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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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,ドル 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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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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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,ドル such that $\tan\angle BAC = t$. Then the envelope of the ...
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Envelope of the function $f(x)=\frac{\sin(x)}{\sin(x/N)}$
Given a positive integer $N,ドル let us consider the function
$$
f(x)=
\frac{\sin(x)}{\sin(x/N)}
$$
in the interval 0ドル<x<2\pi N$. What is the envelope function of $f$ ?
My attempt I know that 1ドル/x$...
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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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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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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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In a triangle ABC : 2 externaly tangent circles, also tangent to BC with centers on line segments AB and AC : envelope of their lines of centers?
The figure here gives an illustration of the configuration described in the title in 4 cases ; consider especialy the fourth one, materialized by red circles, red center points, and a red line segment ...