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''of a family of curves in the plane''
 
''of a family of curves in the plane''
  
The curve that at every point touches one of the curves of the family such that the points of contact along the envelope pass from one curve of the family to another. For example, the family of circles of the same radius with centres on a straight line has an envelope consisting of two parallel lines. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e0358401.png" /> is the parameter of the family, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e0358402.png" /> is the parameter along the envelope and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e0358403.png" /> the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e0358404.png" /> for one of the curves of the family touching the envelope at the point with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e0358405.png" />, then it is assumed that it is possible to choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e0358406.png" /> such that the function is not constant on any part of the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e0358407.png" />.
+
The curve that at every point touches one of the curves of the family such that the points of contact along the envelope pass from one curve of the family to another. For example, the family of circles of the same radius with centres on a straight line has an envelope consisting of two parallel lines. If $  C $
 +
is the parameter of the family, $  t $
 +
is the parameter along the envelope and $  C ( t) $
 +
the value of $  C $
 +
for one of the curves of the family touching the envelope at the point with parameter $  t $,  
 +
then it is assumed that it is possible to choose $  C ( t) $
 +
such that the function is not constant on any part of the range of $  t $.
 +
 
 +
For the family of curves given by  $  f ( x, y, C) = 0 $,
 +
where  $  f \in C  ^ {1} $
 +
and  $  | f _ {x} | + | f _ {y} | \neq 0 $,
 +
a necessary condition for the existence of an envelope is that  $  x ( t) $,
 +
$  y ( t) $,
 +
$  C ( t) $
 +
satisfy the condition
 +
 
 +
$$ \tag{1 }
 +
= 0,\ \
 +
f _ {C}  = 0.
 +
$$
  
For the family of curves given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e0358408.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e0358409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584010.png" />, a necessary condition for the existence of an envelope is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584013.png" /> satisfy the condition
+
The system (1) serves to determine the points of the envelope, but other singular points of the family may also satisfy (1). A sufficient condition for a point to belong to the envelope is that $  f \in C  ^ {2} $
 +
and satisfies, in addition to (1), the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{2 }
 +
f _ {CC}  \neq  0,\ \
  
The system (1) serves to determine the points of the envelope, but other singular points of the family may also satisfy (1). A sufficient condition for a point to belong to the envelope is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584015.png" /> and satisfies, in addition to (1), the condition
+
\frac{D ( f, f _ {C} ) }{D ( x, y) }
 +
  \neq  0.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
For a family of plane curves given by a  $  C  ^ {1} $-
 +
function
  
For a family of plane curves given by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584017.png" />-function
+
$$
 +
\mathbf r ( u, C)  = \
 +
\{ x ( u, C),\
 +
y ( u, C) \} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584018.png" /></td> </tr></table>
+
where  $  C $
 +
is the parameter of the family and  $  u $
 +
the parameter along its curves, a necessary condition for a point to be on the envelope is that  $  \mathbf r _ {u}  \|  \mathbf r _ {C} $,
 +
or, which is the same thing,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584019.png" /> is the parameter of the family and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584020.png" /> the parameter along its curves, a necessary condition for a point to be on the envelope is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584021.png" />, or, which is the same thing,
+
$$ \tag{3 }
 +
\phi  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac{D ( x, y) }{D ( u, C) }
 +
  = 0.
 +
$$
  
A sufficient condition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584023.png" /> and that, in addition to (3),
+
A sufficient condition is that $  \mathbf r \in C  ^ {2} $
 +
and that, in addition to (3),
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\mathbf r _ {u} \phi _ {C} -
 +
\mathbf r _ {C} \phi _ {u}  \neq  0.
 +
$$
  
 
Violation of conditions (2) and (4) is most often related to the appearance of cusps on the envelope.
 
Violation of conditions (2) and (4) is most often related to the appearance of cusps on the envelope.
  
The envelope of a family of surfaces in space depending on one parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584025.png" /> is the surface that at each of its points with intrinsic parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584026.png" /> touches the surface of the family with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584027.png" />, and is such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584028.png" /> is not constant on any domain in domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584029.png" />. For example, the envelope of spheres with the same radius and centres on a straight line is a cylinder. For a family given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584032.png" />, a necessary condition for the envelope is satisfaction of the system of equations
+
The envelope of a family of surfaces in space depending on one parameter $  C $
 +
is the surface that at each of its points with intrinsic parameters $  ( u , v) $
 +
touches the surface of the family with parameter $  C ( u , v) $,  
 +
and is such that the function $  C ( u , v) $
 +
is not constant on any domain in domain of definition of $  ( u , v) $.  
 +
For example, the envelope of spheres with the same radius and centres on a straight line is a cylinder. For a family given by $  f ( x, y, z, C) = 0 $,  
 +
where $  f \in C  ^ {1} $
 +
and $  | f _ {x} | + | f _ {y} | + | f _ {z} | \neq 0 $,  
 +
a necessary condition for the envelope is satisfaction of the system of equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
= 0,\ \
 +
f _ {C}  = 0,
 +
$$
  
and a sufficient condition is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584034.png" /> and, in addition to (5), the conditions
+
and a sufficient condition is: $  f \in C  ^ {2} $
 +
and, in addition to (5), the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
f _ {CC}  \neq  0,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584036.png" /></td> </tr></table>
+
$$
 +
\left |
 +
\frac{D ( f, f _ {C} ) }{D ( x, y) }
  
For the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584039.png" />, a necessary condition for the envelope is satisfaction of the equation
+
\right | + \left |
 +
\frac{D ( f, f _ {C} ) }{D ( y,
 +
z) }
 +
\right | + \left |
 +
\frac{D ( f, f _ {C} ) }{D ( z, x) }
 +
\right |  \neq  0.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
For the family  $  \mathbf r ( u , v, C) $,
 +
where  $  \mathbf r \in C  ^ {1} $
 +
and  $  \mathbf r _ {u} \times \mathbf r _ {v} \neq 0 $,
 +
a necessary condition for the envelope is satisfaction of the equation
  
and a sufficient condition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584041.png" /> and that, besides (7), the following conditions are satisfied:
+
$$ \tag{7 }
 +
\phi  = \
 +
( \mathbf r _ {u} \mathbf r _ {v} \mathbf r _ {C} )  = 0,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
and a sufficient condition is that  $  \mathbf r \in C  ^ {2} $
 +
and that, besides (7), the following conditions are satisfied:
 +
 
 +
$$ \tag{8 }
 +
\left |
 +
 
 +
\begin{array}{ccc}
 +
\phi _ {u}  &\phi _ {v}  &\phi _ {C}  \\
 +
\mathbf r _ {u}  ^ {2}  &\mathbf r _ {u} \mathbf r _ {v}  &\mathbf r _ {u} \mathbf r _ {C}  \\
 +
\mathbf r _ {u} \mathbf r _ {v}  &\mathbf r _ {v}  ^ {2}  &\mathbf r _ {v} \mathbf r _ {C}  \\
 +
\end{array}
 +
\
 +
\right |  \neq  0,\ \
 +
| \phi _ {u} | +
 +
| \phi _ {v} |  \neq  0.
 +
$$
  
 
Violation of the first of the conditions in (6) and (8) is most often related to the appearance of a cuspidal edge on the envelope. The line of contact of the envelope with one of the surfaces of the family is called a characteristic. A cuspidal edge on the envelope in turn is usually the envelope of the characteristics.
 
Violation of the first of the conditions in (6) and (8) is most often related to the appearance of a cuspidal edge on the envelope. The line of contact of the envelope with one of the surfaces of the family is called a characteristic. A cuspidal edge on the envelope in turn is usually the envelope of the characteristics.
  
The envelope of a family of surfaces in space depending on two parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584044.png" /> is the surface touching at each of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584045.png" /> the surface of the family with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584046.png" />, and such that there is no function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584047.png" /> on any domain of the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584048.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584049.png" />. For the family given by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584050.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584052.png" />, a necessary condition for the envelope is satisfaction of the system of equations
+
The envelope of a family of surfaces in space depending on two parameters $  A $
 +
and $  B $
 +
is the surface touching at each of its points $  ( u , v) $
 +
the surface of the family with parameters $  A ( u , v), B ( u , v) $,  
 +
and such that there is no function $  \Phi \in C  ^ {1} $
 +
on any domain of the range of $  ( u , v) $
 +
with $  A ( u , v) = \Phi ( B ( u , v)) $.  
 +
For the family given by the equation $  f ( x, y, z, A, B) = 0 $,  
 +
where $  f \in C  ^ {1} $
 +
and $  | f _ {x} | + | f _ {y} | + | f _ {z} | \neq 0 $,  
 +
a necessary condition for the envelope is satisfaction of the system of equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>
+
$$ \tag{9 }
 +
= 0,\ \
 +
f _ {A}  = 0,\ \
 +
f _ {B}  = 0,
 +
$$
  
and a sufficient condition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584054.png" /> and that, as well as (9), the following conditions holds:
+
and a sufficient condition is that $  f \in C  ^ {2} $
 +
and that, as well as (9), the following conditions holds:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584055.png" /></td> </tr></table>
+
$$
  
For a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584058.png" />, a necessary condition is
+
\frac{D ( f, f _ {A} , f _ {B} ) }{D ( x, y, z) }
 +
  \neq  0,\ \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>
+
\frac{D ( f _ {A} , f _ {B} ) }{D ( A, B) }
 +
  \neq  0.
 +
$$
  
and a sufficient condition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584060.png" /> and, as well as (10), satisfaction of
+
For a family  $  \mathbf r ( u , v, A, B) $,
 +
where  $  \mathbf r \in C  ^ {1} $
 +
and $  \mathbf r _ {u} \times \mathbf r _ {v} \neq 0 $,
 +
a necessary condition is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584061.png" /></td> </tr></table>
+
$$ \tag{10 }
 +
\phi  = \
 +
( \mathbf r _ {u} \mathbf r _ {v} \mathbf r _ {A} )  = 0,\ \
 +
\psi  = \
 +
( \mathbf r _ {u} \mathbf r _ {v} \mathbf r _ {B} )  = 0,
 +
$$
  
The more complicated concept of the envelope of a family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584062.png" />-dimensional submanifolds depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584063.png" /> parameters in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035840/e03584064.png" />-dimensional manifold can be introduced (see [[#References|[1]]]) based on the theory of singularities of differentiable mappings as a special form of singularity of a family.
+
and a sufficient condition is that  $  \mathbf r \in C  ^ {2} $
 +
and, as well as (10), satisfaction of
  
====References====
+
$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Zalgaller,  "The theory of envelopes" , Moscow (1975) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Favard,  "Cours de géométrie différentielle locale" , Gauthier-Villars (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.P. Tolstov,  ''Uspekhi Mat. Nauk'' , '''7''' : 4 (1952) pp. 173–179</TD></TR></table>
+
\left |
 +
 
 +
\begin{array}{cccc}
 +
\phi _ {u}  &\phi _ {v}  &\phi _ {A}  &\phi _ {B}  \\
 +
\psi _ {u}  &\psi _ {v}  &\psi _ {A}  &\psi _ {B}  \\
 +
\mathbf r _ {u}  ^ {2}  &\mathbf r _ {u} \mathbf r _ {v} &\mathbf r _ {u} \mathbf r _ {A}  &\mathbf r _ {u} \mathbf r _ {B}  \\
 +
\mathbf r _ {v} \mathbf r _ {u} &\mathbf r _ {v} ^ {2} &\mathbf r _ {v} \mathbf r _ {A} &\mathbf r _ {v} \mathbf r _ {B} \\
 +
\end{array}
 +
  \
 +
\right | \neq 0,\ \
  
 +
\frac{D ( \phi , \psi ) }{D ( A, B) }
 +
  \neq  0.
 +
$$
  
 +
The more complicated concept of the envelope of a family of  $  m $-
 +
dimensional submanifolds depending on  $  k $
 +
parameters in an  $  n $-
 +
dimensional manifold can be introduced (see [[#References|[1]]]) based on the theory of singularities of differentiable mappings as a special form of singularity of a family.
 +
 +
====References====
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Zalgaller, "The theory of envelopes" , Moscow (1975) (In Russian) {{MR|0159267}} {{ZBL|0139.14702}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Favard, "Cours de géométrie différentielle locale" , Gauthier-Villars (1957) {{MR|0083155}} {{ZBL|0077.15002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.P. Tolstov, ''Uspekhi Mat. Nauk'' , '''7''' : 4 (1952) pp. 173–179</TD></TR></table>
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Thom,   "Sur la théorie des enveloppes" ''J. de Math. Pures Appl.'' , '''56''' (1962) pp. 177–192</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Weil,   "Collected papers" , '''1''' , Springer (1980) pp. 133</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall (1976)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Thom, "Sur la théorie des enveloppes" ''J. de Math. Pures Appl.'' , '''56''' (1962) pp. 177–192 {{MR|0141041}} {{ZBL|0105.16102}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Weil, "Collected papers" , '''1''' , Springer (1980) pp. 133 {{MR|0537937}} {{MR|0537936}} {{MR|0537935}} {{MR|0465761}} {{MR|0465760}} {{ZBL|0428.01014}} {{ZBL|0428.01013}} {{ZBL|0428.01012}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) {{MR|}} {{ZBL|0326.53001}} </TD></TR></table>

Latest revision as of 19:37, 5 June 2020


of a family of curves in the plane

The curve that at every point touches one of the curves of the family such that the points of contact along the envelope pass from one curve of the family to another. For example, the family of circles of the same radius with centres on a straight line has an envelope consisting of two parallel lines. If $ C $ is the parameter of the family, $ t $ is the parameter along the envelope and $ C ( t) $ the value of $ C $ for one of the curves of the family touching the envelope at the point with parameter $ t $, then it is assumed that it is possible to choose $ C ( t) $ such that the function is not constant on any part of the range of $ t $.

For the family of curves given by $ f ( x, y, C) = 0 $, where $ f \in C ^ {1} $ and $ | f _ {x} | + | f _ {y} | \neq 0 $, a necessary condition for the existence of an envelope is that $ x ( t) $, $ y ( t) $, $ C ( t) $ satisfy the condition

$$ \tag{1 } f = 0,\ \ f _ {C} = 0. $$

The system (1) serves to determine the points of the envelope, but other singular points of the family may also satisfy (1). A sufficient condition for a point to belong to the envelope is that $ f \in C ^ {2} $ and satisfies, in addition to (1), the condition

$$ \tag{2 } f _ {CC} \neq 0,\ \ \frac{D ( f, f _ {C} ) }{D ( x, y) } \neq 0. $$

For a family of plane curves given by a $ C ^ {1} $- function

$$ \mathbf r ( u, C) = \ \{ x ( u, C),\ y ( u, C) \} , $$

where $ C $ is the parameter of the family and $ u $ the parameter along its curves, a necessary condition for a point to be on the envelope is that $ \mathbf r _ {u} \| \mathbf r _ {C} $, or, which is the same thing,

$$ \tag{3 } \phi = \ \frac{D ( x, y) }{D ( u, C) } = 0. $$

A sufficient condition is that $ \mathbf r \in C ^ {2} $ and that, in addition to (3),

$$ \tag{4 } \mathbf r _ {u} \phi _ {C} - \mathbf r _ {C} \phi _ {u} \neq 0. $$

Violation of conditions (2) and (4) is most often related to the appearance of cusps on the envelope.

The envelope of a family of surfaces in space depending on one parameter $ C $ is the surface that at each of its points with intrinsic parameters $ ( u , v) $ touches the surface of the family with parameter $ C ( u , v) $, and is such that the function $ C ( u , v) $ is not constant on any domain in domain of definition of $ ( u , v) $. For example, the envelope of spheres with the same radius and centres on a straight line is a cylinder. For a family given by $ f ( x, y, z, C) = 0 $, where $ f \in C ^ {1} $ and $ | f _ {x} | + | f _ {y} | + | f _ {z} | \neq 0 $, a necessary condition for the envelope is satisfaction of the system of equations

$$ \tag{5 } f = 0,\ \ f _ {C} = 0, $$

and a sufficient condition is: $ f \in C ^ {2} $ and, in addition to (5), the conditions

$$ \tag{6 } f _ {CC} \neq 0, $$

$$ \left | \frac{D ( f, f _ {C} ) }{D ( x, y) } \right | + \left | \frac{D ( f, f _ {C} ) }{D ( y, z) } \right | + \left | \frac{D ( f, f _ {C} ) }{D ( z, x) } \right | \neq 0. $$

For the family $ \mathbf r ( u , v, C) $, where $ \mathbf r \in C ^ {1} $ and $ \mathbf r _ {u} \times \mathbf r _ {v} \neq 0 $, a necessary condition for the envelope is satisfaction of the equation

$$ \tag{7 } \phi = \ ( \mathbf r _ {u} \mathbf r _ {v} \mathbf r _ {C} ) = 0, $$

and a sufficient condition is that $ \mathbf r \in C ^ {2} $ and that, besides (7), the following conditions are satisfied:

$$ \tag{8 } \left | \begin{array}{ccc} \phi _ {u} &\phi _ {v} &\phi _ {C} \\ \mathbf r _ {u} ^ {2} &\mathbf r _ {u} \mathbf r _ {v} &\mathbf r _ {u} \mathbf r _ {C} \\ \mathbf r _ {u} \mathbf r _ {v} &\mathbf r _ {v} ^ {2} &\mathbf r _ {v} \mathbf r _ {C} \\ \end{array} \ \right | \neq 0,\ \ | \phi _ {u} | + | \phi _ {v} | \neq 0. $$

Violation of the first of the conditions in (6) and (8) is most often related to the appearance of a cuspidal edge on the envelope. The line of contact of the envelope with one of the surfaces of the family is called a characteristic. A cuspidal edge on the envelope in turn is usually the envelope of the characteristics.

The envelope of a family of surfaces in space depending on two parameters $ A $ and $ B $ is the surface touching at each of its points $ ( u , v) $ the surface of the family with parameters $ A ( u , v), B ( u , v) $, and such that there is no function $ \Phi \in C ^ {1} $ on any domain of the range of $ ( u , v) $ with $ A ( u , v) = \Phi ( B ( u , v)) $. For the family given by the equation $ f ( x, y, z, A, B) = 0 $, where $ f \in C ^ {1} $ and $ | f _ {x} | + | f _ {y} | + | f _ {z} | \neq 0 $, a necessary condition for the envelope is satisfaction of the system of equations

$$ \tag{9 } f = 0,\ \ f _ {A} = 0,\ \ f _ {B} = 0, $$

and a sufficient condition is that $ f \in C ^ {2} $ and that, as well as (9), the following conditions holds:

$$ \frac{D ( f, f _ {A} , f _ {B} ) }{D ( x, y, z) } \neq 0,\ \ \frac{D ( f _ {A} , f _ {B} ) }{D ( A, B) } \neq 0. $$

For a family $ \mathbf r ( u , v, A, B) $, where $ \mathbf r \in C ^ {1} $ and $ \mathbf r _ {u} \times \mathbf r _ {v} \neq 0 $, a necessary condition is

$$ \tag{10 } \phi = \ ( \mathbf r _ {u} \mathbf r _ {v} \mathbf r _ {A} ) = 0,\ \ \psi = \ ( \mathbf r _ {u} \mathbf r _ {v} \mathbf r _ {B} ) = 0, $$

and a sufficient condition is that $ \mathbf r \in C ^ {2} $ and, as well as (10), satisfaction of

$$ \left | \begin{array}{cccc} \phi _ {u} &\phi _ {v} &\phi _ {A} &\phi _ {B} \\ \psi _ {u} &\psi _ {v} &\psi _ {A} &\psi _ {B} \\ \mathbf r _ {u} ^ {2} &\mathbf r _ {u} \mathbf r _ {v} &\mathbf r _ {u} \mathbf r _ {A} &\mathbf r _ {u} \mathbf r _ {B} \\ \mathbf r _ {v} \mathbf r _ {u} &\mathbf r _ {v} ^ {2} &\mathbf r _ {v} \mathbf r _ {A} &\mathbf r _ {v} \mathbf r _ {B} \\ \end{array} \ \right | \neq 0,\ \ \frac{D ( \phi , \psi ) }{D ( A, B) } \neq 0. $$

The more complicated concept of the envelope of a family of $ m $- dimensional submanifolds depending on $ k $ parameters in an $ n $- dimensional manifold can be introduced (see [1]) based on the theory of singularities of differentiable mappings as a special form of singularity of a family.

References

[1] V.A. Zalgaller, "The theory of envelopes" , Moscow (1975) (In Russian) MR0159267 Zbl 0139.14702
[2] J. Favard, "Cours de géométrie différentielle locale" , Gauthier-Villars (1957) MR0083155 Zbl 0077.15002
[3] G.P. Tolstov, Uspekhi Mat. Nauk , 7 : 4 (1952) pp. 173–179

Comments

References

[a1] R. Thom, "Sur la théorie des enveloppes" J. de Math. Pures Appl. , 56 (1962) pp. 177–192 MR0141041 Zbl 0105.16102
[a2] A. Weil, "Collected papers" , 1 , Springer (1980) pp. 133 MR0537937 MR0537936 MR0537935 MR0465761 MR0465760 Zbl 0428.01014 Zbl 0428.01013 Zbl 0428.01012
[a3] M. do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) Zbl 0326.53001
How to Cite This Entry:
Envelope. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Envelope&oldid=12188
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article