DESIGN OF CURVES 101

andc lie on the parabola as well. To find the tangents at a and c, Eq. (4.27) may be differentiated with
respect to uas

dr/du = –2(1 – u)a + (2 – 4u)d + 2uc (4.28)

using which dr/du at u= 0 is 2(d – a) and that at u= 1 is 2(c–d). Thus, adanddcare tangents to
the parabola at aandc, respectively. Rearranging Eq. (4.28) yields

dr/du = –2{(1 – u)a + ud} + 2{(1 – u)d + uc}

= –2e + 2f (4.29)

This implies that efis a tangent to the parabola at rfor some value of u. Thus, the three tangent
theorem for a parabola is verified and in the process, a procedure for constructing a parabola with
three given points (a,d and c) is evolved. The construction which is known as the de Casteljau’s
algorithm involves two levels of repeated linear interpolation given by Eqs. (4.25) and (4.26).

4.2.1 Generalized de Casteljau’s Algorithm

The above algorithm can be generalized for use with n+ 1 data points to generate a curve of degree
n. Given data points b 0 ,b 1 ,... , bn, compute bij such that

bbbij = (1 – )uujninjuij–1 + ij+1–1, = 1,... , ; = 0,... , – ; [0, 1]∈ (4.30)

4.2 Three-Tangent Theorem

Stated without proof, consider three points a,r and c on a parabola. Let the tangents at aandc
intersect at d. Also, let the tangent at rintersect the previous two tangents at eandf, respectively. Then

| |

| |

= | |

| |

=

| |

| |

ae

ed

er

rf

df

fc

(4.24)

Based on this theorem, a parabola may be constructed to verify the aforementioned conditions. Let
pointe be chosen on adsuch that

e = (1 – u)a + ud for some u ∈ [0, 1] (4.25)

This implies | |
| |

1 –

ae ,

ed

u

u

so that Eq. (4.24) is satisfied. Choose f and r on dc and ef, respectively,

such that

Figure 4.11 Geometric construction of a parabola

a

r

c

u

u

u

d

f

1 – u

1 – u

1 – u

f = (1 – u)d + uc e

r = (1 – u)e + uf(4.26)

Substituting for e and f in Eq. (4.26) in terms
ofa,c and d, we have

r= (1 – u) {(1 – u)a + ud} + u{(1 – u)d + uc}

= (1 – u)^2 a + 2u (1 – u)d + u^2 c (4.27)

This is the equation of a parabola on which r lies
for some parameter value of u. From Eq. (4.27),
atu= 0, r = a and at u= 1, r=c implying that a