TR/04/90 March 1990 CONVEXITY OF BẺZIER NETS ON SUB-TRIANGLES John A. Gregory and Jianwei Zhou With minor revisions, November 1990. To appear in CAGD.

TR/04/90 March 1990 CONVEXITY OF BẺZIER NETS ON SUB-TRIANGLES

John A. Gregory and Jianwei Zhou

With minor revisions, November 1990.

To appear in CAGD.

Convexity of Bẻzier Nets on Sub-triangles

by

John A. Gregory and Jianwei Zhou

Department of Mathematics and Statistics, Brunei University, Uxbridge, UB8 3PH, England

Abstract: This note generalizes a result of Goodman[3], where it is shown that the convexity of Bèzier nets defined on a base triangle is preserved on sub-triangles obtained from a mid-point subdivision process. Here we show that the convexity of Bèzier nets is preserved on and only on sub-triangles that are "parallel" to the base triangle.

1.Introduction

Let T be a triangle, called the base triangle (see [1]), with vertices V

1

,V

2

andV

3

. ( Here, and elsewhere in the paper, we assume that triangles are non-degenerate, that is, their vertices are not colinear.) Then each point P of the plane determined by V

1

,V

2

andV

3

.can be represented by its barycentric coordinates (u,v,w) with respect to the base triangle T as

(1) P = uV

₁

+ vV

₂

+ wV

₃

, u+v+w = 1.

Denote f

n

as a set of (n+1)(n+2)/2 values

(2) ^f

n

= { ^f

ijk

∈ ^{R i} + ^j + ^k = ^{n i} ≥ ^{0, j} ≥ ^{0, k} ≥ ^0, } ^.

-1-

The Bernstein polynomial of f

n

over T is then given by (3) ∑

= + +

=

n k j i

n ijk ijk

n

; u, v, w) f B (u, v, w) ,

B(f Where

(4)

_ijkⁿ

u

ⁱ

v

^j

w

^k

k!

j!

i!

w) n!

v, (u,

B =

are Bernstein basis functions. If we let shift operators E

1

, E

2

and E

3

with respect to T be defined by

(5) E

₁

f

_ijk

= f

_i+1,j,k

, E

₂

f

_ijk

= f

_i,j+1,k

, E

₃

f

_ijk

= f

_i,j,k+1

, then the Bernstein polynomial can be represented symbolically as (6) B(f

_n

; u, v, w) = (uE

₁

+ vE

₂

+ wE

₃

)

ⁿ

f

₀₀₀

.

Now we consider (n+l)(n+2)/2 points of T with barycentric coordinates (i/n,j/n,k/n), namely,

. n k j i , (k/n)V (j/n)V

(i/n)V

P

_ijk

=

₁

+

₂

+

₃

+ + =

Connecting

, P , P ,

P

_{i+1,j,k i,j+1,k i,j,k+1}

we obtain a triangle, denoted U

_ijk

for i+j+k=n-l. Similarly, a triangle W

_ijk

is obtained for i+j+k=n+l with

1 k j, i, k 1, j i, k j, 1,

i

, P , P

P

_{− − −}

as its vertices. A piecewise linear function L ( f

_n

; u, v, w ) , or L ( ) f

_n

for short, is defined on T such that it satisfies

(7) ^L ( ^f

n

^{; P}

ijk

) = ^f

ijk

^{, i} + ^j + ^k = ^{n ,}

and is linear on each triangle U

_ijk

or W

_ijk

. We call L ( f

_n

; u, v, w ) the Bėzier net of f

_n

. The approximation theory of Bernstein polynomials and their applications in CAGD, Computer-aided Geometric Design, indicate that the Bėzier net L ( f

_n

) is closely related to

w) v, u,

;

B(f

_n

and reflects certain features of B(f

_n

; u, v, w ) . As this note is concerned with the convexity, we state the following results (see [1,2]):

-2-

Theorem 1 (i) If the Bėzier net L(f

n

;u,v,w) is convex with respect to T, so is the Bernstein polynomial B(f

_n

; u, v, w) .

(ii) The Bėzier net L(f

_n

; u, v, w) is convex with respect to T if and only if

(8)

⎪ ⎩

⎪ ⎨

⎧

+

≥ +

+

≥ +

+

≥ +

+ + + + +

+ +

+ + +

+ + + +

+ + + + + + +

, f

f f

f

, f

f f

f

, f

f f

f

1 k 1, j i, 1 k j, 1, i k 1, j 1, i 2 k j, i,

1 k 1, j i, k 1, j 1, i 1 k j, 1, i k 2, j i,

1 k j, 1, i k 1, j 1, i 1 k 1, j i, k j, 2, i

for i+j+k=n-2.

Using the shift operators, one may rewrite (8) as

(9) ( ^E

i₁

− ^E

i₂

) ( ^E

i₁

− ^E

i₃

) ^f

ijk

≥ ^{0 , i} + ^j + ^k = ⁿ − ^{2 ,} for any permutation { ⁱ

₁

^{, i}

₂

^{, i}

₃

} { ^{of 1,2,3} } ^.

be the vertices of another triangle in the same plane as T and let

* 3

* 2

*

1

, V and V

V

Let T

^*

(10) P= u

^*

V

₁^*

+ v

^*

V

₂^*

+ w

^*

V

₃^*

, u

^*

+ v

^*

+ w

^*

= 1 ,

define the barycentric coordinates ( ^u

^*

^{, v}

^*

^{, w}

^*

) of P with respect to T

^*

. Assume that V

_i^*

has barycentric coordinates (u

_i

, v

_i

, w

_i

) with respect to T, that is,

(11) V

_i^*

= u

_i

V

₁

+ v

_i

V

₂

+ w

_i

V

₃

, u

_i

+ v

_i

+ w

_i

= 1 for i=l, 2, 3. Then

(12)

⎪ ⎩

⎪ ⎨

⎧

+ +

=

+ +

=

+ +

=

. w w w v w u w

, v w v v v u v

, u w u v u u u

3

* 2

* 1

*

3

* 2

* 1

*

3

* 2

* 1

*

A Bernstein polynomial on T

^*

is defined symbolically by (13) ^B ( ^{f ; u , v , w} ) ( ^{u E v E w E} ) ^f

000^*

^,

* n 3

*

* 2

*

* 1

*

*

*

*

*

n

= + +

Where

(14) ^f { ^f

i^*jk

^{R i j k n, i 0, j 0, k 0} }

*

n

= ∈ + + = ≥ ≥ ≥

and E

^*_i

define the shift operators on T

^*

. We then have:

Theorem 2 (Chang and Davis[l]) Let

(15) f

_ijk^*

= ( u

₁

E

₁

+ v

₁

E

₂

+ w

₁

E

₃

) (

ⁱ

u

₂

E

₁

+ v

₂

E

₂

+ w

₂

E

₃

) (

^j

u

₃

E

₁

+ v

₃

E

₂

+ w

₃

E

₃

)

^k

f

₀₀₀

,

-3-

for i+j+k=n. Then B ( f

n^*

; u

^*

, v

^*

, w

^*

) is the Bernstein representation of B( ;u,v,w) with respect to

f

n

. T

^*

Proof: From (12) and (15) we obtain, by equating coefficients,

( ) ( )

000

n 3 2 1

n

; u, v, w uE vE wE f

f

B = + +

( ) ( ) (

[ ]

000

n 3 3 2 3 1 3

* 3 2 2 2 1 2

* 3 1 2 1 1 1

*

u E v E w E v u E v E w E w u E v E w E f

u + + + + + + + +

= )

= ( )

000^*

* n 3

*

* 2

*

* 1

*

E v E w E f

u + +

if and only if

( ^{u E v E w E} ) ( ^{u E v E w E} ) ( ^{u E v E w E} ) ^{f .}

f E E

E

^*₁^{i *}₂^{j *}₃^k ₀₀₀^*

=

_{1 1}

+

_{1 2}

+

_{1 3} ⁱ

+

_{2 1}

+

_{2 2}

+

_{2 3} ^j

+

_{3 1}

+

_{3 2}

+

_{3 3} ^k ₀₀₀

Remark In the above proof we note the identities (16) E

^*_i

= u

_i

E

₁

+ v

_i

E

₂

+ w

_i

E

₃

, i = 1,2,3 ,

where, in symbolic manipulation involving these operators, indices in operator expressions must sum to n and the operator expressions are applied to f

₀₀₀^*

= f

₀₀₀

.

The set f

_n^*

determines a new Bezier net L ( f

n^*

; u

^*

, v

^*

, w

^*

) which is a piecewise linear function on T ^* . We call L ( ) f

n^*

the restricted Bezier net of L(f

n

) on T

^*

. Naturally, T

^*

is called a sub-triangle of T if V

₁^*

, V

₂^*

and V

₃^*

, the vertices of T

^*

are all inside or on the boundary of T.

In this case u

_i,

v

_i,

w

_i

≥ 0, i = 1,2,3.

Let T

^*

be a sub-triangle of T. Then if B ( f

_n

; u, v, w ) is convex with respect to T, so is

( f

n^*

; u

^*

, v

^*

, w

^*

)

B with

respect to

One would ask whether or not a similar result holds for the Bėzier nets. Grandine[4] showed a negative result but in [3], Goodman shows that if is a

. T

^*

T

*

sub-triangle obtained from a "mid-point" subdivision process, then the convexity of L(f

n

) does guarantee the convexity of the restricted Bėzier net L ( ) f

n^*

with respect to . In the next section, we generalize this result and show that only a very limited class of sub-triangles have

T

*

the property of preserving the convexity of Bėzier nets.

-4-

2. Main Result

Let T

^*

be a non-degenerate triangle that lies on the plane determined by the base triangle T. We say T

^*

is parallel to T if each of the edges of T

^*

is parallel to one of those of T. We now make the following definition:

Definition A non-degenerate sub-triangle T

^*

is called "Bėzier net convexity preserving" if, for all Bėzier nets L(f

n

) that are convex with respect to T, the restricted Bėzier nets L ( ) f

n^*

on T

^*

are also convex with respect to T

^*

Noting the barycentric coordinate representation of the vertices of T

^*

with respect to T in (11), we have the following lemma.

Lemma The following statements are equivalent, (i) T

^*

is parallel to T.

(ii) There exist a non-zero scalar ρ and a permutation { i

1

, i

2

, i

3

} of {1,2,3} such that (17) V

_i^*

V

_i^*

( V

₁

V

₂

), V

_i^*

V

_i^*

( V

₃

V

₁

), V

_i^*

V

_i^*

( V

₂

V

₃

).

3 2 1

3 2

1

− = ρ − − = ρ − − = ρ −

(iii) None of the sets { ^u

₁

^{, u}

₂

^{, u}

₃

} { ^{, v}

₁

^{, v}

₂

^{, v}

₃

} ^and { ^w

₁

^{, w}

₂

^{, w}

₃

} has all distinct members.

Figure. Examples of parallel triangles

This lemma is illustrated by the two examples of parallel triangles labelled as shown in the -5-

figure. Property (ii) is simply a statement that the edges are parallel and property (iii) is a

restatement of this property in terms of the barycentric coordinates of V

_i^*

= u

_i

V

₁

+ v

_i

V

₂

+ w

_i

V

₃

, i = 1,2,3.

Thus, in the examples u

₂

= u

₃

, v

₃

= v

₁

and w

₁

= w

₂

. .

We now present our main theorem:

Theorem 3 Let T

^*

be a non-degenerate sub-triangle of T. Then T

^*

is Bėzier net convexity preserving if and only if it is parallel to T.

Proof: Suppose T

^*

is parallel to T and let L(f

n

) be any Bėzier net that is convex with respect to T. Comparing (16) and (11), it follows by statement (ii) of the lemma that there exist a non-zero scalar ρ and a permutation { i

1

, i

2

, i

3

} of {1,2,3} such that

).

E ρ(E E

E ), E ρ(E E

E ), E ρ(E E

E

^*_i1

−

^*_i2

=

₁

−

₂ ^*_i3

−

^*_i1

=

₃

−

₁ ^*_i2

−

^*_i3

=

₂

−

₃

Thus, by (16), we have for i+j+k=n-2

( ^{E E} )( ^{E E} ) ^{f E E E} ( ^{E E} )( ^{E E}

^*i

) ^f

^*

⁰⁰⁰

* i

* i

* i k

* 3 j

* 2 i

* 1

* ijk

* i

* i

* i

*

i₂

−

_{2 2}

−

₃

=

₂

−

_{3 2}

−

₃

= ( ) ( ) ( ) (

1 2

)(

1 3

)

000

k 3 3 2 3 1 3 j 3 2 2 2 1 2 i 3 1 2 1 1 1

2

u E v E w E u E v E w E u E v E w E E E E E f

ρ + + + + + + − −

=

^{2 i}_rst

(

_{1 1 1}

)

_αβγ^j

(

_{2 2 2}

)

^k_λµν

(

_{3 3 3}

)

k v µ λ i

t s

r α β y l

w , v , u B w v , u B w , v , u B

∑ ρ

∑ ∑

= + +

= +

+ + + =

( E

₁

− E

₂

)( E

₁

− E

₃

) f

_{r+α+λ,δ+β+µ,t+y+ν}

≥ 0 .

Similarly,

( ^{E E} )( ^{E E} ) ^{f 0 ,} ( ^{E E} )( ^{E E} ) ^f

ijk^*

^{0 .}

* i

* i

* i

* i

* ijk

* i

* i

* i

*

i₂

−

_{1 2}

−

₃

≥

₃

−

_{1 3}

−

₂

≥

We therefore conclude that L ( ) f is convex with respect to T*.

n^*

W e n ow suppose that . is not parallel to T. By statement (iii) of the lemma, at least one

of the sets { } ^{, and}

T

* 3 2 1

, u , u

u { v

1

, v

2

, v

3

} { w

1

, w

2

, w

3

} has all distinct members. Without loss of

generality, we assume u

₂

> u

₁

> u

₃

≥ 0. Then we have

( ^u

₁

^{− u}

₂

)( ^u

₁

^{− u}

₃

) ^{< 0.}

Consider the situation where

-6-

(18) f

_ijk

= δ

_i,n−1

, i + j + k = n ,

that is, f

_n,0,0

= 1 and

f

ijk

=0 if i ≤ − 1 . Obviously, the Bezier net L ( ) f

_n

defined by (18) is convex with respect to T. Using (15) we obtain

n k j i , u u u

f

_ijk^*

=

₁ⁱ ₂^j ₃^k

+ + = Hence for i+j+k=n-2 we have

( )( ) (

1 2

)(

1 3

)

k 3 i 2 i i

* ijk

* 3

* 1

* 2

*

1

E E E f u u u u u u u

E − − = − −

and, particularly for n 2, ≥

( E

^*_i1

− E

^*_i2

)( E

^*_i1

− E

^*_i3

) f

_0,^*_n−2,0

= u

ⁿ₂⁻²

( u

₁

− u

₂

)( u

₁

− u

₃

) < 0.

Thus the restricted Bėzier net L ( ) f

n^*

is not convex with respect to T*.

Acknowledgements: This work was supported by the ACME Directorate of the Science and Engineering Research Council and by Siemens AG.

References

[1] Chang, G. and P.J.Davis(1984), The convexity of Bernstein polynomials over triangles, J.

Approximation Theory 40, 11-28.

|2] Dahmen, W. and C.A.Micchelli(1985), Convexity of multivariate Bernstein olynomials and box spline surfaces, Research Report RC 11176, T.J.Watson Research Center.

[3] Goodman, T.N.T.(1989), Convexity of Bėzier nets on triangulations, AA/899, Department of Mathematics and Computer Science, University of Dundee.

[4] Grandine, T.A.(1989), On convexity of piecewise polynomial functions on triangulations,

Computer Aided Geometric Design 6, 181-187.

-7-