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, G¹, 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. The ellipse with center (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 result. Convert this trigonometric parameterization to a rational one. Does your parameterization contain circles as special cases?
5. 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.
6. We showed in Rational Curves that a quadric polynomial parametric form can only represent a parabola through calculation. There are some minor flaws in the calculation. Please fill these gaps by answering the following questions:
   • We assume that a and p are both non-zero. What would happen if a and p are zero? What curve will you get?
   • What if only one of a and p, say a, is non-zero? What curve will you get?
   • In solving for u, the denominator is bp-aq. What if bp-aq is zero? Note that bp-aq=0 is equivalent to a/p=b/q? What curve will you get?