#### Family of surface

An equation

\( F( x,y,z;a )=0 \)

is a surface, where \( a \) is constant.

If \( a \) is parameter then,

\( F( x,y,z;a )=0 \)

is a family of surface.

For example

\( F( x,y,z;a )=3a^2x+3ay+z-a^3=0 \)

where \( a \) is parameter, is one parameter family of surface.

Similarly, an equation of the form

\( F( x,y,z;a,b )=0 \)

where\( a \) and\( b \) are parameters, is a two parameter family of surface.

For example

\( F( x,y,z;a,b )=( x-a cosb )^2+( y-a sinb )^2+z^2=a^2 \)

is a family of spherical surfaces.

NOTE

By assigning different values of the parameter(s) we can get different surfaces of the family.

#### Characteristic curve

Let \( F( x,y,z;a )=0 \) be a family of surface.

Then two neighboring surfaces with parameter values \( a \) and \( a+\Delta a \) intersects in a curve, called characteristic curve of the family.

The characteristic curve is given by the equation

\( F( x,y,z;a )=0 \) and \( F( x,y,z;a+\Delta a )=0 \)

or\( F( x,y,z;a )=0;\frac{\partial F( x,y,z;a )}{\partial a}=0 \)

or\( F=0; \frac{\partial F}{\partial a}=0 \)

Thus characteristic curve of the family of surface is obtained by intersection of two neighboring surfaces.

It is given by the equation

\( F=0; \frac{\partial F}{\partial a}=0 \) (i) \( a \) is parameter

By assigning different values of \( a \), (i) will give different characteristic curves of the family.

#### Envelop

Let \( F( x,y,z;a )=0 \) be a family of surface.

Then characteristic curve of the family is given by

\( F=0; \frac{\partial F}{\partial a}=0 \) (i) \( a \) is parameter

By assigning different values of \( a \), (i) will give different characteristic curves of the family.

Now,

Locus of all characteristic curves generates a surface, called the envelope of the family.

The equation of the envelope is obtained by eliminating parameter a from the equations

\( F=0; \frac{\partial F}{\partial a}=0 \) (ii) a is functions of \( x,y,z \)

#### Example

Find envelope of a family \( F( x,y,z;a )=3a^2x-3ay+z-a^3=0\).
Solution

The family of surface is

\( F( x,y,z;a )=3a^2x-3ay+z-a^3=0\)

In short, we write

\( F:3a^2x-3ay+z-a^3=0\) (i)

By differentiation of (i) w. r. to. \( a \), we get

\( \frac{\partial F}{\partial a}:6ax-3y-3a^2=0\) (ii)

The equation of envelope is

\( F=0; \frac{\partial F}{\partial a}=0 \) \( a \) is functions of \( x,y,z \)

or\( F=0; \frac{\partial F}{\partial a}=0 \) remove \( a \)

Now, eliminating \( a \) between (i) and (ii), we get envelope of the family

Thus multiplying (i) by 3, (ii) by \( a \) and then subtracting we get

\( 9a^2x-9ay+3z-3a^3=0 \)

\( 6a^2x-3ay-3a^3=0 \)

\( (-) \)

=================================================

\( 3a^2x-6ay+3z=0\) (iii)

Now, solving (ii) and (iii) for \( a \) we get

\( \frac{a^2}{2zx-2y^2}=\frac{a}{-xy+z}=\frac{1}{2y-2x^2} \)

Hence, by value of \( a \) is

\( a= \frac{2zx-2y^2}{-xy+z} \) and \( a= \frac{-xy+z}{2y-2x^2} \)

Hence, by eliminating \( a \), the required envelope is

\( \frac{2zx-2y^2}{z-xy}=\frac{z-xy}{2y-2x^2} \)

or\( ( xy-z )^2=4( x^2-y )( y^2-zx ) \)

#### An important property

Show that Envelope touches each member of the family at all points of characteristic curves.

Proof

We know, normal to the surface \( F( x,y,z;a )=0 \) ...(i) is parallel to the vector

\( (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \) (A)

Let \( F( x,y,z;a )=0 \) be a family of surface, then envelope is

\( F=0,\frac{\partial F}{\partial a}=0 \) [eliminate a]

Thus, envelope is considered as

\( F=0 \) (ii) ; where \( a \) is functions of \( x,y,z \) given by \( \frac{\partial F}{\partial a}=0 \)

Here, normal to the envelope (i) is parallel to the vector

\( ( \frac{\partial F}{\partial x}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial x},\frac{\partial F}{\partial y}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial y},\frac{\partial F}{\partial z}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial z} ) \)

Since

\( \frac{\partial F}{\partial a}=0 \)

Then, normal to the envelope (ii) is parallel to the vector

\( ( \frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \) (B)

From (A) and (B), it shows that, normal to the envelope (ii) and normal to the member of the family (i) are same.

Thus, envelope touches each member of the family.

#### An important property

Show that envelope touches each member of the family at all corresponding points of characteristic curves.

Proof

We know, normal to the surface \( F( x,y,z;a,b )=0 \) ...(i) is parallel to the vector

\( (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \) (A)

Let \( F( x,y,z;a,b )=0 \) be family of surface, then envelope is

\( F=0,\frac{\partial F}{\partial a}=0,\frac{\partial F}{\partial b}=0 \) [eliminate a and b]

Thus, envelope is considered as

\( F=0 \)(ii); where \( a \) and \( b \) are functions of \( x,y,z \) by \( \frac{\partial F}{\partial a}=0,\frac{\partial F}{\partial b}=0 \)

Here, normal to the envelope (ii) is parallel to the vector

\( ( \frac{\partial F}{\partial x}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial x}+\frac{\partial F}{\partial b}\frac{\partial b}{\partial x},\frac{\partial F}{\partial y}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial y}+\frac{\partial F}{\partial b}\frac{\partial b}{\partial y},\frac{\partial F}{\partial z}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial z}+\frac{\partial F}{\partial b}\frac{\partial b}{\partial z} ) \)

Since

\( \frac{\partial F}{\partial a}=0 \) and \( \frac{\partial F}{\partial b}=0 \)

Then, normal to the envelope (ii) is parallel to the vector

\( (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \) (B)

Here, from (A) and (B), it show that, normal to the envelope (ii) and normal to the member of the family (i) is same.

Hence, Envelope touches each member of the family at all corresponding points of characteristic curves.

#### Characteristic point

Let \( F( x,y,z;a )=0 \) be a family of surface

Then characteristic curve of the family is given by

\( F=0; \frac{\partial F}{\partial a}=0 \) (i)\( a \) is parameter

When parameter\( a \) varies, it gives different characteristic curves

Now

Two neighboring characteristic curves

\( F=0; \frac{\partial F}{\partial a}=0 \) and \( \frac{\partial F}{\partial a}=0,\frac{\partial^2 F}{\partial a^2}=0 \)

in general intersect at a point, called characteristic point

It is given by the equations

\( F=0,\frac{\partial F}{\partial a}=0,\frac{{{\partial }^2}F}{\partial a^2}=0 \) (i)\( a \) is parameter

#### Edge of regression

Let \( F( x,y,z;a )=0 \) be a family of surface

Then characteristic point of the family of surface is given by

\( F=0,\frac{\partial F}{\partial a}=0,\frac{{{\partial }^2}F}{\partial a^2}=0 \) (i)\( a \) is parameter

When parameter \( a \) varies, it gives different characteristic points

Now

Locus of all characteristic points generates a curve, called edge of regression of the family

The equation of edge of regression is obtained by eliminating parameter a from the equations

\( F=0,\frac{\partial F}{\partial a}=0,\frac{{{\partial }^2}F}{\partial a^2}=0 \) ; \( a \) is functions of \( x,y,z \)

#### Example

Find edge of regression of a family of surface \(F( x,y,z;a ):3a^2x-3ay+z-a^3=0 \)

Solution

The family of surface is

\( F( x,y,z;a ):3a^2 x-3ay+z-a^3=0\)

In short, we write

\( F( x,y,z;a ):3a^2 x-3ay+z-a^3=0\) (i)

By successive differentiation of (i) w. r. to. a, we get

\( \frac{\partial F}{\partial a}:6ax-3y-3a^2=0 \)

or\( 2ax-y-a^2=0 \) (ii)

Next

\( \frac{\partial ^2 F}{\partial a^2}:2x-2a=0 \)

or\( x-a=0 \) (iii)

Now

The equation of edge of regression is

\( F=0; \frac{\partial F}{\partial a}=0;\frac{\partial^2 F}{\partial a^2}=0 \) where \( a \) is functions of \( x,y,z \)

or\( F=0; \frac{\partial F}{\partial a}=0;\frac{\partial^2 F}{\partial a^2}=0 \) remove \( a \)

Thus, eliminating \( a \) from (i),(ii) and (iii), we get edge of regression

Thus, the edge of regression of the family is

\( 3a^2 x-3ay+z-a^3=0;2ax-y-a^2=0; x-a=0 \)

Putting \( x=a \) we get

\(3x^3-3xy+z-x^3=0,2x^2-y-x^2=0 \)

or\(2x^3-3xy+z=0,x^2=y \)

#### An important property

Show that each characteristic curves touches the edge of regression.

Proof

We know that, equation of characteristic curve is

\( F=0,\frac{\partial F}{\partial a}=0 \)

Thus, tangent to the characteristic curve holds

\( \vec{t} || [ (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \times (\frac{\partial^2 F}{\partial a \partial x},\frac{\partial^2 F}{\partial a \partial y},\frac{\partial^2 F}{\partial a \partial z} ) ] \)
(A)

Let \( F( x,y,z;a )=0 \) be family of surface, then edge of regression is

\( F=0,\frac{\partial F}{\partial a}=0,\frac{{{\partial }^{2}}F}{\partial {{a}^{2}}}=0 \) [eliminate a]

Thus, edge of regression is considered as

\( F=0,\frac{\partial F}{\partial a}=0 \); \( a \)is functions of\( x,y,z \) given by \(\frac{{{\partial }^{2}}F}{\partial {{a}^{2}}}=0 \)

Let \( \vec{t} \) is unit tangent to the edge of regression,
then

\( \vec{t} \) .normal to \( F =0 \) and

\( \vec{t} \) . normal to \( \frac{\partial F}{\partial a} =0 \)

or
\( \vec{t} . ( \frac{\partial F}{\partial x}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial x},\frac{\partial F}{\partial y}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial y},\frac{\partial F}{\partial z}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial z} ) =0 \) and

\( \vec{t} . (\frac{\partial^2 F}{\partial a \partial x}+\frac{\partial^2 F}{\partial a^2}\frac{\partial a}{\partial x},\frac{\partial^2 F}{\partial a \partial y}+\frac{\partial^2 F}{\partial a^2}\frac{\partial a}{\partial y},\frac{\partial^2 F}{\partial a \partial z}+\frac{\partial^2 F}{\partial a^2}\frac{\partial a}{\partial z} ) =0 \)

Since

\( \frac{\partial F}{\partial a}=0,\frac{\partial ^2 F}{\partial a^2}=0 \)

We have

\( \vec{t} . (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) =0 \) (i) and

\( \vec{t} .(\frac{\partial^2 F}{\partial a \partial x},\frac{\partial^2 F}{\partial a \partial y},\frac{\partial^2 F}{\partial a \partial z} )=0 \) (ii)

From (i) and
(ii), we see that

\( \vec{t} || [ (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \times (\frac{\partial^2 F}{\partial a \partial x},\frac{\partial^2 F}{\partial a \partial y},\frac{\partial^2 F}{\partial a \partial z} ) ] \)
(B)

Here, from (A) and (B), it shows that tangent to the edge of regression and tangent to the characteristic curve are same.

Hence, characteristic curve touches the edge of regression.

#### Exercise

- Find envelope of a family F(x,y,z ;a )=3a^2 x-3ay+z=0.
- Find envelope of a family F(x,y,z ;a )=3a^2 x-3ay+z-a^3=0.
- Find characteristic curves of a family F(x,y,z ;a )=3a^2 x-3ay+z=0.
- Find characteristic curves of a family F(x,y,z ;a )=3a^2 x-3ay+z-a^3=0.
- Find characteristic point of a family F(x,y,z ;a )=3a^2 x-3ay+z=0.
- Find characteristic point of a family F(x,y,z ;a ) \[Colon]3 a^2 x-3ay+z-a^3=0.
- Find edge of regression of a family F(x,y,z ;a ) \[Colon]3 a^2 x-3ay+z-a^3=0.

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