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 $F(x,y,a)=0$, 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 $C_1$, 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 $N$, 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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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 ...
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What is the equation of the evolute of a curve given in implicit equation?
I am trying to find the evolute of a curve given in implicit equation. I tried to find it using the definition, as the envelope of the family of normals, however I didn't come to a conclusion.
Then I ...
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Understanding the derivation of the equation for envelopes.
Given a family of curves, an envelope is defined as a curve that it is tangent to every curve in the family of curves at some point on it.
To derive the equation for it, the first step is to ...
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Solving Envelope Equation [closed]
I have a family of curves: $$F(x, y, t) = \sin(2t)x-\cos(2t)y-sin(t)$$
I'm trying to solve the equation for the envelope, that is this systems of equations:
$$ \text{} \left\{ \begin{align} F(x, y, \...
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Halving planes of a tetrahedron
This nice article
Megan Martin, Cornelia A. Van Cott & Qiyu Zhang (2024) The Beauty of Halving it All, Math Horizons, 31:2, 14-17, DOI: 10.1080/10724117.2023.2249357.
shows that the envelope of ...
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Prove the envelope of lines is a hyperbola
$A$ and $C$ are fixed points on fixed circles $O_1$ and $O_2$.
Point $B$ is moving on circle $O_1$.
Point $D$ is the intersection of the circle through $A,B,C$ with circle $O_2$.
Point $F,G$ are the ...
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Conceptually Understanding the Mathematical Definition for an Envelope of Family of Curves
Assuming we have a one-parameter, two-dimensional, family of curves, given by $f(x, y, p) = 0$, there are two requirements for the envelope (see https://en.wikipedia.org/wiki/Envelope_(mathematics)#) ...
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Understanding proof for cardioid into coffee cup
I've been trying to learn more about envelope of lines and came over this proof for the cardioid inside a cup. I have arrived at the two parametric equations which I've written in matrix form.
$$\...
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An Ambiguous Definition of Envelopes with reference to Differential Equations and Singular Solutions.
Recently, I was, going through a definition of envelope. I know that the definitions can be written and the exposition of it, might vary, but the following definition, which I am hereby mentioning, ...
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The area between parabolic lines inside a square
The following square has edges of size $1$ and I'm trying to find the area of the blue region trapped between the parabolic curves created by the straight lines (number of lines is technically ...
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Envelope of a Sliding circle [closed]
"Find the envelope of the circle whose diameter is a line of constant length which slides between two fixed straight lines at right angles."
I could not figure out equation exactly, may be I ...
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Is it just me or does it look like this is graphing shapes similar to $r^t = |x|^t+|y|^t$?
I wrote a program to cast some points out to a shape defined by lines and then connect any point to a certain amount of points down the shape with a line, and it looks like this is producing shapes on ...
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Two definitions of Evolute of a curve?
Recall, when the tangents to a curve $\gamma$ are normal to another curve, the second curve is called an involute of $\gamma.$ In literature, there are two seemingly different dual notions for ...
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How to find the enveloping curve of this family of polynomials?
I was studying the Rule 90 cellular automaton and came across a family of polynomials defined by
\begin{equation}
D_n(x)=\begin{cases}
\displaystyle\sum_{k=0}^{m}(-1)^{m+k}\binom{m+k}{m-k}x^{2k}\ , &...
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Intuition behind enveloping functions
The following are examples of enveloping functions.
$$f(x) = x\cdot\sin(x)$$
$$f(x) = x^2 \cdot \sin(x)$$
$$f(x) = \frac{1}{x} \cdot \sin(x)$$
It seems that given a function $f(x) = g(x)\cdot \sin(...
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The envelope for the extremals of $\cos((n+1) \arccos x)-\cos((n-1) \arccos x)$ forms an ellipse.
The Chebyshev polynomial of the first kind is defined on $[-1, 1]$ by
$$T_n(x) = \cos(n \arccos x).$$
Prove that the envelope for the extremals of $T_{n+1}(x)-T_{n-1}(x)$ forms an ellipse.
The ...
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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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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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