Family of surface
An equation
\( F( x,y,z;a )=0 \) where \( a \) is constant.
is a surface,
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 parameter
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\( F=0; \frac{\partial F}{\partial a}=0 \) (i) \( a \) is parameter
By assigning different values of \( a \), (i) will give different characteristic curves of the family.
Slide the value of a, to get different characteristic curves.
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 \( x,y,z \)
Slide the value of a, to get the envelope.
Example
Find envelope of a family \( F( x,y,z;a )=3a^2x-3ay+z=0\).
Solution
The family of surface is
\( F:3a^2x-3ay+z=0\) (i)
By differentiation of (i) w. r. to. \( a \), we get
\( \frac{\partial F}{\partial a}:6ax-3y=0\)
or\( \frac{\partial F}{\partial a}:y=2ax\) (ii)
The equation of envelope is
\( F=0; \frac{\partial F}{\partial a}=0 \) \( a \) is functions of \( x,y,z \)
or \( 3a^2x-3ay+z=0 \) and \( a=\frac{y}{2x} \)
or \( 3(\frac{y}{2x})^2x-3(\frac{y}{2x})y+z=0 \)
or\( 3y^2=4xz \)
Example
Find envelope of a family \( F( x,y,z;a )=3a^2x-3ay+z-a^3=0\).
Solution
The family of surface is
\( F:3a^2x-3ay+z-a^3=0\) (i)
By differentiation of (i) w. r. to. \( a \), we get
\( \frac{\partial F}{\partial a}:6ax-3y-3a^2=0\)
or\( \frac{\partial F}{\partial a}:2ax-y-a^2=0\) (ii)
The equation of envelope is
\( F=0; \frac{\partial F}{\partial a}=0 \) \( a \) is functions of \( x,y,z \)
Now, eliminating \( a \) between (i) and (ii), we get envelope of the family
Thus multiplying (ii) by \( a \) and then subtracting from (i), we get
\( 3a^2x-3ay+z-a^3=0 \)
\( a^2x-ay-a^3=0 \)
\( (-) \)
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\( a^2x-2ay+z=0\) (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
\( \frac{2zx-2y^2}{-xy+z}=\frac{-xy+z}{2y-2x^2} \)
or\( ( z-xy )^2=4( xz-y^2 )( y-x^2 ) \)
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
\( (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \) (A)
The envelope of the family is
or \( F=0 \) (ii) ; where \( a \) is functions of \( x,y,z \) given by \( \frac{\partial F}{\partial a}=0 \)
Here, normal to the envelope (ii) is
\( ( \textcolor{red}{ \frac{\partial F}{\partial x}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial x}},\frac{\partial F}{\partial y}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial y},\textcolor{blue}{\frac{\partial F}{\partial z}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial z}} ) \)
Since
\( \frac{\partial F}{\partial a}=0 \)
Then, normal to the envelope (ii) is
\( ( \frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \) (B)
From (A) and (B), it shows that, normal to the envelope (ii) and normal to the family (i) is same.
Thus, envelope touches each member of the family at all points of characteristic curves.
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
\( (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \) (A)
Now, the envelope to the family is
\( F=0,\frac{\partial F}{\partial a}=0,\frac{\partial F}{\partial b}=0 \) [eliminate a and b]
or \( F=0 \)(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
\( (\textcolor{red}{ \frac{\partial F}{\partial x}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial x}+\frac{\partial F}{\partial b}\frac{\partial b}{\partial x}},\frac{\partial F}{\partial y}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial y}+\frac{\partial F}{\partial b}\frac{\partial b}{\partial y},\textcolor{blue}{ \frac{\partial F}{\partial z}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial z}+\frac{\partial F}{\partial b}\frac{\partial b}{\partial z} }) \)
Since
\( \frac{\partial F}{\partial a}=0 \) and \( \frac{\partial F}{\partial b}=0 \)
Then, normal to the envelope (ii) is
\( (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \) (B)
Here, from (A) and (B), it show that, normal to the envelope (ii) and normal to the 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, (i) 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 equation
\( F=0,\frac{\partial F}{\partial a}=0,\frac{{{\partial }^2}F}{\partial a^2}=0 \) (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, (i) gives different characteristic points
Now
Locus of all characteristic points generates a curve, called the edge of regression
The equation of edge of regression is obtained by eliminating parameter a from the equation
\( F=0,\frac{\partial F}{\partial a}=0,\frac{{{\partial }^2}F}{\partial a^2}=0 \) \( a \) 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
\( F:3a^2 x-3ay+z-a^3=0\) (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\( 3a^2 x-3ay+z-a^3=0;2ax-y-a^2=0; x-a=0 \)
or\(3x^3-3xy+z-x^3=0,2x^2-y-x^2=0 \)
or\(2x^3-3xy+z=0,x^2=y \)
An important property
Show that each characteristic curves touches the edge of regression.
Proof
The characteristic curve is
\( F=0,\frac{\partial F}{\partial a}=0 \)
So, tangent to the characteristic curve is parallel to the vector
\( \left (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} \right) \times \left (\frac{\partial^2 F}{\partial a \partial x},\frac{\partial^2 F}{\partial a \partial y},\frac{\partial^2 F}{\partial a \partial z} \right) \)
(A)
As we know, the edge of regression is
\( F=0,\frac{\partial F}{\partial a}=0,\frac{{{\partial }^{2}}F}{\partial {{a}^{2}}}=0 \) \( a \) is functions of\( x,y,z \)
or \( F=0,\frac{\partial F}{\partial a}=0 \) \( a \) is functions of\( x,y,z \) given by \(\frac{{{\partial }^{2}}F}{\partial {{a}^{2}}}=0 \)
Then tangent to the edge of regression is parallel to the vector
\( \left ( \frac{\partial F}{\partial x}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial x},\frac{\partial F}{\partial y}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial y},\frac{\partial F}{\partial z}+\frac{\partial F}{\partial a}\frac{\partial a}{\partial z} \right ) \times \left (\frac{\partial^2 F}{\partial a \partial x}+\frac{\partial^2 F}{\partial a^2}\frac{\partial a}{\partial x},\frac{\partial^2 F}{\partial a \partial y}+\frac{\partial^2 F}{\partial a^2}\frac{\partial a}{\partial y},\frac{\partial^2 F}{\partial a \partial z}+\frac{\partial^2 F}{\partial a^2}\frac{\partial a}{\partial z} \right ) \)
Since \( \frac{\partial F}{\partial a}=0,\frac{\partial ^2 F}{\partial a^2}=0 \), tangent to the edge of regression is parallel to the vector
\( \left (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \times (\frac{\partial^2 F}{\partial a \partial x},\frac{\partial^2 F}{\partial a \partial y},\frac{\partial^2 F}{\partial a \partial z} \right )\) (B)
Here, from (A) and (B), it shows that tangent to the edge of regression and tangent to the characteristic curve are same.
Hence, each characteristic curve touches the edge of regression.
Exercise
- Find envelope
- \(F(x,y,z ;a )=3a^2 x-3ay+z=0\). \(3y^2=4xz\).
- \(F(x,y,z ;a )=3a^2 x-3ay+z-a^3=0\). \( ( z-xy )^2=4( xz-y^2 )( y-x^2 ) \) .
- Find characteristic curves of a family
- \(F(x,y,z ;a )=3a^2 x-3ay+z=0\) \( \{t, 2 a t, 3 a^2 t\} \) .
- \(F(x,y,z ;a )=3a^2 x-3ay+z-a^3=0\).\( \{ t, -a^2 + 2 a t, -2 a^3 + 3 a^2 t\}\) .
- Find characteristic point of a family
- \(F(x,y,z ;a )=3a^2 x-3ay+z=0\). \((0,0,0)\).
- \(F(x,y,z ;a ) :3 a^2 x-3ay+z-a^3=0\). \( \{a, a^2, a^3\} \) .
- \(F(x,y,z ;a ) :3 a^2 x^2-3ay+z-a^3=0\). \( \{\sqrt{a}, a^2, a^3\} \) .
- Find edge of regression of a family \(F(x,y,z ;a ) :3 a^2 x-3ay+z-a^3=0\). \( \{t, t^2, t^3\} \) .
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