Curves on a Parametric Surface

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 \(v\), then for Any Curve on this Surface, the Parameters \(u\) and \(v\) are themselves Functions of a Single Variable Parameter. Hence, Any Curve on this Surface is given in form of a Parametric Position Vector Function of a Single Variable Parameter \(t\) as

\(\vec{R}=\vec{f(t)}=f_x(u(t),v(t))\hspace{1mm}\hat{\mathbf{i}} + f_y(u(t),v(t))\hspace{1mm}\hat{\mathbf{j}} + f_z(u(t),v(t))\hspace{1mm}\hat{\mathbf{k}}\) ...(2)
Since Vector Function of the Curve \(\vec{R}\) as given in equation (2) has Parameters that are themselves Functions of a Variable, the Tangent/Velocity Vector of the Curve (denoted by \(\vec{R\hspace{1mm}'}\) or \(\vec{V}\)) is calculated using the Multivariable Chain Rule of Derivatives as

\(\vec{V}=\vec{R\hspace{1mm}'}={\Large\frac{d\vec{R}}{dt}}= \begin{bmatrix} {\Large \frac{du}{dt}} & {\Large \frac{dv}{dt}}\end{bmatrix} \begin{bmatrix} {\Large \frac{\partial \vec{R}}{\partial u}} \\ {\Large \frac{\partial \vec{R}}{\partial v}}\end{bmatrix} ={\Large \frac{du}{dt}\frac{\partial \vec{R}}{\partial u}} + {\Large\frac{dv}{dt}\frac{\partial \vec{R}}{\partial v}}= \vec{R_u} {\Large\frac{du}{dt}} + \vec{R_v} {\Large\frac{dv}{dt}}\) ...(3)
Multiplying equation (3) with Differential of Parameter Variable \(t\) (denoted by \(dt\)), we get the Differential Change in Position Vector \(\vec{R}\) along the Curve on the Surface denoted by \(d\vec{R}\) as

\(d\vec{R}= \vec{V} dt= \vec{R\hspace{1mm}'} dt= {\Large \frac{\partial \vec{R}}{\partial u}\frac{du}{dt}}dt + {\Large\frac{\partial \vec{R}}{\partial v}\frac{dv}{dt}}dt= \vec{R_u} {\Large\frac{du}{dt}}dt + \vec{R_v} {\Large\frac{dv}{dt}}dt\) ...(4)
The Unit Vector Normal to the Tangent Plane (denoted by \(\hat{R_n}\)) is also a Function of 2 Variable Parameters \(u\) and \(v\). However, Since along the Curve the Parameters \(u\) and \(v\) are themselves Functions of a Variable \(t\) , the Rate of Change of Unit Vector Normal to the Tangent Plane of the Surface along the Curve is calculated using the Multivariable Chain Rule of Derivatives as

\(\hat{R_n\hspace{1mm}'}={\Large\frac{d\hat{R_n}}{dt}}= \begin{bmatrix} {\Large \frac{du}{dt}} & {\Large \frac{dv}{dt}}\end{bmatrix} \begin{bmatrix} {\Large \frac{\partial \hat{R_n}}{\partial u}} \\ {\Large \frac{\partial \hat{R_n}}{\partial v}}\end{bmatrix} ={\Large \frac{du}{dt}\frac{\partial \hat{R_n}}{\partial u}} + {\Large\frac{dv}{dt}\frac{\partial \hat{R_n}}{\partial v}}=\vec{R_{n_{u}}}{\Large \frac{du}{dt}} + \vec{R_{n_{v}}}{\Large \frac{dv}{dt}}\) ...(5)
Multiplying equation (13) with Differential of Parameter Variable \(t\) (denoted by \(dt\)), we get the Differential Change in Unit Vector Normal to the Tangent Plane of the Surface \(\hat{R_n}\) along the Curve denoted by \(d\hat{R_n}\) as

\(d\hat{R_n}= \hat{R_n\hspace{1mm}'} dt= {\Large \frac{\partial \hat{R_n}}{\partial u}\frac{du}{dt}}dt + {\Large\frac{\partial \hat{R_n}}{\partial v}\frac{dv}{dt}}dt=\vec{R_{n_{u}}}{\Large \frac{du}{dt}}dt + \vec{R_{n_{v}}}{\Large \frac{dv}{dt}}dt\) ...(6)
The Acceleration Vector of the Curve (denoted by \(\vec{R\hspace{1mm}''}\) or \(\vec{A}\)) given by the Derivative of the Tangent/Velocity Vector as

\(\vec{A}=\vec{R\hspace{1mm}''}={\Large\frac{d\vec{V}}{dt}}={\Large\frac{d\vec{R\hspace{1mm}'}}{dt}}={\Large\frac{d^2\vec{R}}{dt^2}}= {\Large \frac {d (\vec{R_u} {\Large\frac{du}{dt}} + \vec{R_v} {\Large\frac{dv}{dt}}) }{dt}} = (\vec{R_u} {\Large\frac{du}{dt}} + \vec{R_v} {\Large\frac{dv}{dt}})' \)   ...(7)

\(\Rightarrow \vec{A}=\vec{R\hspace{1mm}''}= \vec{R_u\hspace{1mm}'} {\Large\frac{du}{dt}} + \vec{R_u}({\Large\frac{du}{dt}})' + \vec{R_v\hspace{1mm}'} {\Large\frac{dv}{dt}} + \vec{R_v}({\Large\frac{dv}{dt}})'\)   ...(8)

Since \(\vec{R_u}\) and \(\vec{R_v}\) are both functions of Variables \(u\) and \(v\), which are themselves functions of Variable \(t\), \(\vec{R_u\hspace{1mm}'}\) and \(\vec{R_v\hspace{1mm}'}\) get calculated using Multivariable Chain Rule of Derivatives as

\(\vec{R_u\hspace{1mm}'} = {\Large \frac{d \vec{R_u}}{dt}} = \begin{bmatrix} {\Large \frac{du}{dt}} & {\Large \frac{dv}{dt}}\end{bmatrix} \begin{bmatrix} {\Large \frac{\partial \vec{R_u}}{\partial u}} \\ {\Large \frac{\partial \vec{R_u}}{\partial v}}\end{bmatrix} ={\Large \frac{du}{dt}\frac{\partial \vec{R_u}}{\partial u}} + {\Large\frac{dv}{dt}\frac{\partial \vec{R_u}}{\partial v}} =\vec{R_{uu}} {\Large\frac{du}{dt}} + \vec{R_{uv}} {\Large\frac{dv}{dt}}\)   ...(9)

\(\vec{R_v\hspace{1mm}'} = {\Large \frac{d \vec{R_v}}{dt}} = \begin{bmatrix} {\Large \frac{du}{dt}} & {\Large \frac{dv}{dt}}\end{bmatrix} \begin{bmatrix} {\Large \frac{\partial \vec{R_v}}{\partial u}} \\ {\Large \frac{\partial \vec{R_v}}{\partial v}}\end{bmatrix} ={\Large \frac{du}{dt}\frac{\partial \vec{R_v}}{\partial u}} + {\Large\frac{dv}{dt}\frac{\partial \vec{R_v}}{\partial v}} =\vec{R_{vu}} {\Large\frac{du}{dt}} + \vec{R_{vv}} {\Large\frac{dv}{dt}}\)   ...(10)

Also,

\(({\Large\frac{du}{dt}})' = {\Large\frac{d}{dt}}({\Large\frac{du}{dt}})={\Large\frac{d^2u}{dt^2}}\)   ...(11)

\(({\Large\frac{dv}{dt}})' = {\Large\frac{d}{dt}}({\Large\frac{dv}{dt}})={\Large\frac{d^2v}{dt^2}}\)   ...(12)

Substituting the values of \(\vec{R_u\hspace{1mm}'}\), \(\vec{R_v\hspace{1mm}'}\), \(({\Large\frac{du}{dt}})'\) and \(({\Large\frac{dv}{dt}})'\) from equations (9), (10), (11) and (12) in equation (8) we get

\(\vec{A}=\vec{R\hspace{1mm}''}= (\vec{R_{uu}} {\Large\frac{du}{dt}} + \vec{R_{uv}} {\Large\frac{dv}{dt}}) {\Large\frac{du}{dt}} + \vec{R_u}{\Large\frac{d^2u}{dt^2}} + (\vec{R_{vu}} {\Large\frac{du}{dt}} + \vec{R_{vv}} {\Large\frac{dv}{dt}}) {\Large\frac{dv}{dt}} + \vec{R_v}{\Large\frac{d^2v}{dt^2}}\)

\(\Rightarrow \vec{A}=\vec{R\hspace{1mm}''}= \vec{R_{uu}} ({\Large\frac{du}{dt}})^2 + \vec{R_{uv}} {\Large\frac{dv}{dt}}{\Large\frac{du}{dt}} + \vec{R_{vu}} {\Large\frac{du}{dt}}{\Large\frac{dv}{dt}} + \vec{R_{vv}} ({\Large\frac{dv}{dt}})^2 + \vec{R_u}{\Large\frac{d^2u}{dt^2}} + \vec{R_v}{\Large\frac{d^2v}{dt^2}}\)

\(\Rightarrow \vec{A}=\vec{R\hspace{1mm}''}= \vec{R_{uu}} ({\Large\frac{du}{dt}})^2 + 2\vec{R_{uv}} {\Large\frac{du}{dt}}{\Large\frac{dv}{dt}} + \vec{R_{vv}} ({\Large\frac{dv}{dt}})^2 + \vec{R_u}{\Large\frac{d^2u}{dt^2}} + \vec{R_v}{\Large\frac{d^2v}{dt^2}}\)    (Since \(\vec{R_{uv}}= \vec{R_{vu}}\))   ...(13)

Setting \(u'={\Large\frac{du}{dt}}\), \(v'={\Large\frac{dv}{dt}}\), \(u''={\Large\frac{d^2u}{dt^2}}\) and \(v''={\Large\frac{d^2v}{dt^2}}\) in equation (13) we get

\(\vec{A}=\vec{R\hspace{1mm}''}= \vec{R_{uu}} (u')^2 + 2\vec{R_{uv}} u'v' + \vec{R_{vv}} (v')^2 + \vec{R_u}(u'')^2 + \vec{R_v}(v'')^2\)   ...(14)
The equations (11) and (12) above give the formula for Acceleration Vector of a Curve on a Parametric Surface.

The terms \(\vec{R_u}(u'')^2\), \(\vec{R_v}(v'')^2\) and their Sum (\(\vec{R_u}(u'')^2 + \vec{R_v}(v'')^2\)) all lie on the Tangent Plane of the Surface and hence are Pure Tangential Components of the Acceleration Vector of the Curve on the Surface.

The terms \(\vec{R_{uu}} (u')^2\), \(2\vec{R_{uv}} u'v'\), \(\vec{R_{vv}} (v')^2\) and their Sum (\(\vec{R_{uu}} (u')^2 + 2\vec{R_{uv}} u'v' + \vec{R_{vv}} (v')^2\)) all lie Somewhere Between the Tangent Plane of the Surface and the Normal to the Surface and hence are Mixed Components of the Acceleration Vector of the Curve on the Surface.

Home | TOC | Calculators