Tangent Plane and Normal Line

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Tangent Plane

Let \( \vec{r}=\vec{r}(u,v) \) be a surface Then,\( \vec{r_1} \) and \( \vec{r_2} \) are the tangent to u-curve and v-curve respectively.
Now, a plane spanned by \( \vec{r_1} \) and \( \vec{r_2} \) is called tangent plane.

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Normal Line

Let \(S: \vec{r}=\vec{r}(u,v) \) be a surface
Then
\( \vec{r_1} \times \vec{r_2} \) is a line perpendicular to the tangent plane to the surface S
So,
\( \vec{r_1} \times \vec{r_2} \) is called normal line to the surface
Since \( \vec{r_1} \times \vec{r_2} \) is normal line, we denote the unit vector along normal line by \( \vec{N} \) and defined as
\( \vec{N}= \frac{\vec{r_1} \times \vec{r_2}}{|\vec{r_1} \times \vec{r_2}|} \)
or \( \vec{N}= \frac{\vec{r_1} \times \vec{r_2}}{\textbf{H}} \) where \( |\vec{r_1} \times \vec{r_2}| =\textbf{H} \)

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Equation of Tangent Plane

Let \( \vec{r}=\vec{r}(u,v) \) be a surface and P be a point with
\( \overrightarrow{OP}= \vec{r}\)
Now
\( \vec{r_1} \) and \( \vec{r_2} \) are the tangents to the surface.
Let \( \Gamma \) be tangent plane at P, then
\( \vec{r_1} \) and \( \vec{r_2} \) lies in \( \Gamma \).
Let \( \vec{R} \) be position vector of arbitrary point T on the tangent plane \( \Gamma \)
Then,
\( \overrightarrow{PT} \) lies in the tangent plane \( \Gamma \)
or\( \overrightarrow{PT} , \vec{r_1} , \vec{r_2} \) lies in the tangent plane \( \Gamma \)
or\( [\overrightarrow{PT} , \vec{r_1}, \vec{r_2}]=0 \) is the equation of tangent plane at P.
or\( [\overrightarrow{OT}-\overrightarrow{OP}, \vec{r_1}, \vec{r_2}]=0 \) is the equation of tangent plane at P.
or\( [\vec{R}-\vec{r}, \vec{r_1}, \vec{r_2}]=0 \) is the equation of tangent plane at P.
Equivalently, the equation of tangent plane are

1. \( (\vec{R}-\vec{r}). (\vec{r_1} \times \vec{r_2}) =0 \)
   or \( (\vec{R}-\vec{r}). H \vec{N} =0 \)
   or \( (\vec{R}-\vec{r}). \vec{N} =0 \)
2. \( \vec{R}- \vec{r}= \lambda \vec{r_1}+ \mu \vec{r_2} \)
3. \( \vec{R}= \vec{r} + \lambda \vec{r_1}+ \mu \vec{r_2} \)

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Equation of Tangent Plane in Cartesian Form

Let \( \vec{r}=\vec{r}(u,v) \) be a surface then equation of tangent plane
\( (\vec{R}-\vec{r}). (\vec{r_1} \times \vec{r_2}) =0 \)
Since, \( F(x,y,z)=0 \) is given, we have
\( GradF =(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \)
We also know that
\( GradF || (\vec{r_1} \times \vec{r_2}) \)
Thus, equation of tangent plane is
\( (\vec{R}-\vec{r}). \text{ GradF} =0 =0 \)
or\( (X-x,Y-y,Z-z). (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) =0 \)
or\( (X-x)\frac{\partial F}{\partial x} + (Y-y)\frac{\partial F}{\partial y}+(Z-z). \frac{\partial F}{\partial z} =0 \)

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Equation of Normal Line

Let \( \vec{r}=\vec{r}(u,v) \) be a surface and P be a point with
\( \vec{OP}= \vec{r}\)
Now
\( \vec{r_1} \) and \( \vec{r_2} \) are the tangents to the surface.
Then
\( \vec{r_1} \times \vec{r_2} \) is Normal line
Let \(l\) be the normal line at P, and \( \vec{R} \) be position vector of arbitrary point T on this normal line,
Then,
\( \vec{PT} \) and \( \vec{r_1} \times \vec{r_2} \) are parallel
or\( \vec{PT} || \vec{r_1} \times \vec{r_2}\)
or\( \vec{OT}-\vec{OP}= \lambda (\vec{r_1} \times \vec{r_2}) \)
or\( R-\vec{r}= \lambda (\vec{r_1} \times \vec{r_2}) \)
Equivalently, the equation of normal line is
\( \vec{R}=\vec{r}+ \lambda ( \vec{r_1} \times \vec{r_2}) \)

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Equation of Normal Line in Cartesian Form

Let \( \vec{r}=\vec{r}(u,v) \) be a surface then equation of normal line is
\( (\vec{R}-\vec{r})=\lambda (\vec{r_1} \times \vec{r_2}) \)
Since, \( F(x,y,z)=0 \) is given, we have
\( GradF =(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} ) \)
We also know that
\( GradF || (\vec{r_1} \times \vec{r_2}) \)
Thus, equation of normal line is
\( (\vec{R}-\vec{r})=\lambda \text{ GradF} \)
or \( [(X,Y,Z)-(x,y,z)]= \lambda \left ( \frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} \right ) \)
or \( (X-x,Y-y,Z-z)=\lambda \left (\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} \right ) \)
or \( \frac{X-x}{\frac{\partial F}{\partial x}} = \frac{Y-y}{\frac{\partial F}{\partial y}} = \frac{Z-z}{\frac{\partial F}{\partial z}} =\lambda\)

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Exercise

Find the equation of tangent plane and normal line to the surface

1. \( x^2+y^2-z =0 \) at (1,-1,2).
   
2. \( z= 5+(x-1)^2 + (y+2)^2 \) at (2,0,10).
   
3. \( z=\log (2x+y) \) at (-1,3).
   
4. \( z=3+ \frac{x^2}{16}+ \frac{y^2}{9} \) at (-4,3).
   

Theorem

Prove that a proper parametric transformation either leaves every normal unchanged or reverses every normal.
Proof
Let \( \vec{r}=\vec{r}(u,v) \) be a surface with parameters \((u,v) \).
Let transformation of the parameter \((u,v) \) into \((U,V) \) is done 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 \( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}= (\frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V}) J \)
or \( H \vec{N}= H^* \vec{N^*} \times J \) where \( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} = H \vec{N} \) and \(\frac{\partial \vec{r}}{\partial U} \times \frac{\partial \vec{r}}{\partial V} =H^* \vec{N^*} \)
Case1 If \( J > 0 \) then normal have same sign
Case 2 If \( J < 0 \) then normal have opposite sign
Therefore, proper parametric transformation either leaves every normal unchanged or reverses every normal.

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