MA 227 Standard S09

Surface Parametrization

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)