Computer Aided Engineering Design

306 COMPUTER AIDED ENGINEERING DESIGN

(a) Point cloud (b) Approximating surface (c) Error plot

Figure 10.13 An approximating B-spline patch for a segmented point cloud.

discussed above. The second is to fit B-spline curves at the boundaries of the segmented cloud and
later define a Coon’s patch using the same. Following the concepts developed in Chapters 5 and 7,
givenM control vertices Bi (i = 0, 1,...,M−1), a B-spline curve of order k is defined as

rB( ) = ( ), [ , ]

=0

–1

uNuuuu,+ min max

i

M

Σ i kk i ∈ (10.2)

Let a smooth parametric curve r(u) defined by the above equation pass through a sequence of data
points {Pi,i= 0,..., j}. If a data point lies on the B-spline curve, it must satisfy Eq. (10.2). Writing
the same for each of j data points yields

P 1 (u 1 ) = Nk,k(u 1 )B 0 + Nk,k+1(u 1 )B 1 +... + Nk,k+M–1(u 1 )BM–1

P 2 (u 2 ) = Nk,k(u 2 )B 0 + Nk,k+1(u 2 )B 1 +... + Nk,k+M–1(u 2 )BM–1

....
Pj(uj) = Nk,k(uj)B 0 + Nk,k+1(uj)B 1 +... + Nk,k+M–1(uj)BM–1 (10.3)

where 2 ≤k≤M≤j. This set of equations is written in matrix form as

P = CB (10.4)

with C

=

() ()

() ()

, 1 ,+ 1 1

,,+1

Nu N u

Nu N u

kk k k M

kk j kk M j

LL

MM M

MMM

LL

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

whereP,CandB are the point data, basis and defining polygon matrices, respectively. In case of
curve approximation, C is not a square matrix. The problem is over-specified and can be solved using
some mean sense. Noting that a matrix times its transpose is square, the defining polygon vertices for
a B-spline curve that smoothes the data is given by

B = [CTC]−^1 CTP (10.5)

Least square fitting technique of B-spline curves described above assumes that Cis known. Given the
orderkof the B-spline basis, the number of defining polygon vertices M, the parameter values