DIFFERENTIAL GEOMETRY OF CURVES 75
For a generic curve seqment, the scalar functions x(u),y(u) and z(u) are preferred to be polynomials
of a lower degree. With regard to position, slope and/or curvature continuity of a composite curve
overall, differential properties of curves in parametric form are discussed below.
3.4 Differential Geometry of Curves
Consider two closely adjacent points P(r(u)) and Q(r(u+Δu)) on a parametric curve r=r(u) in
Figure 3.6, Δu, the change in parameter being small. The length of the segment Δs between P and Q
may be approximated by the chord length |Δr| = |r(u+Δu) – r(u)|. Taylor series expansion gives
rr
r r
( + ) = ( ) + +^1
2!
( ) +... higher order terms...
2
2
uu ud^2
du
u d
du
ΔΔΔu (3.18)
For very small Δu, only the first order term may be retained. Thus
ΔΔsuuuΔ Δ
d
du
| ≈≈rr | = | ( + ) – ( ) | r u
r
(3.19)
Figure 3.5 Viviani’s curve shown in one octant
Figure 3.6 Parametric curve represented in vector form
y
x
z
P Q
T
r(u)
Δu
r(u + Δu)
x
z
y
Trace of cylinder on
x-y plane
Profile of the sphere
Curve of intersection between a
cylinder and sphere