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Lemma 3 Let a , b , c be distinct, nonzero points in Euclidean space and v a nonzero vector. Then there exists t R such that v Span { a + t v , b + t v , c + t v } .

The scalar triple product

D ( t ) = ( ( a + t v ) × ( b + t v ) ) · ( c + t v ) = ( a × b ) · c + t ( ( a - b ) × ( b - c ) ) · v

is constant in t if v Span { a , b , c } and is nonzero for all but one t otherwise. In the former case, setting t = 0 while in the latter choosing t so that D ( t ) 0 and consequently Span { a + t v , b + t v , c + t v } = R 3 then implies the lemma.

Combining the above lemmas, we have

Theorem 1 A necessary and sufficient condition that a point force field f = f 1 δ ( a 1 ) + + f ν δ ( a ν ) be balanced is that it is equilibrated by a truss T .

(Necessity) Suppose f is balanced by the truss T . Let v R 3 and φ ( x ) = v for all x R 3 . Then ( ) implies

0 = ( δ T , φ ) = ( f , φ ) = i = 1 μ f i · v = 0 .

Since v was arbitrary, we infer ( ). For w R 3 , define the skew symmetric matrix

w ^ = 0 - w 3 w 2 w 3 0 - w 1 - w 2 - w 1 0

One then checks that w ^ v = w × v . Let v R 3 and φ ( x ) = x ^ v . Then

( δ T , φ ) = i = 1 μ ω i ( φ ( a i ) - φ ( b i ) ) · a i - b i | a i - b i | = i = 1 μ ω i ( a ^ i - b ^ i ) v · a i - b i | a i - b i | = i = 1 μ ω i ( a i - b i ) × v · a i - b i | a i - b i | = 0 .

By ( ), this implies

0 = ( δ T , φ ) = ( f , φ ) = i = 1 ν f i · φ ( a i ) = i = 1 ν f i · a ^ i v = i = 1 ν f i · a i × v = i = 1 ν f i × a i · v .

Since v is arbitrary, we infer ( ). Thus f is balanced.

(Sufficiency) We proceed by induction on ν . If ν = 2 , then f is equilibrated by the truss T guaranteed in lemma . If ν = 3 and a 1 , a 2 and a 3 do not all lie on a line, then f is equilibrated by the truss T guaranteed in lemma . If ν = 3 and a 1 , a 2 and a 3 lie on a line, consider the equivalent point force field g = f 1 δ ( a 1 ) + f 2 δ ( a 2 ) + f 3 δ ( a 3 ) + 0 δ ( a 4 ) where a 4 is a arbitrary point chosen off the line containing a 1 , a 2 and a 3 . The result for ν = 4 proves that g is equilibrated by a truss T . However, since ( g , φ ) = ( f , φ ) for all continuous vector fields φ , this implies that T also equilibrates f . Assume that sufficiency holds for ν 3 . If f ν + 1 = 0 , then we are done. Otherwise, according to lemma , there is ρ R so that

f ν + 1 Span ( a 1 - ( a ν + 1 + ρ f ν + 1 ) , a 2 - ( a ν + 1 + ρ f ν + 1 ) , a 3 - ( a ν + 1 + ρ f ν + 1 ) ) .

Let b = a ν + 1 + ρ f ν + 1 and let

f ν + 1 = Ω 1 ( a 1 - b ) + Ω 2 ( a 2 - b ) + Ω 3 ( a 3 - b ) .

Define the point force field g = g 1 δ ( b 1 ) + + g ν δ ( b ν ) by

g i = f i + ( a i - b ) Ω i , i = 1 , 2 , 3 , g i = f i , i = 4 , , ν , b i = a i , i = 1 , , ν .

We claim that g is balanced; by ( )

i = 1 ν g i = i = 1 ν f i + i = 1 3 ( a i - b ) Ω i = i = 1 ν f i + f ν + 1 = 0

and by ( )

i = 1 ν g i × b i = i = 1 ν f i × a i + i = 1 3 Ω i ( a i - b ) × a i = i = 1 ν f i × a i + i = 1 3 Ω i ( a i - b ) × ( a i - b ) + i = 1 3 Ω i ( a i - b ) × b = i = 1 ν f i × a i + f ν + 1 × ( a ν + 1 + ρ f ν + 1 ) = 0 .

By induction, there exists a truss R equilibrating g . Let S be the truss consisting of the collection of beams a 1 - b , a 2 - b , a 3 - b and b - a ν + 1 with weights Ω 1 , Ω 2 , Ω 3 and | f ν | resp. Arguing as in lemma , we find that T = R + S equilibrates f .

Some economical trusses

After proving the question of existence we turn to the question of economy. We say that a truss T is economical if it satisfies

Cost ( T ) Cost ( S )

whenever δ S = δ T . That is to say, the cost of T is less than or equal to that of any truss which equilibrates the same force system as T . We begin by describing some global statements about economical trusses that can proven by choosing special test vector fields φ in ( ). Then we make local perturbations on trusses with corners to find a necessary conditionfor economy.

Some global properties

A surprising and easily proven fact is

Lemma 4 A truss is economical if the weights of the beams are all of the same sign. Such a truss lies in the closed convex hull of the support of the point forcesit equilibrates.

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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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