Questions tagged [envelope]

In geometry, an Envelope of a family of curves is a curve that touches each member of that family at same family. Therefore it is the limiting curve of intersection of contiguous members the initial family.

Filter by
Sorted by
Tagged with
0
votes
0answers
23 views

Is there a function that is the envelope of the sum of ceilings of reciprocal functions

TL;DR: Given a sum of ceilings of reciprocal functions $$y_1 = T = \sum^{n-1}_i \Big\lceil \frac{p_i}{k} \Big\rceil$$ is there a corresponding form for a function that envelopes the $T$ on the left? ...
1
vote
0answers
32 views

Verifying solution of the envelop theorem on an example

im struggling with understanding if my solution is correct. Given: $F(x,y,z)=25x+125y+5z$ on $xyz=1000$ Where the criticial points are: $x=10, y=2$ and $z=50$ where the target value is 750. Adjusting ...
0
votes
0answers
22 views

Find the envelope of a two-parameter family of surfaces

I need to find the envelope of the family of surfaces described by: x/a + y/b + z/(10-a-b) = 1 Background information: At the moment I'm doing a self study course in vectorcalculus and differential ...
0
votes
0answers
30 views

Plotting tight bounds for simple Wiener Brownian motion - problems with classic definitions

I am trying to plot the standard bounds of simple Brownian motion (implemented as a Wiener process), but I have found some difficulties when drawing the typical equations: When trying to plot the ...
0
votes
1answer
48 views

Find equation of envelope of a family of Arrhenius-like exponential curves

I have an equation for an Arrhenius-like exponential curve: $y = t\exp(-1000t/x)$ Where t is some scaling parameter. If I allow $t$ to vary from 1 to 20 in steps of 0.1, I obtain the family of curves ...
3
votes
2answers
176 views

Why is enveloping algebra called enveloping algebra?

What does the enveloping algebra of $\mathfrak{g}$ have to do with envelopes? If $\mathfrak{g}$ is a Lie algebra, we take tensor algebra on $\mathfrak{g}$ and make quotient through ideal of T, so we ...
2
votes
1answer
106 views

How do I find the equation of an envelope?

I read that you must solve the two equations $$g(x,y,c)=0\\\frac{\partial g}{\partial c}=0$$ for $x$ and $y$ as a function of $c$, but how exactly do you go about doing this? The specific example I am ...
0
votes
2answers
57 views

What do they mean by lower envelope of parabolas?

I'm studying an algorithm on distance transforms and there's a part which confuses me. Let $G = \{0, . . . , n − 1\}$ be a one dimensional grid, and $f :G →R$ an arbitrary function on the grid. The ...
0
votes
0answers
31 views

Continuity of pointwise infimum (of a special function); Moreau-Yosida regularisation

I'm seeking how to prove that Moreau-Yosida regularisation provides a continuous function: Given a convex function $f:X \rightarrow \mathbb{R}$ where $X$ is a compact subspace of an Euclidean space (...
0
votes
0answers
22 views

How do we invert the process of finding envelopes?

The curve “stitched” by the “stitches”, the lines, $$f(x,y,t)=yt-kx-ht^2=0$$ Can be shown to have the envelope $$y=\sqrt{4hkx}$$ And all we do is find the partial derivative of $f$ w.r.t. $t$ and set ...
0
votes
1answer
80 views

What is upper convex envelope of a function

Only the definition of lower convex envelope I can find in Wiki. Following the definition of lower convex envelope, I guess the definition of upper convex envelope is $$ \hat f(x) = \sup\{ g(x): g \...
3
votes
0answers
96 views

Envelope of a real function consisting of a complex function and its conjugate

For a real function $f(x)=A(x)e^{ix}+\overline{A}(x)e^{-ix}$, where $\overline{A}$ represents complex conjugate of $A$. Note that $A(x)$ itself is a complex function $A(x)=A_r(x)+i A_i(x)$. It seems ...
0
votes
1answer
70 views

Lack of clarity in complete integrals and envelopes in Partial Differential Equations?

The Complete integrals is used in the analysis of non linear first order PDE. I came across it in the books Partial Differential Equations by Evans Lawrence. $$F(Du,u,x)=0$$ Where Du is the gradient ...
1
vote
1answer
37 views

Description of curve arising from rotation of a cube

Take a unit cube, and place it so that one of its body diagonals lies along the $z$-axis. For symmetry, assume that the vertices of the cube are at $(0,0,\pm\sqrt{3}/2)$. Then rotate it about the z-...
0
votes
0answers
15 views

Eliminate a solution for envelop curve

In the Cartesian coordinate system, we have a family of circles with a radius 1 and these circles center at the circle x^2+y^2=4 Mathematical, if we solve the ...
2
votes
2answers
252 views

Envelope of family of ellipses tangent to x-axis and y-axis

I wanted to find the ellipse of the largest area that can pass through a hallway that makes a 90 degrees turn.The vertex of this hallway is at (c , d). In order to do that, I tried to find the ...
1
vote
1answer
117 views

How to calculate explicitly this convex envelope?

Define a function $F:[0, \infty) \to [0,\infty)$ by $$F(s) := \begin{cases} \sqrt 2|\sqrt{s}-1|, & \text{ if }\, s \ge \frac{1}{4} \\ \sqrt{1-2s}, & \text{ if }\, s \le \frac{1}{4} \end{...
1
vote
0answers
62 views

First Order Condition with 1 decision variable and 2D state space

TL;DR: I'm trying to find the first-order condition (FOC) for an optimization problem with two state variables and one control variable. I don't want the value function $V$ to appear in the FOC but ...
0
votes
1answer
36 views

Enveloping and Multiplication

In the context of the Fourier Transform, when we multiply the rotating vector tracing out a circle, $e^{-2\pi i f t}$, by an input function $g(t)$ in the complex plane, the output graph is wound ...
1
vote
1answer
125 views

Relation between general solutions and singular solution of Clairaut’s equation.

So I'm trying to do this proof, The form of Clairauts equation is $$y(x) = xy' + f(y')$$ You differentiate once to get $$ y' = y' + xy'' + f'(y')y''$$ You rearrange and get two solutions The general ...
1
vote
0answers
18 views

Can a one parameter family of surfaces be only once-differentiable?

Reviewing this and that resource, I see that a one parameter family of curves has equation $$f(x,y,z,t)=0 \tag{1}$$ where $t$ is the parameter. And I see that the equation of the envelope of the ...
5
votes
2answers
423 views

circle envelope tangent in another circle

As the picture shows, One big circle ,$(0,0)$ ,radius=R, there is a small circle in it, $(m,0)$ ,radius=r . G is on the big circle. From G ,we can do two tangent lines about the small circle. Get ...
-5
votes
3answers
380 views

How do you find the area of an egg shape formed from the union of circles whose diameters are horizontal chords of the unit circle? [closed]

This egg is the union of the red circles, whose diameters are horizontal chords of (the upper half of) the unit circle. How do you find its area? Some background information on this egg is shown ...
0
votes
1answer
219 views

Concave envelope: equivalent definitions

Let $f: X\to \mathbb{R}$ be bounded. Consider the following definitions: Definition 1: $f^{*}(x)=\inf\{h(x): h\in \mathbb{R}^{X} \;\;\text{affine and}\;\; h\ge f\}$. Definition 2: $f^{**}(x)=\inf\{h(...
3
votes
1answer
205 views

Proof that Singular Solution of Clairaut's Equation is the envelope of the family of General Solutions [duplicate]

I want to show that the singular solution is the envelope for the general solutions. Proof Outline Both solutions pass from the same point $(a,b)$ Both solutions have the same gradient at that ...
2
votes
1answer
35 views

Find $\bigcup_{r\in R}\ \{(x,y): (x-r)^2 + (y+2r)^2 < r^2+1\}$

I have to obtain an union: $\bigcup_{r\in R}\ \{(x,y): (x-r)^2 + (y+2r)^2 < r^2+1\}$ I know this is a series of circles that is limited by two hyperbolic functions that are symmetrical in ...
1
vote
1answer
113 views

Calculus of variations and envelope theorem application

My Euler-Lagrange equation is a non-linear differential equation and I am convinced that it cannot be expressed in a closed-form solution. My question is: Is there a theorem in the calculus of ...
4
votes
0answers
238 views

Find the envelope of ellipses defined by a difficult function

The problem sounds easy, that is, if I would have had an easy function $\phi(k_p, k_i, \omega)$ that defines these ellipses. This function, $\phi$ depends on which transfer system $G(s)$ I am trying ...
1
vote
1answer
72 views

Running time of computing upper and lower envelope

What is the running time of computing the upper and lower envelope of a set of piecewise functions (curves)? I am looking for an O(n log n) algorithm for this problem, but couldn't find the paper.
1
vote
1answer
133 views

Envelopes In Mathematics

How can I make sure that when we eliminate the parameter from the curve \begin{align*} F(t,x,y) &= 0 \\ \frac{\partial F}{\partial t}(t,x,y) &= 0\,, \end{align*} the equation obtained is the ...
0
votes
1answer
71 views

How is this integral computed??

I'm reading this book about electrical properties of materials where the electron is introduced as a wave. Using the equation of a wave, they bring about the "envelope" of a wave. So here is how the ...
1
vote
1answer
90 views

Continuity of an upper envelope

I'm now approaching for the first time to the definition of upper envelope of a family of functions, and I just wonder to know some basic properties. Suppose $\Omega $ is an open subset of $\mathbb{R}...
3
votes
1answer
154 views

Exponential PDF Mystery

The envelope function of the family of exponential PDFs of the form $$f_\lambda(x)=\lambda e^{-\lambda x}$$ is $$g(x)=\frac{1}{ex}$$ for $x > 0, \lambda > 0$. The point of tangency between $...
1
vote
2answers
519 views

Are higher-order Bézier curves envelopes?

I only realized from this question and the answers to it that quadratic Bézier curves are the envelopes of the lines used to compute them iteratively. That is, if a quadratic Bézier curve for points $...
1
vote
0answers
247 views

Envelope of the family of solutions.

If a complete integral of the pde $x(p^2+q^2)=zp$, passes through the curve $x=0, z^2=4y$, then the envelope of this family passing through $x=1$ and $y=1$ has $z=-2$ $z=2$ $z=\sqrt{2+\sqrt{2}}$ $...
3
votes
0answers
54 views

Envelope of a family of curves given in complex form

Does anybody know how to compute the envelope of a family of curves given in complex form $$F:\mathbb {R} \times \mathbb{R} \rightarrow \mathbb{C}$$ $$(w,c) \mapsto F(w,c)$$ without decomposing $F(w,...
0
votes
1answer
18 views

Solving the second example problem from the Wikipedia page on evelopes

While reading the Wikipedia page on Envelopes there are some examples given. In the second example a jump is made from a linear equation to $F(x,y,t)=0$ form. For context the transformation is made ...
4
votes
0answers
472 views

Graph Envelope Constraint puzzles from The Witness game.

The computer game "The Witness" contains various puzzles based on a finite square grid graph arranged in the usual way. A path must be found from a given point on the edge to another. Each square can ...
3
votes
1answer
308 views

Tractrix & limits

I'm in calculus 1 and I'm having quite a bit of trouble with this problem. Any advice as to how to obtain a solution or anything would be much appreciated. Thanks! To determine the points on the ...
2
votes
1answer
129 views

Envelope of a family of lines in the plane

Let $A$ and $B$ be two bugs lying on two distinct points $a_0, b_0$ on a fixed circle. They start to walk along the circle in the same direction such that at time $t$ their coordinates $a_t, b_t$ ...