Envelope curve

Parabola (y-x-13)2 = 52x [¹] which is straight line group y = (t-13)(x-t)/t and the envelope curve

The envelope curve (the envelope that one does not take its ease) is a curve contacting with curve to share a tangent line with a given curve group namely all curves that were given (the generally infinite unit). In the imminent place, the envelope curve of the electric wave signal of the amplitude modulation used for AM radiobroadcast is a voice signal.

I do the envelope curve as follows and am demanded.

Curve group {F_t(x₁, on the n dimension Euclidean space Rⁿ added an additional character to in parameter t ... The envelope curve for ,x_n) = 0}_{t ∈ R} is a simultaneous equation

${\displaystyle {\begin{cases}F_{t}(x_{1},\dots ,x_{n})=0\\{\cfrac {\partial }{\partial t}}F_{t}(x_{1},\dots ,x_{n})=0\end{cases}}}$

Curve φ(x₁, which removes から t, and is provided ... It is equal to ,x_n) = 0.

Table of contents

Envelope example

Family {L_t}_{t ∈ R} Lt of the straight line added an additional character to in parameter t of the actual number: I try to really demand an envelope curve of y = sin(t)x + cos(t) by an upper method. At first if I transform L_t in form of sin(t)x - y + cos(t) = 0 and do partial differentiation about t,

${\displaystyle {\frac {\partial }{\partial t}}(\sin(t)x-y+\cos(t))=\cos(t)x-\sin(t)}$ 

Simultaneous equation to demand となるから, an envelope curve

${\displaystyle {\begin{cases}\sin(t)x-y+\cos(t)=0,\\\cos(t)x-\sin(t)=0\end{cases}}}$ 

I get を. Transform an expression; and x2 - y2 +1 = 0. This is an envelope curve of straight line group {L_t}_{t ∈ R}. In this case the envelope curve understands that it is a hyperbola.

Allied item

• Curve
• Parameter
• I detect an envelope curve

Footnote

1. ^ ハイラー & ヴァンナー 2012, II.3 envelope curve and curvature.

References

• ハイラー, E., ヴァンナー, G., G. In ヴァンナー "analysis course of study," it is Yukihiro Kanie reason, Maruzen, 2012. ISBN 978-4-621-06203-6。

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This article is taken from the Japanese Wikipedia Envelope curve

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