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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