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.

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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 ...
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Finding a Lower Envelope of a Compact Convex Set in $R^2$

Consider the compact convex set $S$ in $\mathbb{R}^2$ with three extreme points $(0, 0)$, $(1, 1)$, and $(0, 1)$, i.e., $S$ is a triangle. To find the lower convex envelope $f(t)$ of $S$, where $t\in[...
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Envelopes for the Gauss function with auxiliary periodic signal for the argument

We have a Gaussian function with an auxiliary periodic signal for the argument: $g(x) = exp(-\frac{(x+a \cdot sin(\omega \cdot x)-b)^2}{2 \cdot c^2})$ where $x$ - argument, $b$ - shift for x-axis, $c$ ...
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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 ...
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What is the 2D equivalent of of a channel surface?

I was reading about channel surfaces on Wikipedia (also know as canal and pipe surfaces) and it reminded me of something I had independently investigated some time ago which is the 2D equivalent. ...
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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 ...
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Envelope of two-parameter family of curves

Consider a two-parameter family of curves $F(x,y,u,v)=0$, where $u,v$ are two parameters. We may have two ways to calculate the envelope of the family of curves. We can solve $\begin{cases} F(x,y,u,v)...
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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 ...
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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 ...
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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 ...
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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(...
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Proof that Singular Solution of Clairaut's Equation is the envelope of the family of General Solutions

So I'm trying to do this proof. The form of Clairauts equation is y(x) = xy' + f(y') You differentiate once to get ...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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}...
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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 $...
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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 $...
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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 $z=-2$ $z=2$ $z=\sqrt{2+\sqrt{2}}$ $...
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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,...
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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 ...
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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 ...
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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 ...
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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 $a_t, b_t$ ...