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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What differential equations describe absolute squares of Hankel-like functions of real numbers?

The linearly independent solutions of Bessel equation can be combined into two complex functions, which would represent running radial cylindrical waves. These are the Hankel functions $H_\alpha^{(1)}$...
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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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Envelope of a family of data plots

I have a transfer function $$M(s;\theta)=\frac{K(\theta) s (s+z(\theta))}{(s^2+2\xi\omega_n s+w_n^2)^2}$$ where $K$ and $z$ are constants that depend on some parameters $\theta$. I plotted the step ...
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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 $a_t, b_t$ ...