Computer Aided Engineering Design

94 COMPUTER AIDED ENGINEERING DESIGN

To obtain C^2 continuous curve, Eq. (4.17) may be employed to get the intermediate slopes. Using
the end slopes as T 0 = (1, 1) and T 3 = (1, –1), the system in Eq. (4.17) results in

(1, 1) + 4T 1 + T 2 = 3(3, 2) – 3(0, 0)

T 1 + 4T 2 + (1, –1) = 3(6, –1) – 3(1, 2)

4 T 1 + T 2 = 3(3, 2) – 3(0, 0) – (1, 1) = (8, 5)

or T 1 + 4T 2 = 3(6, – 1) – 3(1, 2) – (1, –1) = (14, – 8)

We can solve the above system individually for x and y components of the slopes to get T 1 =
(1.2, 1.87) and T 2 = (3.2, – 2.47). The process of obtaining the individual segments is then identical
to that described for C^1 continuous Ferguson curves. The polynomials for the curve segments are

r 11 (uuuuuuu) = [0.2 13 – 0.4 12 + , – 1.13 1 13 + 2.13 12 + ] 1

r 22 (uuuu u uu) = [0.4 23 + 0.4^22 + 1.2 2 + 1, – 0.6^32 – 1.27 22 + 1.87 + 2] 2

r 33 (uuuu uuu) = [–1.8 33 + 1.6^23 + 3.2 3 + 3, 2.53 – 3.06^3332 – 2.47 + 2] 3

For data point C(3, 2) to be relocated to (1.5, 4), the intermediate tangents must be re-computed. Thus,

4 T 1 + T 2 = 3(1.5, 4) – 3(0, 0) – (1, 1) = (3.5, 11)

T 1 + 4T 2 = 3(6, –1) – 3(1, 2) – (1, –1) = (14, – 8)

the solution for which is T 1 = (0.0, 3.47) and T 2 = (3.5, – 2.87). Using these slopes, Eq. (4.17) can
be employed to generate the new Ferguson segments as

r 1 new(uuuuuuu 1 ) = [–3.5 13 + 3.5 12 + , 0.47 1 13 + 0.53 12 + 1 ]

r 2 new(uuu uuu 2 ) = [2.5^32 – 2.0 22 + 1, – 3.4^32 + 1.93 22 + 3.47 2 + 2]

r 3 new(uuuu uuu 3 ) = [– 4.5^33 + 5.5^23 + 3.5 + 1.5, 6.13 3 33 – 8.26 32 – 2.87 3 + 4]

Figure 4.5 (b) shows C^2 continuous composite curves before (solid lines) and after (dashed lines)
pointC(3, 2) is moved to a new location. The recorded change in curve shape is global.
When a data point (or a slope vector) in a C^1 continuous composite Ferguson curve is altered, a
maximum of two segments having that data point (or slope) at the junction get reshaped. In other
words, a C^1 continuous composite Ferguson curve possesses local shape control properties. For a C^2
continuous curve, however, altering a data point requires re-computing the slopes using Eq. (4.17).
There is an overall change in the composite curve.

4.1.2 Curve Trimming and Re-parameterization

In many design situations, a user may require to trim a curve that has been sketched. Commonly,
trimming is performed at the intersection of two curves. Figure 4.6 shows a segment to be trimmed
with the geometric matrix G as [Pi Pi+1TiTi+1]T. Let trimming be performed at u = ui (0 < ui < 1) and
u = uj(0 < uj < 1). The segments 0 ≤u < ui and uj < u≤ 1 are to be removed. The resultant curve BC
lies in the parametric interval ui≤u≤uj. Sometimes, it is useful to express this trimmed curve using
a new parameter interval 0 ≤v≤ 1.