Envelope

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

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