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

1. Given a Curve in form of a Parametric Position Vector Function as
   
   \(\vec{R}=\vec{f(t)}=f_x(t)\hspace{1mm}\hat{\mathbf{i}} + f_y(t)\hspace{1mm}\hat{\mathbf{j}} + f_z(t)\hspace{1mm}\hat{\mathbf{k}}\)   ...(1)
   
   where \(f_x(t)\), \(f_y(t)\) and \(f_z(t)\) are Functions of a Single Variable Parameter \(t\), and its Tangent Vector given as
   
   \(\vec{R\hspace{1mm}'}={\Large\frac{d \vec{R}}{dt}}={\Large\frac{d (\vec{f(t)})}{dt}} = {\Large \frac{d (f_x(t))}{dt}}\hat{\mathbf{i}} + {\Large \frac{d (f_y(t))}{dt}}\hat{\mathbf{j}} + {\Large \frac{d (f_z(t))}{dt}}\hat{\mathbf{k}} =\vec{f\hspace{1mm}'(t)}= f_x\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{i}} + f_y\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{j}} + f_z\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{k}}\)   ...(2)
   
   then Any Perpendicular Projection from Any Point having Position Vector \(\vec{P}\) given by Coordinates (\(x_0,y_0,z_0\)) on the Curve will be Perpendicular to the Tangent to the Curve at the Point of Projection. That means at the Point of Projection
   
   \((\vec{R} - \vec{P}).\vec{R\hspace{1mm}'}=0\)
   
   \(\Rightarrow ((f_x(t) - x_0)\hspace{1mm}\hat{\mathbf{i}} +(f_y(t) - y_0)\hspace{1mm}\hat{\mathbf{j}} + (f_z(t) - z_0)\hspace{1mm}\hat{\mathbf{k}}) \cdot (f_x\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{i}} + f_y\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{j}} + f_z\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{k}}) = 0\)
   
   \(\Rightarrow (f_x(t) - x_0)f_x\hspace{1mm}'(t) +(f_y(t) - y_0)f_y\hspace{1mm}'(t) + (f_z(t) - z_0)f_z\hspace{1mm}'(t)=0\)   ...(3)
   
   
2. Perpendicular Projections of \(\vec{P}\) on \( \vec{R}\) can be found out by First Solving equation (3) given above for One or More Values of \(t\) and then Finding the Corresponding Points of Projection on the Curve by Putting those Values of \(t\) in equation (1).
   
   If equation (3) is Non Linear, then Newton Raphson Method can be used for finding value(s) of \(t\).
3. The Distance \(D\) of \(\vec{P}\) from each of the Projection Point on the Curve can be calculated by using the Distance Formula as follows
   
   \(D=| \vec{R} - \vec{P} | = \sqrt{(f_x(t) - x_0)^2 +(f_y(t) - y_0)^2 + (f_z(t) - z_0)^2}\)   ...(4)
   
   The one with the Minimum Value of Distance \(D\) is the Distance Between the Point and the Curve.
4. The equation (3) can also be obtained by Equating the Derivative of Square of the Distance of Point from the Curve with respect to the Parameter Variable \(t\) with 0 as follows
   
   \(D^2=| \vec{R} - \vec{P} |^2 = (f_x(t) - x_0)^2 +(f_y(t) - y_0)^2 + (f_z(t) - z_0)^2\)
   
   \(\Rightarrow {\Large \frac{dD^2}{dt}}= 2(f_x(t) - x_0)f_x\hspace{1mm}'(t) +2(f_y(t) - y_0)f_y\hspace{1mm}'(t) + 2(f_z(t) - z_0)f_z\hspace{1mm}'(t)\)   ...(5)
   
   Equating equation (5) with \(0\) we get
   
   \(2(f_x(t) - x_0)f_x\hspace{1mm}'(t) +2(f_y(t) - y_0)f_y\hspace{1mm}'(t) + 2(f_z(t) - z_0)f_z\hspace{1mm}'(t)=0\)   ...(6)
   
   Dividing equation (6) by \(2\) on both sides we get
   
   \((f_x(t) - x_0)f_x\hspace{1mm}'(t) + (f_y(t) - y_0)f_y\hspace{1mm}'(t) + (f_z(t) - z_0)f_z\hspace{1mm}'(t)=0\)   ...(Same as equation (3) above)