# 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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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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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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### 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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### 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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### 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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### 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 ...
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 $a_t, b_t$ ...