Lecture No.36 Space Curves
A space curve is not confined to a plane. It is free to twist through space. To define a
space curve we must use parametric functions that are cubic polynomials. For x (u) we
write
xxxx ducubuaux 23 (1)
With similar expressions for y(u) and z(u). Again the a, b, c and d terms are constant
coefficients. As we did with Equation for a plane curve, we combine the x(u), y(u) , and
z(u) expressions into a single vector equation :
dcubuauup 23 (2)
If a = 0, then this equation is identical to Equation discussed in plane curves
To define a specific curve in space, we use the same approach as we did for a plane curve.
This time, though, there are 12 coefficients to be determined. We specify four points
through which we want the curve to pass, which provides all the information we need to
determine a, b, c, and d. but which four points? Two are obvious:
0p and
1p, the end
points at 0 u and 1 u. For various reasons beyond our scope, it turns out to be
advantageous to use two intermediate points that we assign parametric values of 3
1
u
and 3
2
u, or ¸
¹
·
¨
©
§
3
1
p and ¸
¹
·
¨
©
§
3
2
p. So we now have the four points we need:
Figure 1 Four points define a cubic space curve
0p,¸
¹
·
¨
©
§
3
1
p and ¸
¹
·
¨
©
§
3
2
p and
1p, which we can rewrite as the more convenient p1, p2,
p3, and p4 (Figure 1).
Substituting each of the values of the parametric variable
1,32,31,0 u into
Equation 1, we obtain the following four equations in four unknowns:
x
dx
1
xxxx dcbax 3
1
9
1
27
1
2
xxxx dcbax 3
2
9
4
27
8
3 (3)
x
y
z
P1
P1
/3
P
0
P2
/3
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