Tricubic Solids
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Transcript of Tricubic Solids
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The indices of the basis functions of a term are all less
then 3, then the associated bvector denotes a point
When one and only one of the indices of a term is 3 or 4then the associated b vector denotes a tangent vector
When two and only two of the indices are 3 or 4 then the
associated b vector denotes a twist vector
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When in the case of tricubic solid indices of a term are 3or 4 then the associated b vector denotes the vector
defined by the third order mixed partial derivatives of the
function p(u,v,w)
Our current method of constricting the arrays ofgeometric coefficients suggests the following empiricalinterpretation
For each odd index on b assign a zero and for eacheven index assign a one
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The transformation form geometric to algebraic
coefficients is given by
More compactly as
hereMis the Hermite basis transformation matrix and
the various subscripted indices denote specific element of
the matrix
4
1
4
1
4
1l m n
lmnknjmilijk bMMMa
lmnknjmilijk bMMMa
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Stanton and Crain developed a slightly different approchin 1974
Treating the 64 hyper patch parameters as 4 sets of 16
parameters
This allows us to understand more easily the basic
function properties and the relationship between patchesand hyper patches
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Algebraic form:
Geometric form: Point form:
are basic functions applied to the
algebraic coefficients defined by
Similarly for vand w
ijkp
kp
jp
i
ijkbkbjbi
ijka
ka
ja
i
pwFvFuFwvup
bwFvFuFwvup
awFvFuFwvup
)()()(),,(
)()()(),,(
)()()(),,(
)(),(),( wFvFuF aka
j
a
i
.1)(
,)(
,)(
,)(
4
3
2
2
3
1
uF
uuF
uuF
uuF
a
a
a
a
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are basic functions applied to the
point coefficients defined by
)(),(),( wFvFuF pkp
j
p
i
.2
9
2
9)(
,2
918
2
27)(
,92
45
2
27)(
,12
119
2
9)(
23
4
23
3
23
2
23
1
uuuuF
uuuuF
uuuuF
uuuuF
p
p
p
p