In numerical analysis, a blossom is a functional that can be applied to any polynomial, but is mostly used for Bézier and spline curves and surfaces.
The blossom of a polynomial ƒ, often denoted
B
[
f
]
,
{\displaystyle {\mathcal {B}}[f],}
is completely characterised by the three properties:
It is a symmetric function of its arguments:
B
[
f
]
(
u
1
,
…
,
u
d
)
=
B
[
f
]
(
π
(
u
1
,
…
,
u
d
)
)
,
{\displaystyle {\mathcal {B}}[f](u_{1},\dots ,u_{d})={\mathcal {B}}[f]{\big (}\pi (u_{1},\dots ,u_{d}){\big )},\,}
(where π is any permutation of its arguments).
It is affine in each of its arguments:
B
[
f
]
(
α
u
+
β
v
,
…
)
=
α
B
[
f
]
(
u
,
…
)
+
β
B
[
f
]
(
v
,
…
)
,
when
α
+
β
=
1.
{\displaystyle {\mathcal {B}}[f](\alpha u+\beta v,\dots )=\alpha {\mathcal {B}}[f](u,\dots )+\beta {\mathcal {B}}[f](v,\dots ),{\text{ when }}\alpha +\beta =1.\,}
It satisfies the diagonal property:
B
[
f
]
(
u
,
…
,
u
)
=
f
(
u
)
.
{\displaystyle {\mathcal {B}}[f](u,\dots ,u)=f(u).\,}
