is tangent to a curve 1) Imagine a point T(g, f) on a Cartesian plane, where g, f ≠ 0. Deduce, in terms of g and f, the equation of the line, in which its x-intercept and y-intercept have midpoint T, which means that when ... 0answers 24 views Penrose Singularity Theorem Proof I have problems understanding the first part of the proof for the Penrose singularity theorem in the book "Leonor Godinho José Natário An Introduction to Riemannian Geometry": I know that \langle\... 1answer 24 views condition of curvature continuity - Is direction of the bending important? What is the condition of curvature continuity? Do we only consider the value of the curvature \kappa, or do we have to also take account of the direction of the curvature vector \mathbf{k} as ... 2answers 40 views How to calculate the derivative of this curve? How do I go about calculating the derivative of this curve from \mathbb{R} to \mathbb{R}^3?$$γ(t) = (t\cdot\cos(2t), t\cdot\sin(2t), t)$$I have tried simply taking the derivative of the three ... 2answers 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 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 ... 0answers 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? ... 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 ... 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 ... 0answers 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 ... 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 ... 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 ) ... 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 ... 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 ... 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 ... 0answers 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 ... 0answers 21 views 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 ... 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 (\... 0answers 8 views 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 ... 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 ... 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 ... 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" ... 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? ... 0answers 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 ... 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 ... 0answers 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 ... 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 ... 0answers 17 views 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 ... 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 ... 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 ... 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 ... 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. ... 0answers 18 views 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 ... 0answers 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 ... 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^{'... 0answers 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}$$This field turned out to be quite special ... 2answers 70 views How to solve a differential equation when the right-hand side of the equation is a non-parametric curve? I just asked another question (see here: How to calculate the shape of a curve given y coordinates and slope?) and was advised by the user who answered my question to ask a new question. I would ... 1answer 37 views How to calculate the shape of a curve given y coordinates and slope? I apologise in advance if my description of the problem does not use the correct terminology but I'm still learning! Let me know if something is ambiguous or not clear and I'll try to rephrase it. ... 0answers 33 views How to draw astroid given by Parametric equation a(t)=(\cos^3(t),\sin^3(t) ) I wanted to draw Asteroid with parametric equation a(t)=(\cos^3(t),\sin^3(t) ) I know that x^{2/3}+y^{2/3}=1 is equation . Also x^2+y^2=1 is equation of circle. By using online app I had ... 0answers 48 views Determine value for which a curve is geodesic Let (N,g_N) be Riemannian manifold, I \subset \mathbb{R} open with coordinate r and M:= I \times N with metric g=dr^2+f^2 g_N. Now let \gamma:J \rightarrow N be a geodesic curve. I need ... 1answer 22 views Using Green's theorem to compute integral on curve Prove:$$\ \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 ...
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 ...