基于对称样条的均匀插值曲线与曲面构建
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"Uniform interpolation curves and surfaces based on a family of symmetric splines" 是一篇研究论文，发表在"Computer Aided Geometric Design"期刊的第30卷，2013年，页码范围为844–860。该论文由Ren-Jiang Zhang撰写，来自中国杭州的浙江工商大学数学系。
文章主要探讨了一种基于对称样条的新方法，用于构造通过给定数据点集的任意阶连续曲线。这种方法引入了一种新的对称插值样条家族，这些样条具有多种优良性质，如单位分割、插值性质、局部支持和二阶精度等。通过对这些新样条的应用，可以构建曲线和表面，并通过移动一些插值点或插入新的插值点来局部调整所构建的曲线和表面的形状。
关键词包括：插值、曲线和表面构造、对称样条、分段多项式基、细分。这些关键词揭示了论文的核心内容，即研究如何利用对称性来优化曲线和表面的构造过程，以及如何通过细分算法实现数据点的插值，以创建出满足特定需求的几何形状。
在计算机辅助几何设计中，对称样条是一种重要的工具，因为它允许在保持几何形状的平滑性和精确性的前提下，进行灵活的修改。通过对称性，可以确保样条在构造过程中保持左右对称，这对于设计具有对称性的物体特别有用。局部支持特性意味着改变某一部分的数据点只会影响样条的相应部分，而不影响全局，这在设计中提供了高效的编辑能力。而二阶精度则意味着样条曲线能够更精确地拟合数据点，提高模型的细节表现。
这篇论文为计算机图形学和几何建模领域提供了一个新的、灵活的工具，用于创建和修改复杂几何形状，特别是在需要精确控制和局部调整的情况下。通过对称样条的深入研究和应用，可以期待在CAD软件和相关工程应用中看到更多创新和改进。
R.-J. Zhang / Computer Aided Geometric Design 30 (2013) 844–860 847
d
dt
i
H
j
(0) = δ
i, j
, i = 0, 1,...,r; j = 0, 1,...,r,
and
d
dt
i
H
j
(1) = δ
i,2r+1−j
, i = 0, 1,...,r; j = r +1, r +2,...,2r + 1.
Note:
(a) Conditions (5) can
assume that the constructed curve with symmetry about the point P
i
if the set of points P
i
is
symmetric about P
i
. This is a natural and rational requirement.
(b) If P
i
is a set of data with dimension larger than 1, then d
(r)
i
can be viewed as a derivative vector. In the later, if P
i
is
a set of data with dimension one, we denote it by y
i
.
The above method to construct curves has the following properties.
(2.1) The constructed curve P
(t) is an interpolatory curve which exactly passes through all the given points, and is
symmetric about the point P
i
if the set of points P
i
is symmetric about the point P
i
.
(2.2) The curve P
(t) is at least C
r
continuous. According to the construction procedure, Properties (1) and (2) are obvious.
(2.3) Local shape adjustable. If we move one point P
i
, the curve only changes its shape which lies between P
i−T
and
P
i+T
locally. Where
T =
max
r
(
r
2
+ s(r) + 1), r is even;
max
r
(
r+1
2
+ s(r) + 1), r is odd.
(7)
It is easily observed from Eq. (4) that moving the point P
i
only changes d
( j)
i−T +1
,...,d
( j)
i
,...,d
( j)
i+T −1
, j = 1, 2,...,r,sothe
conclusion is valid.
We now introduce the symbol
{P
i
}={...,P
i−1
, P
i
, P
i+1
,...}, which denotes a given set of points, and then define the
addition of the two sets of
{P
1
i
}={...,P
1
i
−1
, P
1
i
, P
1
i
+1
,...} and {P
2
i
}={...,P
2
i
−1
, P
2
i
, P
2
i
+1
,...} as
P
1
i
+
P
2
i
=
...,
P
1
i
−1
+ P
2
i
−1
, P
1
i
+ P
2
i
, P
1
i
+1
+ P
2
i
+1
,...
.
(2.4) Additivity. If we construct two curves P
1
(t) and P
2
(t) by the method above from the two sets of data points {P
1
i
}
and {P
2
i
} respectively, then P
1
(t) + P
2
(t) is the curve passing through the set of data points {P
1
i
}+{P
2
i
}.
Proof. Le
t d
(r)
i,1
and d
(r)
i,2
, which defined by Eq. (4), refer to the set of points P
1
i
and P
2
i
respectively. d
(r)
i
refer to the set of
points P
1
i
+ P
2
i
. It is easy to see that
d
(r)
i
=d
(r)
i,1
+d
(r)
i,2
.
Then the conclusion derives from Eq. (6).
(2.5) If P
(t) is a curve passing through a set of {P
i
},thenc · P (t) is a curve passing through a set of c ·{P
i
}, where c is
a constant.
Notice that d
(r)
i
is a linear combination of {P
i
},byEq.(6), the conclusion is obvious.
(2.6) Second order precision. If the set of data points P
i
is evenly taken from the straight line y = x or the parabolic line
y
= x
2
, then the curve P(t) reproduces the curve if r 2inthemethod.Notingd
(1)
i
= constant for P
i
= i, and d
(1)
i
= 2i
and d
(2)
i
= 2forP
i
= i
2
. So the parabolic line can be reproduced exactly. The cubic curve y = x
3
can only be approximately
reconstructed because its various order derivatives do not be estimated exactly. Fig. 2 shows these results.
Applying the formula of transforming Hermite polynomials to Bernstein basis (Chang,
1984)
H
2r+1
k
(t) =
(
2r +1 −k)!
(2r + 1)!
r
i=k
B
2r+1
i
(t)
i
k
,
k = 0, 1,...,r (8)
and
H
2r+1
k
(t) = (−1)
k+1
k!
(2r + 1)!
r
i=2r+1−k
B
2r+1
2r
+1−i
(t)
i
2r
+1 −k
,
k = r + 1, r + 2,...,2r +1, (9)
where B
n
i
(t) =
n
i
(
1 − t)
n−i
t
i
are Bernstein basis functions. We can rewrite curve equation (6) as in the Bézier form. For
later use, we only give two formulas in the matrix form according to Eqs. (8) and (9) for r
= 1 and r = 2, respectively:
⎛
⎜
⎜
⎜
⎝
H
3
0
H
3
1
H
3
2
H
3
3
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
11 0 0
0
1
3
00
00
−
1
3
0
00 1 1
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎝
B
3
0
B
3
1
B
3
2
B
3
3
⎞
⎟
⎟
⎟
⎠
(10)
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