Equidistance 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), 48
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], equidistance theory (EDT) 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, essence, and algorithm of equidistance theory (EDT) are as follows:
1) determine all the intersections of the bisectors of the adjacent corners of the polygon. If all the n 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 . The own quantity of each of these intersections is the product of the own quantities of the both bisectors (of the adjacent corners of this polygon) building this intersection. Using namely a quantiset and a quantiunion with a quantity of each element is also necessary (because those different intersections can contain coinciding elements) to precisely consider the quantity (multiplicity in the simplest case) of such an element. If there are coinciding intersections, then we have s < n and by each coincidence, simply add the own quantities of the coinciding intersections. Namely, by each coincidence, simply add the own quantities of the coinciding intersections. pointwise consider the above quantiset which contains at all s (s ∈ N+ = {1, 2, ...}) intersections Ri which are points with their own quantities r(i) and coordinates
r(1)R1(x1 , y1), r(2)R2(x2 , y2), ... , r(s)Rs(xs , ys);
2) for each intersection Ri , determine its distance di from the polygon side (edge) Ei (naturally, with the same index) connecting the both corresponding adjacent corners of the polygon. If all these distances are different, then their number is s . If there are coinciding distances, then we have t < s different distances and by each coincidence, simply add the own quantities of the coinciding distances. Pointwise consider the obtained quantiset
r'(1)d1 , r'(2)d2 , ... , r'(s)ds
which contains at all t (t ∈ N+ = {1, 2, ...}) distances Ri with their own quantities r'(i);
3) in this finite quantiset, determine the unique minimum distance dmin and all the m (m ∈ N+ = {1, 2, ...}) polygon sides (edges) providing this minimum distance;
4) move each polygon side (edge) in the polygon interior direction so that the length of the movement vector is namely this minimum distance dmin whereas the initial and the end positions of this polygon side (edge) are the opposite sides of the corresponding movement rectangle;
5) consider the new polygon built by intersecting all the initial polygon sides (edges) in their end positions. This new polygon completely lies in the initial polygon interior and is less than the initial polygon. Moreover, all the m polygon sides (edges) providing that minimum distance dmin transform to points. Therefore, the new polygon has n - m sides (edges) only whereas the initial polygon has n sides (edges);
6) fix the end position of each initial polygon side (edge) providing this minimum distance dmin ;
7) for the new polygon with n - m sides (edges) only, continue this process with further reducing the polygon sides (edges) number. This process is finite because this reducing by a positive integer can take place at most n times;
8) finish this process at once by obtaining a triangle;
9) for this triangle, determine the inscribed circumference center which is the intersection of the bisectors of all the internal corners of this triangle whose case is trivial.
Nota bene: The Cartesian coordinates of the incenter (inscribed circumference center) W of a triangle are weighted averages of the corresponding coordinates of the three corners (vertices), the lengths of the triangle sides opposite to these corners (vertices) being the weights of the corresponding corners (vertices) coordinates [1]. Namely, if the three corners (vertices) and their coordinates are
A(xA , yA), B(xB , yB), C(xC , yC)
and the lengths of the triangle sides opposite to these corners (vertices) are
a , b , c ,
respectively, then the Cartesian coordinates of the incenter (inscribed circumference center) of a triangle are
xW = (axA + bxB + cxC)/(a + b + c),
yW = (ayA + byB + cyC)/(a + b + c).
Give this center the quantity which is the sum of the quantities of the three initial polygon sides (edges) whose end positions after all the steps of reducing the initial polygon build this triangle;
10) for each of the remaining n - 3 sides (edges) of the initial polygon, determine the base of the perpendicular from this center to the straight line including the fixed end position of this initial polygon side (edge) by its transformation to a point. Give this perpendicular base the quantity of the corresponding side (edge) of the initial polygon;
11) for the obtained quantiset
s(1)S1(x1 , y1), s(2)S2(x2 , y2), ... , s(n-2)Sn-2(xn-2 , yn-2)
of n - 2 points (the inscribed circumference center and the n - 3 perpendicular bases) with their own quantities s(i) and coordinates, determine their center (of gravity, mass, etc.)
x = Σi=1n-2 s(i)xi / Σi=1n-2 s(i) ,
y = Σi=1n-2 s(i)yi / Σi=1n-2 s(i)
and consider this point (x , y) to be the unique best quasisolution [2-5] to a finite overdetermined set of n linear equations (Ei).
Nota bene: It is also possible that by finishing this process of reducing the initial polygon sides (edges) number, we obtain either a straight line segment or a point.
8a) if by finishing this process of reducing the initial polygon sides (edges) number, we obtain a straight line segment, then
9a) determine its middle point and give this center the quantity which is the sum of the quantities of the two initial polygon sides (edges) whose end positions after all the steps of reducing the initial polygon build this segment;
10a) for each of the remaining n - 2 sides (edges) of the initial polygon, determine the base of the perpendicular from this center to the straight line including the fixed end position of this initial polygon side (edge) by its transformation to a point. Give this perpendicular base the quantity of the corresponding side (edge) of the initial polygon;
11a) for the obtained quantiset
s(1)S1(x1 , y1), s(2)S2(x2 , y2), ... , s(n-1)Sn-1(xn-1 , yn-1)
of n - 1 points (the inscribed circumference center and the n - 2 perpendicular bases) with their own quantities s(i) and coordinates, determine their center (of gravity, mass, etc.)
x = Σi=1n-1 s(i)xi / Σi=1n-1 s(i) ,
y = Σi=1n-1 s(i)yi / Σi=1n-1 s(i)
and consider this point (x , y) to be the unique best quasisolution [2-5] to a finite overdetermined set of n linear equations (Ei).
8b) if by finishing this process of reducing the initial polygon sides (edges) number, we obtain a point, then
9b) give this center point the quantity which is the sum of the quantities of all the u (u ∈ N+ = {1, 2, ...}) initial polygon sides (edges) whose end positions after all the steps of reducing the initial polygon build this point;
10b) for each of the remaining n - u sides (edges) of the initial polygon, determine the base of the perpendicular from this center point to the straight line including the fixed end position of this initial polygon side (edge) by its transformation to a point. Give this perpendicular base the quantity of the corresponding side (edge) of the initial polygon;
11b) for the obtained quantiset
s(1)S1(x1 , y1), s(2)S2(x2 , y2), ... , s(n-u+1)Sn-u+1(xn-u+1 , yn-u+1)
of n - u + 1 points (the inscribed circumference center and the n - u perpendicular bases) with their own quantities s(i) and coordinates, determine their center (of gravity, mass, etc.)
x = Σi=1n-u+1 s(i)xi / Σi=1n-u+1 s(i) ,
y = Σi=1n-u+1 s(i)yi / Σi=1n-u+1 s(i)
and consider this point (x , y) to be the unique best quasisolution [2-5] to a finite overdetermined set of n linear equations (Ei).
Compare applying equidistance theory (EDT), opposite sides bisectors theory (OSBT), adjacent corners bisectors theory (ACBT), 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).
Equidistance theory (EDT) 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
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[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