Transformation and its Geometric Significance

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Transformation and its geometric significance

Let \( \vec{r}=\vec{r}(u,v) \) be a surface with parameters \((u,v) \)
If a new set of parameters \((U,V) \) is defined as functions of \((u,v) \) such that
\(U=\phi(u,v),V=\psi((u,v)) \)
then this defining process is called parameter transformation on the surface.

Example

If a surface is given by
\( x=u+v,y=u-v, z=uv \) (i)
If we define \((U,V) \) as
\( U=u+v \) and \( V=u-v \)
Then surface (i) can be written as
\( x=U,y=V, z=\frac{1}{4} (U^2-V^2) \) (i)
Here, set of parameters \((u,v) \) is transformed into another set \((U,V) \)
This process is called parameter transformation.

Proper Transformation

Let \( \vec{r}=\vec{r}(u,v) \) be a surface with parameters \((u,v) \)
The transformation of the parameter \((u,v) \) into \((U,V) \) by
\(U=\phi(u,v),V=\psi((u,v)) \)
is called proper transformation if
\( J ( \text{jacobian}) \ne 0 \)
or\( \begin{vmatrix} \frac{\partial U}{\partial u} & \frac{\partial U}{\partial v} \\ \frac{\partial V}{\partial u} & \frac{\partial V}{\partial v} \end{vmatrix} \neq 0 \)
or\( \frac{\partial (U, V)}{\partial (u, v)} \neq 0 \)
or\( \frac{\partial (\phi, \psi)}{\partial (u, v)} \ne 0 \)

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Theorem

Proper transformation maps regular point to regular point
Let \( \vec{r}=\vec{r}(u,v) \) be a surface with parameters \((u,v) \)
The transformation of the parameter \((u,v) \) into \((U,V) \) by
\(U=\phi(u,v),V=\psi((u,v)) \)
Then
\( \frac{\partial \vec{r}}{\partial u} = \frac{\partial \vec{r}}{\partial U} \frac{\partial U}{\partial u}+ \frac{\partial \vec{r}}{\partial V} \frac{\partial V}{\partial u} \) and
\( \frac{\partial \vec{r}}{\partial v} = \frac{\partial \vec{r}}{\partial U} \frac{\partial U}{\partial v}+ \frac{\partial \vec{r}}{\partial V} \frac{\partial V}{\partial v} \)
Hence
\( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}= (\frac{\partial \vec{r}}{\partial U} \frac{\partial U}{\partial u}+ \frac{\partial \vec{r}}{\partial V} \frac{\partial V}{\partial u}) \times (\frac{\partial \vec{r}}{\partial U} \frac{\partial U}{\partial v}+ \frac{\partial \vec{r}}{\partial V} \frac{\partial V}{\partial v}) \)
or \( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}= ( \frac{\partial \vec{r}}{\partial U} \frac{\partial U}{\partial u} \times \frac{\partial \vec{r}}{\partial V} \frac{\partial V}{\partial v})+ ( \frac{\partial \vec{r}}{\partial V} \frac{\partial V}{\partial u} \times \frac{\partial \vec{r}}{\partial U} \frac{\partial U}{\partial v}) \)
or \( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}= (\frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V})( \frac{\partial U}{\partial u} \frac{\partial V}{\partial v})+ (\frac{\partial \vec{r}}{\partial V} \times \frac{\partial \vec{r}}{\partial U}) (\frac{\partial V}{\partial u} \frac{\partial U}{\partial v}) \)
or \( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}= (\frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V}) (\frac{\partial U}{\partial u} \frac{\partial V}{\partial v})-(\frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V})( \frac{\partial V}{\partial u} \frac{\partial U}{\partial v}) \)
or \( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}= (\frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V}) ( \frac{\partial U}{\partial u} \frac{\partial V}{\partial v} - \frac{\partial V}{\partial u} \frac{\partial U}{\partial v}) \)
or \( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}= (\frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V}) \begin{vmatrix} \frac{\partial U}{\partial u} & \frac{\partial U}{\partial v} \\ \frac{\partial V}{\partial u} & \frac{\partial V}{\partial v} \end{vmatrix} \)
or \( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}= (\frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V}) J \)
If \( P(u,v) \) is regular point then
\( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \neq 0 \)
Since the transformation is proper
\( J \ne 0 \)
Therefore, we have
\( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}= (\frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V}) J \)
or \( \frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V} \ne 0 \)
This shows that
\( P(U,V) \) is regular point
Thus, proper transformation maps regular point to regular point.
Note
The property of being an ordinary point (a regular point) is that it is unaltered by a proper parametric transformation.

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Parametric Curves

Let \( \vec{r}=\vec{r}(u,v) \) be a surface
Now locus of the points
\( u=u(t) \) , where v is constant
is called u-parameter curve, u-curve or v=constant curve.
Similarly
The locus of points
\( v=v(t) \) , where u is constant
is called v-parameter curve, v-curve or u=constant curve.

u-curve

   r=[4cos(u), 4sin(u),  v]
   

 
  

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v-curve

   r=[4cos(u), 4sin(u),  v]
   


  

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