Problems

  1. Compute the curvature, normal vector or binormal vector of the following parabola:
    f(u) = ( u, 1 + u2, u + u2 )
  2. Consider the following two curve segments with the origin their joining point:
    f(u) = ( u, -u2, 0 )
    g(v) = ( v, 0, v2 )
    where u and v are in [-1,0] and [0,1], respectively, Are they C1, C2 or G2 continuous at the origin? Are they curvature continuous?
  3. Consider the following two circular arcs joining at the origin:
    f(u) = ( cos(u + PI/2), -(1 + sin(u + PI/2)), 0 )
    g(v) = ( -cos(v + PI/2), 0, 1 - sin(v + PI/2) )
    where both u and v are in the range of 0 and PI. Note that circular arcs f(u) and g(v) lie on the xy- and xz-coordinate planes, respectively. Analyze the continuity at the origin.
  4. Eliminate the parameter u from the following equations:
    x = f(u) = au2 + bu + c
    y = g(u) = pu2 + qu + r
    What is the type of the curve? It must be a conic; but which one? Some calculations is necessary. (Hint: If p is not zero, then solving for u from the second equation and plugging the result to the first equation would eliminate u. If both a and p are non-zero, multiplying the first and the second equations by p and a, respectively, and subtracting the second from the first will eliminate u2. From this result, solving for u and plugging the result back to one of the original equation would eliminate u completely. Thus, you will obtain a polynomial p(x,y) = 0. What does it represent? Why?
  5. The ellipse with center at (p, q), axes parallel to the coordinate axes, and semi-major and semi-minor axis lengths a and b has an equation
    (x-p)2/a2 + (y-q)2/b2 = 1
    It can be parameterized with trigonometric functions by x = a cos(t) + p and y = b sin(t) + q. Please verify this relation. Convert this trigonometric parameterization to a rational one. Does your parameterization contain circles as special cases?
  6. Analyze the relationship between u and the two branches of the hyperbola parameterized with the following:
    x = a (1 + u2) / (2u)
    y = b (1 - u2) / (2u)
    Plot several points that correspond to different u will be very helpful.