Questions tagged [envelope]
In geometry, an envelope of a continuous family of differentiable curves is a curve that touches each member of that family at some point, and these points of tangency together form the whole envelope. Therefore it is the limiting curve of the intersection of contiguous members of the initial family.
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Upper envelope of a family of binomial–tail functions with linear penalties
Let $n,m$ be positive integers with $1 \le m < n$.
For each pair of integers $z,r$ with $1 \le z \le n$ and $0 \le r \le z$,
define the function
$$
u_{z,r}(x)
:= \Pr[\operatorname{Bin}(n-z,\,x)\...
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Can any superellipse $|x|^p+|y|^p=1$ arise as the envelope of a one-parameter family of curves?
I am interested in representing $C_p$ as envelopes of curves $F(x,y,a)=0$, where\begin{align}
C_p=\{(x,y)\in\mathbb{R}^2:|x|^p+|y|^p=1\}.
\end{align}
“Interesting” means it is not something like $F(x,...
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Normals at three parabolic points P,Q,R on $y^2=4ax$ meet on a point on the line $y=k,$ then prove that sides of $\Delta$PQR touch $x^2=2ky$
Normal at a point on the parabola $y^2=4ax$ is given as
$$y=mx-am^3-2am,$$ if normals at three points meet at a point $(x_1,k)$ on the line $y=k$
then we have: $$k=mx_1-am^3-2am \tag{1}.$$
This can ...
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Is $z=0$ the envelope of two-parameter family of surfaces $z = ax + by - 2a - 3b$?
In the book "Ordinary and Partial Differential Equations" by Dr. M. D. Raisinghania, you can see it is written that $z=0$ is the envelope of $z=ax+by-2a-3b$. But the surfaces do not touch $...
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Help me prove convexity for this envelope
Let $Q \subset \mathbb{E} \times \mathbb{R}$ be a set, where $\mathbb{E}$ represents the Euclidean space. I have the following:
We know that $E_Q(x)=\inf\{r:(x,r) \in Q\}$ by definition. Furthermore, $...
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The limit of envelope equation passing from the ellipse/hyperbola to parabola
Proposition. Point $A(x_0, y_0)$ is a fixed point on the conic $C_1:ax^2 + by^2 = 1$ ($ab \neq 0$). Points $B$ and $C$ are moving on $C_1$, such that $\tan\angle BAC = t$. Then the envelope of the ...
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A chord of a conic which subtends a constant angle at a given point on the curve envelopes a conic having double contact with the given conic.
I'm trying to generalize this proof ($\alpha=90^\circ$ then $C_2$ is a point) to prove the following
A chord $BC$ of a conic $C_1$ which subtends a constant angle $\angle BAC=\alpha$ at a given point $...
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Envelope of the function $f(x)=\frac{\sin(x)}{\sin(x/N)}$
Given a positive integer $N$, let us consider the function
$$
f(x)=
\frac{\sin(x)}{\sin(x/N)}
$$
in the interval $0<x<2\pi N$. What is the envelope function of $f$ ?
My attempt I know that $1/x$...
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What is the condition for the envelope of an $n$-parameter family of curves?
Suppose we have an $n$-parameter family of curves specified implicitly with $G\colon\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$ by the $n$ equations $G(\mathbf{x},t,\mathbf{x}_0) =...
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Finding the envelope of a family of curves: How to resolve assumptions made during the extraction of the parameter?
I'm asked to find the envelope of the family of the curves represented by $$\begin{cases}x = v_0t\cos{\alpha}\\y=-\dfrac12gt^2+v_0t\sin{\alpha}\end{cases}$$
where $g, v_0 > 0$. In this problem, $t$ ...
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Envelope of lines giving a conic section
This problem originates from another question, which was closed for lack of context. I found a solution but some details are still missing, as explained below.
Given an angle and a point $P$ inside ...
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Constant quantities on the curve of contact between envelope and particular solution of a first order PDE
Given the partial differential equation
$$F(x_1,...,x_n,u,p_1,...,p_n)=0$$
and the complete integral
$$u=\phi(x_1,...,x_n,a_1,...a_n)$$
depending on the $n$ parameters $a_1,...,a_n$ we can get an ...
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Envelope of the set of tangent lines of an immersion
Suppose we have a smooth immersion $f:\mathbb{R}^d\to\mathbb{R}^n$, for $n>d$.
Consider the set of tangent planes of the immersion
$$
\mathcal{P}_f:=\{f(x)+\mathrm{span}(\partial_{x_1}f(x),\ldots,\...
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Help needed in understanding the procedure to calculate the envelope of a two parameter family of surfaces.
I was reading the book Linear Partial Differential Equations for Scientists and Engineers (Fourth Edition) written by Tyn Myint-U and Lokenath Debnath.
In Section 2.3 titled as "Construction of ...
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Area swept by the circumference of an ellipse as it slides such that it is always tangent to the $x$ axis at the origin
You're given the ellipse $\frac{x^2}{a^2} + \frac{(y - b)^2}{b^2} = 1,$
for known $a$ and $b$. Now you slide the ellipse and rotate it such that it remains tangent to the $x$ axis at the origin all ...
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