Estimating the area of a surface

Art Busch
Area, Surface
  • Fixed values of the parameters u and v determine a point Q on the surface.
  • Fixing u but not v creates a parametric curve through Q that lies on the surface, and fixing v but not u creates another curve on the surface through Q.
  • Choosing two small step sizes defines two additional curves that enclose a small region on the surface.
  • By using the step sizes to scale the tangent vectors, we can define a parallelogram in the plane tangent to S at Q.
  • The area of that parallelogram closely approximates the area of the enclosed region on the surface.
  • We can calculate the area of this parallelogram using cross products:

  • Summing over all the "cells" of the surface gives an estimate of the total surface ares:

  • Taking the limit as Δu and Δv go to zero gives: