\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard S09

Surface Parametrization
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At the end of the course, each student should be able to…

  • S09: ParamSurf. Parametrize surfaces in three-dimensional Euclidean space.

S09: Surface Parametrization

  • Just like curves, surfaces have orientation.
    • The orientation of a surface is given by which side is “top” and which is “bottom”, identified as a continuous vector field of normal unit vectors.
    • Some surfaces, such as the Mobius strip, are non-orientable.
  • A surface parameterization maps parameters \(u,v\) to points/vectors \(\vect r(u,v)\) on a surface in 3D space.
    • The orientation of a surface parametrized by \(\vect r(u,v)\) is given by \(\frac{\vect r_u\times\vect r_v}{\|\vect r_u\times\vect r_v\|}\).
    • Surfaces defined by \(z=f(x,y)\) and oriented upwards may be parameterized by \(\vect r(u,v)=\<u,v,f(u,v)\>\).
    • The plane passing through \(P_0\), parallel to the vectors \(\vect w_1,\vect w_2\), and oriented by the right-hand rule on \(\vect w_1,\vect w_2\) may be parameterized by \(\vect r(u,v)=P_0+u\vect w_2+v\vect w_2\).
    • Surfaces defined by cylindrical/spherical equations may be parameterized by substituting into the appropriate coordinate transformation. For instance, the cone \(z=\sqrt{x^2+y^2}\) is equivalent to \(z=r\) in cylindrical and \(\phi=\frac{\pi}{4}\) in spherical, so \(\vect r(r,\theta)= \vect c(r,\theta,r)=\<r\cos\theta,r\sin\theta,r\>\) and \(\vect r(\rho,\theta)=\vect s(\rho,\pi/4,\theta)= \<\rho\sin(\pi/4)\cos\theta, \rho\sin(\pi/4)\sin\theta,\rho\cos(\pi/4)\>\) are possible parameterizations. Orientations may be checked by inspecting \(\vect r_u,\vect r_v\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 15.5 (exercises 1-16)