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Page 6 of 20 Huang et al. Complex Eng Syst 2023;3:2 I http://dx.doi.org/10.20517/ces.2022.43

(6) represents the equation of the -th segment of a CHS curve. A CHS curve has global continuity [16] .
1

The problem of fitting the -th segment of the curve to points can be transformed into a minimization
problem:

Õ Õ

min err = min
F ( , +1 , , +1 , ) −
(7)

=0 =0
It is evident that in each segment of the curve except the first one, only the endpoint tangent vector +1 needs
to be fitted. In (7), fitting a parametric curve equation needs to consider multiple minimization problems at
the same time.

An algorithm for fitting a piecewise spatial CHS curve is proposed based on the idea of asymptotic approxi-
mation, as shown in the Algorithm 1. The main idea of the algorithm is to cyclically optimize , +1 and .
That is, one parameter is optimized while keeping the other two parameters unchanged until each parameter
is optimized iter times. To achieve global continuity, the vector tangent to the starting point of subsequent
1
curves adopts the endpoint tangent vector of the previous segment. In this paper, F ( ) means that other
quantities are left unchanged, and only is changed, and the same is true for other variables. In this paper,
the optimizer uses L-BFGS-B [31] .

Í
 0 ∥F 0 ( 0 , 1 ) − ∥ = 1
 arg min
 =0
 
0 , 1 (8)
Í
 arg min ∥F ( +1 ) − ∥ = 2, 3, . . .
 =0

+1

Algorithm 1 CHS Curve Fitting with Asymptotic Approximation
: Total number of curve segments.
: The -th curve, ∈ [1, ].
: Point set size.
: The -th point in the set of points, ∈ [1, ].
: The parameter corresponding to the th point.
: The column vector consisting of .
iter: Number of iterations.
for ∈ [1, ] do
Initialization : , +1

for ∈ [1, iter ] do

Í
= L-BFGS-B arg min =1 F ( )

if == 1 then !
Í
= L-BFGS-B arg min F ( )
=1

end if !
Í
+1 = L-BFGS-B arg min F ( +1 )
=1
+1
end for
end for
return F