Envelope - Horakusen (English spelling) envelope

Japanese: 包絡線 - ほうらくせん(英語表記)envelope
Envelope - Horakusen (English spelling) envelope

When there is a family of curves {C t } on a plane with one parameter t, the curve obtained by connecting the limit points of intersection of two curves C t and C t+ Δ t with similar parameter values ​​as Δ t → 0 is called the envelope of this family of curves ( Figures A and B ).

The equation F(x,y,t)=0 represents a curve Ct when the value of t is fixed. Therefore, by changing the value of t, a family of curves { Ct } is obtained. Now, if there is a curve L:x=x(t),y=y(t) with t expressed as a parameter, and for each value of t, the point (x(t),y(t)) is the limit position of the intersection of Ct and Ct + Δt as Δt →0, then L is the envelope of the family of curves { Ct }. In many cases, this can be expressed as saying that for each value of t, Ct and L have a common tangent at the intersection. The equation for the envelope is
F(x,y,t)=0, Ft (x,y,t)=0
This can be obtained by eliminating t from the equation, or by expressing it in the form x=f(t), y=g(t).

The envelope of the normals at each point of a curve C is called the evolute of the curve. The evolute is also the locus of the centers of curvature at each point of C. When C' is the evolute of C, C is called the involute of C' ( Figure C ).

Figure D shows the meaning of an involute. As shown in the figure, if you wrap a string tightly around the curve C' and pull one end of it taut while moving it away from the curve, the figure C drawn by the end point will be an involute.

[Osamu Takenouchi]

Envelope of a family of curves (1) [Figure A]
©Shogakukan ">

Envelope of a family of curves (1) [Figure A]

Envelope of a family of curves (2) [Figure B]
©Shogakukan ">

Envelope of a family of curves (2) [Figure B]

Evolute and involute (Fig. C)
©Shogakukan ">

Evolute and involute (Fig. C)

Meaning of involute (Fig. D)
©Shogakukan ">

Meaning of involute (Fig. D)


Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

平面上に一つのパラメーターtをもつ曲線群{Ct}があるとき、パラメーターの値の近い二つの曲線Ct,Ct+Δtの交点のΔt→0の極限の位置を結んで得られる曲線を、この曲線群の包絡線という(図A図B)。

 方程式F(x,y,t)=0は、tの値を一つ固定すれば、一つの曲線Ctを表す。したがって、tの値を変えると、曲線群{Ct}が得られる。いま、tをパラメーターとして表した一つの曲線L:x=x(t),y=y(t)があって、tのおのおのの値に対して、点(x(t),y(t))が、Ct,Ct+Δtの交点のΔt→0とした極限の位置になっていれば、Lが曲線群{Ct}の包絡線である。多くの場合、これは、tのおのおのの値に対して、CtとLは、その交点において共通の接線をもつ、と表現してもよい。包絡線の式は、
  F(x,y,t)=0, Ft(x,y,t)=0
からtを消去するか、あるいは、これをx=f(t),y=g(t)の形に表して得られる。

 曲線Cの各点における法線の包絡線を、この曲線の縮閉線という。縮閉線はCの各点における曲率中心の軌跡でもある。C′がCの縮閉線であるとき、CをC′の伸開線という(図C)。

 図Dは伸開線の意味を示す。図にみられるように、糸を曲線C′に沿ってきっちりと巻き付けておき、一端をもってぴんと張りながら曲線から離していけば、その端点の描く図形Cが伸開線となる。

[竹之内脩]

曲線群の包絡線(1)〔図A〕
©Shogakukan">

曲線群の包絡線(1)〔図A〕

曲線群の包絡線(2)〔図B〕
©Shogakukan">

曲線群の包絡線(2)〔図B〕

縮閉線と伸開線〔図C〕
©Shogakukan">

縮閉線と伸開線〔図C〕

伸開線の意味〔図D〕
©Shogakukan">

伸開線の意味〔図D〕


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