Adjacent Corners 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), 45

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 corners bisectors theory (ACBT) 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)(a_jx + b_jy = c_j) (j = 1, 2, ... , n) (E_i)

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, a_j , b_j , and c_j . Suppose that these n straight lines build a polygon P₁P₂...P_n with n corners (vertices) P₁ , P₂ , ... , P_n (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...P_n 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 P_i be the intersection of straight lines E_i and E_i+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

P_j+kn = P_j ,

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)P_n .

The quantiset [2-5] of n polygon corners bisectors is

_p(1)B₁ , _p(2)B₂ , ... , _p(n)B_n .

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 is 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, essence, and algorithm of adjacent corners bisectors theory (ACBT) are as follows:

1) determine all the intersections of the bisectors of the adjacent corners of the polygon;

2) keep all the internal (i.e., lying in the polygon area including its interior and boundary) intersections of the bisectors of the adjacent corners of the polygon;

3) replace each external (i.e., lying beyond the polygon area including its interior and boundary) intersection of the bisectors of the adjacent corners of the polygon with the nearest (to this intersection) intersection of the bisector of those bisectors (which intersects the polygon side connecting those adjacent corners) with the polygon boundary (its closed polygonal chain);

4) if there are at all s (s ∈ N⁺ = {1, 2, ...}) internal and replaced external intersections R_i of the bisectors of the adjacent corners of the polygon which are points with their own quantities r(i) and coordinates

_r(1)R₁(x₁ , y₁), _r(2)R₂(x₂ , y₂), ... , _r(s)R_s(x_s , y_s)

where the own quantity of the ith intersection of the ith and the i+1st bisectors of the adjacent corners of the polygon is

r(i) = p(i)p(i+1) ,

then determine their center (of gravity, mass, etc.)

x = Σ_i=1^s r(i)x_i / Σ_i=1^s r(i) ,

y = Σ_i=1^s r(i)y_i / Σ_i=1^s r(i)

and consider this point (x , y) to be the unique best quasisolution [2-5] to a finite overdetermined set of n linear equations (E_i).

Nota bene: If all the internal and replaced external intersections of the bisectors of the adjacent corners of the polygon are different, then there are precisely n such intersections with the one-to-one correspondence between the set of these intersections and the set of the polygon sides (edges) connecting the adjacent corners (vertices) of the polygon and we have s = n . Otherwise, we have s < n and by each coincidence, simply add the own quantities of the coinciding intersections.

It is useful to develop an explicit formula to obtain the equation of the bisector whose initial point P_j is the jth corner (vertex) of the polygon and has coordinates

(x_j , y_j)

whereas an intermediate point P'_j of the bisector has coordinates

(x'_j , y'_j).

The equation of straight line P_jP'_j is

(x - x_j)/(x'_j - x_j) = (y - y_j)/(y'_j - y_j),

(y'_j - y_j)x + (x_j - x'_j)y = x_j(y'_j - y_j) + y_j(x_j - x'_j).

Compare applying adjacent corners bisectors theory (ACBT), adjacent sides bisectors theory (ASBT) with one step only, distance quadrat theory (DQT), 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 corners bisectors theory (ACBT) 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] Gelimson L. G. Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich, 2004

[3] Gelimson L. G. 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] Gelimson L. G. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[5] Gelimson L. G. General Problem Fundamental Sciences System. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[6] Gelimson L. G. 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