Computer Aided Engineering Design

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