MIT
Department of Ocean Engineering

13.016 Introduction to Geometric Modeling and Computation

Spring 1997

Problem Set 1

Assigned: February 6, 1997
Due: February 13, 1997

  1. The upper longitudinal profile curve of an AUV is represented by two cubic planar parametric curves and , where and

    1. Plot the two cubic planar parametric curves using N equally distributed points in parameter space, where N=11,101,1001.
    2. Plot the parametric speed of the two curves.
    3. Approximate the length of the two curves using the three polygonal approximations.
    4. Compute the tangent vectors at u=0 and 1, v=0 and 1 for the two curves.
    5. Find the locations where and become maximum.
    6. Plot the curvature in terms of parameter u,v for the two curves for N=101.
    7. Using the plot values of question (f) estimate the parametric value which gives absolute maximum and minimum of curvature for both curves (using an algorithm for finding the maximum and minimum of N numbers for N=101).

    Write C programs to do the calculation, and use MATLAB on Athena for your plots.

  2. Prove that

    where