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First Fundamental Form of a Parametric Surface

1. Given a Parametric Surface defined by Position Vector Function \(\vec{R}\), the corresponding Tangent Vectors \(\vec{R_u}\) and \(\vec{R_v}\), and the Differential Change in Position Vector \(\vec{R}\) on the Surface denoted by \(d\vec{R}\), the Differential Distance between Any 2 Points on Surface \(\vec{R}\) denoted by \(ds\) is given by the Magnitude of the Differential Change in Position Vector \(\vec{dR}\) as follows
   
   \(ds= |d\vec{R}| = \sqrt{d\vec{R} \cdot d\vec{R}} = \sqrt{(d\vec{R_u}+d\vec{R_v}) \cdot (d\vec{R_u}+d\vec{R_v}) }\)   ...(1)
   
   \(\Rightarrow ds= \sqrt{(\vec{R_u}du +\vec{R_v}dv) \cdot (\vec{R_u}du +\vec{R_v}dv)} = \sqrt{(\vec{R_u} \cdot \vec{R_u}) du^2 + 2(\vec{R_u} \cdot \vec{R_v}) dudv + (\vec{R_v} \cdot \vec{R_v}) dv^2 }\)   ...(2)
   
   Setting \(E=(\vec{R_u} \cdot \vec{R_u})\), \(F=(\vec{R_u} \cdot \vec{R_v})\) and \(G=(\vec{R_v} \cdot \vec{R_v})\) in equation (2) above we get
   
   \(ds = \sqrt{E du^2 + 2 F dudv + G dv^2 }\)   ...(3)
2. Distance between Any 2 Points on Surface is Always measured along a Path specified by a Curve on the Surface. Therefore the Differential Distance between Any 2 Points on Surface Along a Curve \(\vec{R}\) denoted by \(ds\) is given by the Magnitude of the Differential Change in Position Vector \(\vec{dR}\) along the Curve on the Surface as follows
   
   \(ds=|d\vec{R}| = \sqrt{d\vec{R} \cdot d\vec{R}} = \sqrt{(\vec{R_u}{\Large \frac{du}{dt}}dt + \vec{R_v}{\Large \frac{dv}{dt}}dt) \cdot (\vec{R_u}{\Large \frac{du}{dt}}dt + \vec{R_v}{\Large \frac{dv}{dt}}dt)}\)
   
   \(\Rightarrow ds= \sqrt{(\vec{R_u} \cdot \vec{R_u}) {\Large (\frac{du}{dt})}^2 dt^2 + 2(\vec{R_u} \cdot \vec{R_v}) {\Large \frac{du}{dt}}{\Large \frac{dv}{dt}} dt^2 + (\vec{R_v} \cdot \vec{R_v}) {\Large (\frac{dv}{dt})}^2 dt^2 }\)
   
   \(\Rightarrow ds= \sqrt{(\vec{R_u} \cdot \vec{R_u}) {\Large (\frac{du}{dt})}^2 + 2(\vec{R_u} \cdot \vec{R_v}) {\Large \frac{du}{dt}}{\Large \frac{dv}{dt}} + (\vec{R_v} \cdot \vec{R_v}) {\Large (\frac{dv}{dt})}^2 }dt\)   ...(4)
   
   Once again, Setting \(E=(\vec{R_u} \cdot \vec{R_u})\), \(F=(\vec{R_u} \cdot \vec{R_v})\) and \(G=(\vec{R_v} \cdot \vec{R_v})\) in equation (4) above we get
   
   \(\Rightarrow ds= \sqrt{E {\Large (\frac{du}{dt})}^2 + 2 F {\Large \frac{du}{dt}}{\Large \frac{dv}{dt}} + G {\Large (\frac{dv}{dt})}^2 }dt\)   ...(5)
   
   The actual Distance Between 2 Points on the Surface along the Curve (denoted by \(s\)) is found by Integrating the Function given in equation (5) between any 2 Values \(t_1\) and \(t_2\) of Parameter Variable \(t\) as follows
   
   \(s={\Large \int_{t1}^{t2}} ds= {\Large \int_{t1}^{t2}} \sqrt{E {\Large (\frac{du}{dt})}^2 + 2 F {\Large \frac{du}{dt}}{\Large \frac{dv}{dt}} + G {\Large (\frac{dv}{dt})}^2}dt\)   ...(6)
3. The Square of the Differential Distance between Any 2 Points on a Surface (as given in equation (3) above) is called the First Fundamental Form of the Surface and is denoted by \(\mathrm{I}\) as follows
   
   \(\mathrm{I} = ds^2 = E du^2 + 2 F dudv + G dv^2 \)   ...(7)
   
   The quantities \(E\), \(F\) and \(G\) (as mentioned in equations (3), (5), (6) and (7) above) are called the Coefficients of First Fundamental Form of a Surface.
4. The First Fundamental Form of a Surface is also represented in form of a \(2\times 2\) Matrix of its Coefficients as follows
   
   \(\mathrm{I} = \begin{bmatrix}E & F \\ F & G\end{bmatrix} = \begin{bmatrix}\vec{R_u} \cdot \vec{R_u} & \vec{R_u} \cdot \vec{R_v} \\ \vec{R_u} \cdot \vec{R_v} & \vec{R_v} \cdot \vec{R_v}\end{bmatrix}\)   ...(8)
   
   The Determinant of the First Fundamental Form Matrix is calculated as
   
   \(|\hspace{2mm}\mathrm{I}\hspace{2mm}| = \begin{vmatrix}E & F \\ F & G\end{vmatrix} = EG - F^2 = \begin{vmatrix}\vec{R_u} \cdot \vec{R_u} & \vec{R_u} \cdot \vec{R_v} \\ \vec{R_u} \cdot \vec{R_v} & \vec{R_v} \cdot \vec{R_v}\end{vmatrix} =(\vec{R_u} \cdot \vec{R_u})(\vec{R_v} \cdot \vec{R_v}) - (\vec{R_u}\cdot \vec{R_v})^2\)   ...(9)
5. The Differential Area on Surface \(\vec{R}\) denoted by \(dA\) is given by the Magnitude of the Cross Product of Differential Change in Tangent Vectors \(\vec{dR_u}\) and \(\vec{dR_v}\) as follows
   
   \(dA = |d\vec{R_u} \times d\vec{R_v}|= |\vec{R_u}du \times \vec{R_v}dv|\)   ...(10)
   
   \(\Rightarrow dA = \sqrt{(\vec{R_u}du \times \vec{R_v}dv) \cdot (\vec{R_u}du \times \vec{R_v}dv) } \)
   
   \(\Rightarrow dA = \sqrt{(\vec{R_u}du \cdot \vec{R_u}du) (\vec{R_v}dv \cdot \vec{R_v}dv) - (\vec{R_u}du \cdot \vec{R_v}dv)^2 } \)
   
   \(\Rightarrow dA = \sqrt{(\vec{R_u} \cdot \vec{R_u})(\vec{R_v} \cdot \vec{R_v})du^2dv^2 - (\vec{R_u}\cdot \vec{R_v})^2 du^2dv^2 } \)
   
   \(\Rightarrow dA = \sqrt{(\vec{R_u} \cdot \vec{R_u})(\vec{R_v} \cdot \vec{R_v}) - (\vec{R_u}\cdot \vec{R_v})^2 }\hspace{1mm}dudv \)   ...(11)
   
   Equation (11) above can be given using Coefficients of the First Fundamental Form of the Surface as
   
   \(dA = \sqrt{EG - F^2}\hspace{1mm}dudv \)   ...(12)
   
   The actual Area on the Surface \(\vec{R}\) (denoted by \(A\)) is found by Integrating the Function given in equation (12) above bounded by any 2 Values \(u_1\) and \(u_2\) of Parameter Variable \(u\) and any 2 Values \(v_1\) and \(v_2\) of Parameter Variable \(v\) as follows
   
   \(A={\Large \int_{v1}^{v2}}{\Large \int_{u1}^{u2}} dA= {\Large \int_{v1}^{v2}}{\Large \int_{u1}^{u2}} \sqrt{EG - F^2}\hspace{1mm}dudv\)   ...(13)
   
   As given in equation (13) above, Any Area on the Surface \(\vec{R}\) is calculated by Integrating the Square Root of the Determinant of its First Fundamental Form Matrix.