- Compute the curvature, normal vector or binormal vector of the
f(u) = ( u, 1 + u2,
u + u2 )
- Consider the following two curve segments with the origin their
f(u) = ( u, -u2, 0 )
where u and v are in [-1,0] and [0,1], respectively,
Are they C1, G1,
at the origin? Are they curvature continuous?
g(v) = ( v, 0, v2 )
- Consider the following two circular arcs joining at the origin:
f(u) = ( cos(u + PI/2),
-(1 + sin(u + PI/2)), 0 )
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.
g(v) = ( -cos(v + PI/2), 0,
1 - sin(v + PI/2) )
- 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
+ (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?
- Analyze the relationship between u and the two branches of
the hyperbola parameterized with the following:
x = a (1 + u2) / (2u)
Plot several points that correspond to different u will be
y = b (1 - u2) / (2u)
- 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
- 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?