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Distance/Projection of a Point to/on a Parametric Surface

1. Given a Surface in form of a Parametric Position Vector Function as
   
   \(\vec{R}=\vec{f(u,v)}=f_x(u,v)\hspace{1mm}\hat{\mathbf{i}} + f_y(u,v)\hspace{1mm}\hat{\mathbf{j}} + f_z(u,v)\hspace{1mm}\hat{\mathbf{k}}\)   ...(1)
   
   where \(f_x(u,v)\), \(f_y(u,v)\) and \(f_z(u,v)\) are Functions of 2 Variable Parameters \(u\) and \(v\), and its Tangent Vectors given as
   
   \(\vec{R_u}={\Large\frac{\partial \vec{R}}{\partial u}}={\Large\frac{\partial (\vec{f(u,v)})}{\partial u}} = {\Large \frac{\partial (f_x(u,v))}{\partial u}}\hat{\mathbf{i}} + {\Large \frac{\partial (f_y(u,v))}{\partial u}}\hat{\mathbf{j}} + {\Large \frac{\partial (f_z(u,v))}{\partial u}}\hat{\mathbf{k}} =\vec{f_u(u,v)}= f_{xu}(u,v)\hspace{1mm}\hat{\mathbf{i}} + f_{yu}(u,v)\hspace{1mm}\hat{\mathbf{j}} + f_{zu}(u,v)\hspace{1mm}\hat{\mathbf{k}}\)   ...(2)
   
   \(\vec{R_v}={\Large\frac{\partial \vec{R}}{\partial v}}={\Large\frac{\partial (\vec{f(u,v)})}{\partial v}} = {\Large \frac{\partial (f_x(u,v))}{\partial v}}\hat{\mathbf{i}} + {\Large \frac{\partial (f_y(u,v))}{\partial v}}\hat{\mathbf{j}} + {\Large \frac{\partial (f_z(u,v))}{\partial v}}\hat{\mathbf{k}} =\vec{f_v(u,v)}= f_{xv}(u,v)\hspace{1mm}\hat{\mathbf{i}} + f_{yv}(u,v)\hspace{1mm}\hat{\mathbf{j}} + f_{zv}(u,v)\hspace{1mm}\hat{\mathbf{k}}\)   ...(3)
   
   then Any Perpendicular Projection from Any Point having Position Vector \(\vec{P}\) given by Coordinates (\(x_0,y_0,z_0\)) on the Surface will be Perpendicular to the Tangents of the Surface at the Point of Projection. That means at the Point of Projection
   
   \((\vec{R} - \vec{P}).\vec{R_u}=0\)
   
   \(\Rightarrow ((f_x(u,v) - x_0)\hspace{1mm}\hat{\mathbf{i}} +(f_y(u,v) - y_0)\hspace{1mm}\hat{\mathbf{j}} + (f_z(u,v) - z_0)\hspace{1mm}\hat{\mathbf{k}}) \cdot (f_{xu}(u,v)\hspace{1mm}\hat{\mathbf{i}} + f_{yu}(u,v)\hspace{1mm}\hat{\mathbf{j}} + f_{zu}(u,v)\hspace{1mm}\hat{\mathbf{k}}) = 0\)
   
   \(\Rightarrow (f_x(u,v) - x_0)f_{xu}(u,v) +(f_y(u,v) - y_0)f_{yu}(u,v) + (f_z(u,v) - z_0)f_{zu}(u,v)=0\)   ...(3)
   
   Also, at the Point of Projection
   
   \((\vec{R} - \vec{P}).\vec{R_v}=0\)
   
   \(\Rightarrow ((f_x(u,v) - x_0)\hspace{1mm}\hat{\mathbf{i}} +(f_y(u,v) - y_0)\hspace{1mm}\hat{\mathbf{j}} + (f_z(u,v) - z_0)\hspace{1mm}\hat{\mathbf{k}}) \cdot (f_{xv}(u,v)\hspace{1mm}\hat{\mathbf{i}} + f_{yv}(u,v)\hspace{1mm}\hat{\mathbf{j}} + f_{zv}(u,v)\hspace{1mm}\hat{\mathbf{k}}) = 0\)
   
   \(\Rightarrow (f_x(u,v) - x_0)f_{xv}(u,v) +(f_y(u,v) - y_0)f_{yv}(u,v) + (f_z(u,v) - z_0)f_{zv}(u,v)=0\)   ...(4)
   
   
2. Perpendicular Projections of \(\vec{P}\) on \( \vec{R}\) can be found out by First Solving System of Equations give by equation (3) and (4) given above for One or More Set of Values of \(u\) and \(v\) and then Finding the Corresponding Points of Projection on the Surface by Putting those Values of \(u\) and \(v\) in equation (1).
   
   If equations (3) and (4) form a System of Non Linear Equations, then Newton Raphson Method can be used for finding value(s) of \(u\) and \(v\).
3. The Distance \(D\) of \(\vec{P}\) from each of the Projection Point on the Surface can be calculated by using the Distance Formula as follows
   
   \(D=| \vec{R} - \vec{P} | = \sqrt{(f_x(u,v) - x_0)^2 +(f_y(u,v) - y_0)^2 + (f_z(u,v) - z_0)^2} \)   ...(5)
   
   The one with the Minimum Value of Distance \(D\) is the Distance Between the Point and the Surface.
4. The System of Equations given by equations (3) and (4) can also be obtained by Equating the Partial Derivatives of Square of the Distance of Point from the Surface with respect to the Parameter Variables \(u\) and \(v\) with 0 as follows
   
   \(D^2=| \vec{R} - \vec{P} |^2 = (f_x(u,v) - x_0)^2 +(f_y(u,v) - y_0)^2 + (f_z(u,v) - z_0)^2\)
   
   \(\Rightarrow {\Large \frac{\partial D^2}{\partial u}}= 2(f_x(u,v) - x_0)f_{xu}(u,v) +2(f_y(u,v) - y_0)f_{yu}(u,v) + 2(f_z(u,v) - z_0)f_{zu}(u,v)\)   ...(6)
   
   And \(\Rightarrow {\Large \frac{\partial D^2}{\partial v}}= 2(f_x(u,v) - x_0)f_{xv}(u,v) +2(f_y(u,v) - y_0)f_{yv}(u,v) + 2(f_z(u,v) - z_0)f_{zv}(u,v)\)   ...(7)
   
   Equating equations (6) and (7) with \(0\) we get
   
   \(2(f_x(u,v) - x_0)f_{xu}(u,v) +2(f_y(u,v) - y_0)f_{yu}(u,v) + 2(f_z(u,v) - z_0)f_{zu}(u,v)=0\)   ...(8)
   
   \(2(f_x(u,v) - x_0)f_{xv}(u,v) +2(f_y(u,v) - y_0)f_{yv}(u,v) + 2(f_z(u,v) - z_0)f_{zv}(u,v)=0\)   ...(9)
   
   Dividing equations (8) and (9) by \(2\) on both sides we get
   
   \((f_x(u,v) - x_0)f_{xu}(u,v) +(f_y(u,v) - y_0)f_{yu}(u,v) + (f_z(u,v) - z_0)f_{zu}(u,v)=0\)   ...(Same as equation (3) above)
   
   \((f_x(u,v) - x_0)f_{xv}(u,v) +(f_y(u,v) - y_0)f_{yv}(u,v) + (f_z(u,v) - z_0)f_{zv}(u,v)=0\)   ...(Same as equation (4) above)