Adjacent Sides Bisectors Theory in Fundamental Science on General Problem Bisectors
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
Mathematical Journal
of the "Collegium" All World Academy of Sciences
Munich (Germany)
11 (2011), 44
The least square method (LSM) [1] by Legendre and Gauss only usually applies to contradictory (e.g. overdetermined) problems. Overmathematics [2-4] and fundamental science of solving general problems [5] have discovered many principal shortcomings [2-6] of this method, by methods of finite elements, points, etc. Additionally consider its simplest approach which is typical. Minimizing the sum of the squared differences of the alone preselected coordinates (e.g., ordinates in a two-dimensional problem) of the graph of the desired approximation function and of everyone among the given functions depends on this preselection, ignores the remaining coordinates, and provides no coordinate system rotation invariance and hence no objective sense of the result.
In fundamental science on general problem bisectors [5], adjacent sides bisectors theory (ASBT) is valid by coordinate system rotation invariance. Show the essence of this theory in the simplest but most typical case providing linear solving with giving the unique best quasisolution [2-5] to a finite overdetermined quantiset [2-5] of n (n > 2; n ∈ N+ = {1, 2, ...}) linear equations
q(i)(ajx + bjy = cj) (j = 1, 2, ... , n) (Ei)
with their own quantities q(i) and 2 unknown variables x and y in a plane with the same coordinates x , y by any given real numbers q(i) > 0, aj , bj , and cj . Suppose that these n straight lines build a polygon P1P2...Pn with n corners (vertices) P1 , P2 , ... , Pn (and, naturally, n sides, or edges) either in the clockwise order or in the anticlockwise order which can be provided via preliminarily renumbering (reindexing) the equations. If necessary and possible, find and use such renumbering (reindexing) of these n corners that polygon P1P2...Pn is convex. If it is impossible, then provide the minimum number of nonconvex corners (vertices) (i.e., with convexity violation) and hence the maximum number of convex corners (vertices) (i.e., without convexity violation). Let Pi be the intersection of straight lines Ei and Ei+1 with their own quantities q(i) and q(i+1) which has its own quantity
p(i) = q(i)q(i+1).
Because of the clear possible periodicity of the indices i = 1, 2, ... , n with period n , consider
Pj+kn = Pj ,
q(i+kn) = q(i) ,
p(i+kn) = p(i)
for any integer k . Then the quantiset [2-5] of n polygon corners (vertices) is
p(1)P1 , p(2)P2 , ... , p(n)Pn .
The quantiset [2-5] of n polygon corners bisectors is
p(1)B1 , p(2)B2 , ... , p(n)Bn .
Nota bene: A bisector of an angle (and of the corresponding angle sides) divides it into two equal parts and is the set of all the points equidistant from the both angle sides [1]. For two intersecting straight lines, there are two mutually perpendicular bisector straight lines. If for a polygon, there is an inscribed circumference, then the desired unique best quasisolution its simply the inscribed circumference center which is the intersection of the bisectors of all the internal corners of this polygon which in this case is always convex. This all holds for any triangle whose case is, therefore, trivial.
Hence further consider case n > 3 only when for a polygon, there is no inscribed circumference.
The main idea of adjacent sides bisectors theory (ASBT) is subsequently replacing every convex (i.e., without local nonconvexity) corner (vertex) of this polygon with the intersection of the bisector of this corner (vertex) and the straight line segment connecting the both adjacent corners (vertices) whereas every nonconvex (i.e., with local nonconvexity) corner (vertex) of this polygon is conserved at the given step of moving to the polygon center, the own quantity r(i) of each (either convex or nonconvex corner (vertex) being conserved.
The essence of adjacent sides bisectors theory (ASBT) is:
1) at the 1st step, first consider the initial polygon P1P2...Pn and finally obtain a polygon P'1P'2...P'n less than polygon P1P2...Pn via replacing each convex corner (vertex) Pj with certain point P'j by following substeps 1.1, 1.2, 1.3, and 1.4 whereas each nonconvex corner (vertex) Pj is conserved, i.e., P'j = Pj :
1.1) for any j = 1, 2, ... , n , consider the jth corner (vertex) Pj along with its both adjacent corners (vertices) Pj-1 and Pj+1 , as well as the both adjacent sides PjPj-1 and PjPj+1 building this jth corner (vertex) Pj ;
1.2) for any j = 1, 2, ... , n , consider triangle Pj-1PjPj+1 built by the both adjacent sides PjPj-1 and PjPj+1 along with segment Pj-1Pj+1 connecting the both adjacent corners (vertices) Pj-1 and Pj+1 ;
1.3) for any j = 1, 2, ... , n , in this triangle Pj-1PjPj+1 , consider its internal bisector PjP'j of the jth corner Pj between its both adjacent sides PjPj-1 and PjPj+1 from the initial point of the bisector at the jth corner (vertex) Pj to the endpoint of the bisector at some point P'j as the intersection of this bisector and the opposite side Pj-1Pj+1 of this triangle Pj-1PjPj+1 ;
1.4) for any j = 1, 2, ... , n , in this triangle Pj-1PjPj+1 , consider its internal bisector PjP'j of the jth corner Pj between its both adjacent sides PjPj-1 and PjPj+1 from the initial point of the bisector at the jth corner (vertex) Pj to the endpoint of the bisector at some point P'j as the intersection of this bisector and the opposite side Pj-1Pj+1 of this triangle Pj-1PjPj+1 ;
2) at the 2nd step, first consider the initial polygon P'1P'2...P'n . If either it is already convex or there is such reordering of the points P'1 , P'2 , ... , P'n which provides polygon convexity, then preliminary use this reordering and continue this process of subsequently decreasing polygon P1P2...Pn via following substeps 2.1, 2.2, 2.3, and 2.4 similar to substeps 1.1, 1.2, 1.3, and 1.4 and finally obtain a polygon P''1P''2...P''n still less than polygon P'1P'2...P'n . Otherwise, this process can be simply broken. But if necessary and useful, it is also possible to preliminarily minimize the nonconvexity of polygon P'1P'2...P'n via some reordering of the points P'1 , P'2 , ... , P'n and to continue this process of subsequently decreasing polygon P1P2...Pn ;
k) at the kth step (k ∈ N+ = {1, 2, ...}), obtain such a small polygon P(k)1P(k)2...P(k)n with corners (vertices) and their own quantities s(i) = p(k)(i) and coordinates
s(1)P(k)1(x(k)1 , y(k)1), s(2)P(k)2(x(k)2 , y(k)2) , ... , s(n)P(k)n(x(k)n , y(k)n)
that it is clearly sufficient to determine their center (of gravity, mass, etc.)
x = Σj=1n s(i)x(k)j / Σj=1n s(i),
y = Σj=1n s(i)y(k)j / Σj=1n s(i)
and consider this point (x , y) to be the unique best quasisolution [2-5] to the initial overdetermined set of n linear equations (Ei). If at any step k, there is no such reordering of the points
P(k)1(x(k)1 , y(k)1) , P(k)2(x(k)2 , y(k)2) , ... , P(k)n(x(k)n , y(k)n)
which provides polygon convexity, then at once finish this process of subsequently decreasing polygon P1P2...Pn (even if we cannot regard polygon P(k)1P(k)2...P(k)n to be sufficiently small) and, nevertheless, determine the center (of gravity, mass, etc) (x , y) via the same formulae and consider this point (x , y) to be the unique best quasisolution [2-5] to a finite overdetermined set of n linear equations (Ei) as earlier.
Nota bene: Because of conserving the own quantities of the corners (vertices) whereas their renumbering (reindexing) is multiply possible, the final ordered set
s(1), s(1), ... , s(n)
is a rearrangement of the initial ordered set
p(1), p(1), ... , p(n) .
It is very useful to develop explicit formulae to obtain the coordinates of the endpoint P(k+1)j of the bisector at the k+1st step, given the coordinates of all the 3 relevant corners (vertices)
P(k)j-1(x(k)j-1 , y(k)j-1) , P(k)j(x(k)j , y(k)j) , P(k)j+1(x(k)j+1 , y(k)j+1).
The equation of the straight line containing two points
P(k)j-1(x(k)j-1 , y(k)j-1) , P(k)j+1(x(k)j+1 , y(k)j+1)
is
(x - x(k)j-1) / (x(k)j+1 - x(k)j-1) = (y - y(k)j-1) / (y(k)j+1 - y(k)j-1).
It is well-known [1] that in any triangle, a bisector divides the opposite side by their intersection in to two parts whose lengths are proportional to their adjacent sides.
The length of the side
P(k)j(x(k)j , y(k)j)P(k)j-1(x(k)j-1 , y(k)j-1)
is
[(x(k)j - x(k)j-1)2 + (y(k)j - y(k)j-1)2]1/2 .
The length of the side
P(k)j(x(k)j , y(k)j)P(k)j+1(x(k)j+1 , y(k)j+1)
is
[(x(k)j - x(k)j+1)2 + (y(k)j - y(k)j+1)2]1/2 .
Then, applying the above equation to the desired point P(k+1)j(x(k+1)j , y(k+1)j) dividing segment
P(k)j-1(x(k)j-1 , y(k)j-1)P(k)j+1(x(k)j+1 , y(k)j+1)
proportionally the above lengths, we obtain
(x(k+1)j - x(k)j-1) / (x(k)j+1 - x(k)j-1) = (y(k)j+1 - y(k)j-1) / (y(k)j+1 - y(k)j-1) =
[(x(k)j - x(k)j-1)2 + (y(k)j - y(k)j-1)2]1/2 / {[(x(k)j - x(k)j-1)2 + (y(k)j - y(k)j-1)2]1/2 + [(x(k)j - x(k)j+1)2 + (y(k)j - y(k)j+1)2]1/2}.
Nota bene: In this equation proportion itself, the ratio of one of the segment parts to the whole segment is considered.
Finelly, we obtain the desired formulae
x(k+1)j = x(k)j-1 + (x(k)j+1 - x(k)j-1) [(x(k)j - x(k)j-1)2 + (y(k)j - y(k)j-1)2]1/2 / {[(x(k)j - x(k)j-1)2 + (y(k)j - y(k)j-1)2]1/2 + [(x(k)j - x(k)j+1)2 + (y(k)j - y(k)j+1)2]1/2},
y(k+1)j = y(k)j-1 + (y(k)j+1 - y(k)j-1) [(x(k)j - x(k)j-1)2 + (y(k)j - y(k)j-1)2]1/2 / {[(x(k)j - x(k)j-1)2 + (y(k)j - y(k)j-1)2]1/2 + [(x(k)j - x(k)j+1)2 + (y(k)j - y(k)j+1)2]1/2}.
Nota bene: These simple formulae at once give the desired result unlike the well-known formulae [1]
(Ax + By + C)/(A2 + B2)1/2 = ±(Dx + Ey + F)/(D2 + E2)1/2
for the both mutually perpendicular bisector straight lines of any pair of intersecting straight lines
Ax + By + C = 0
and
Dx + Ey + F = 0
in a plane with coordinates x , y .
Compare applying adjacent sides bisectors theory (ASBT) with one step only, distance quadrat theory (DQT), the least square method (LSM), least normed square method (LNSM), unierror equalizing method (EEM), and direct solution method (DSM) [2-4] to test equation set
29x + 21y = 50,
50x - 17y = 33,
x + 2y = 7,
2x - 3y = 0
with the unit own quantities of the equations, see Figures 1, 2:
Figure 1
Figure 2
The LSM gives x ≈ 1.0023, y ≈ 1.0075 practically ignoring the last two equations with smaller factors (unlike DQT, the EEM, DSM, and even LNSM).
Adjacent sides bisectors theory (ASBT) providing simple explicit quasisolutions to even contradictory problems is very efficient by solving many urgent problems.
Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his very useful remarks.
References
[1] Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994
[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich, 2004
[3] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416
[4] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[5] Lev Gelimson. General Problem Fundamental Sciences System. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[6] Lev Gelimson. Corrections and Generalizations of the Least Square Method. In: Review of Aeronautical Fatigue Investigations in Germany during the Period May 2007 to April 2009, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2009-076 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2009, EADS Innovation Works Germany, 2009, 59-60