ALIGN AND SCALE TO FIT FOR PRODUCTS WHERE AS ALIGN NO SCALE TO FIT FOR SUMMATIONS - SanjoyNath/SanjoyNathGeometrifyingTrigonometry GitHub Wiki
x ÷ where ALIGN AND SCALE TO FIT IS MANDATORY
+ - where ALIGN AND STRICTLY NO SCALE TO FIT IS STRICTLY TO FOLLOW
When we try to teach kids regarding problems of Height and Distance , we start with some known values of lengths and derive some other values of lengths or angles and in the process all other pictorial thinking vanish.We can now retrieve all possible conditions of these pictorial arrangements from the given equation and flow of positions of these known or unknown line segments in the process of Trigonometric derivations. This will initiate picture thinking approach in scientific way and computer algorithms for this Geometrifying Trigonometry will eradicate many Heuristic(Trial and Error process) from the show.Large number of Time Series or Stochastic problems will now become Deterministic models which are directly applicable in Economics , Econometrics , Forecasting sciences of Business modeling.Several insights will come for mathematical modeling of scientific researches where dynamics of stability points or chaotic conditions will become more explainable through dynamics of line segments within the NARRAYS BUNCH of Geometrifying Trigonometry systems of analysis.So Machine Learning will become more efficient so is Artificial Intelligence.Computational Geometry , Algebraic Geometry , Combinatorics, Vector Spaces , Affine Transformations will get new lights of analytical models due to implementations of Geometrifying Trigonometry.
On the process of developing every algorithms of Operators in Geometrifying Trigonometry , it was necessary to define the unity (1) which plays a central role in fundamental Trigonometric Identities. Intuitive meaningfulness of the value of 1(Precised Definition of 1 means the INITIATING LINE SEGMENT as per Geometrifying Trigonometry which was never defined in this way before) When we get 1 in Sin^2 (Θ) + Cos ^2 (Θ) =1 Or in Other places like Cosec ^2 (Θ) - Cot ^2 (Θ) = 1 or in Sec ^2 (Θ) - tan ^2 (Θ) = 1 which are pictorially explained in the links https://geometrifyingtrigonometry.quora.com/Corollary-Geometrifying-Trigonometry-Cosec-%CE%98-x-Cosec-%CE%98-Cot-%CE%98-x-Cot-%CE%98-1-represents-PH*PH-PB*PB-P also there are newly thought geometric intuitive insights in case of https://geometrifyingtrigonometry.quora.com/Corollary-Geometrifying-Trigonometry-Cos-%CE%98-x-Sin-%CE%98-Sin-%CE%98-x-Cos-%CE%98-has-25-Chance-to-become-equal . This was a challenge for us to define the value of 1 in terms of line segments . So i had to bring the concepts of Six types of Equality Or congruence conditions , where we are trying to redefine the SCALAR terms present in Trigonometric Expressions which are categorised as LOOSE SCALARS and TIGHT SCALARS , INITIAL LINE SEGMENT of GT_QUERY_STRINGS which forms GT_NARRAYS_BUNCH_OF_LOCKED_SETS_OF_LINE_SEGMENT_OBJECTS .To connect Geometric meaningfulness of every Trigonometric operators , we had to redefine several Geometric Affine Transformations in Intuitive way which shows its results implemented in AUTOLISP , VISUALLISP programs.
One of the most amazing Discovery is the ∞ operator (ALT + 236) [Sometimes i thought ô (ALT 147) looks better than ∞ but that dont come good in printouts so i have rejected that (ALT + 147) . ∞ Operator has LEAST ENERGY ROTATION PRINCIPLE which is a complete new concept of Geometric Symmetry searching and we are following this rule for every situations of ∞ operators ALIGN AND SCALE TO FIT options Dilemma. We know Axiomatic Euclidean Geometry says ALIGNMENT means Rotation + Translation +PIVOTING + Scaling where Rotation is not defined precisely.So i had to bring LEAST ENERGY PRINCIPLE OF ROTATION WHERE ANTICLOCK WISE IS PREFFERED THAN CLOCK WISE ROTATION Where PIVOTING of merging points are decided as per the minimum preorder summation of nodes (incidence points of line segments)which is now the working rule for ∞ operation which replaces multiplications or divisions of Trigonometric Expressions so it is very sensitive for Geometric Reasoning in the whole show of Geometrifying Trigonometry. Definitions of Merged Line and Perpendicular Bisectors of Merged lines from the GT_Query_String * operations needed extensive experience on Autocads Programming ,G CODE , M CODE Programming for CNC systems for long 20 years.Other types of multiplications and Divisions Operations ó(Flip of ∞ output locked set on the merged line),ö(Double Flip operation once on the merged line and other on the perpendicular bisector of merged line taken as mirror line),ò(Flip with respect to perpendicular bisector of Merged lines) . We have defined Semi Integral Symbols (⌠ and ⌡ to identify the start and final node points of merged line segments) This will play a very big roles for Geometrifying Trigonometry Theoritical Consistency every where for * operations.
- , + symbols are well defined as eight symbols each and - is applicable for overlaps (Retracing line segments of one above another as geometric intuitive understanding) + signifies end to end placements of collinier line segments or forcedly collinier line segments.FORCING AFFINE TRansformations are defined with ô(ALT + 147) which is not allowed on loose piece of line segment.When some FORCING is done then that is done on the whole LOCKED SET of Line segments(Pre arranged for given SEED ANGLE) .We have defined the Geometrifying Trigonometry Algebraic Settings with Doublets where left side symbol represents KNOWN LINE SEGMENT WHICH IS INPUT LINE SEGMENT and the right hand side to represent the UNKNOWN_LINE_SEGMENT_OUTPUT_LINE_SEGMENTthese concepts are coming to mind from common sense of height and distance problems of unitary methods systems so we can keep consistency of the whole story aligned with daily use Trigonometric scenarios. Since we are not getting sufficient numbers of symbols in ASCII charts , we are trying to define only four types of unique - and + Symbols (We need to define some fonts in unicode or in LATEX systems to represent these operators) Substractions are defined with ï(ALT + 139) , î(ALT + 140) ,ì(ALT +141) ,í(ALT + 161) and the Four Different Oriented flips of Additions are defined with â(ALT + 131) , ä(ALT + 132) , à(ALT + 133) and á(ALT + 160) where FORCED FREE ROTATION of entire LOCKED SET to align to do substraction and summation for non collinier cases is Defined with the Anticlock Rotation to Align as Ñ(ALT + 165) and Clock wise rotation to align as ñ(ALT + 164) where PLACE TO LEFT is defined with ├(ALT + 195) and PLACE TO RIGHT ┤(ALT + 180) .These defines the FORCED ALIGNMENTS OF LOCKED SETS OR NARRAYS BUNCH to make some specified line segments collinier and fit to end of other output line segments in case of Substractions and Additions in Geometrifying Trigonometry.I have tried to put the details of Symbols to make things consistent mnemonically memorizable and symbolically valid.
It is quite amazing that world has never defined the concept of ALIGN Operation of two line segments as so important factor for ratio arithmatic. i dont have access to journals https://www.researchgate.net/figure/The-reconstruction-of-the-line-segments-a-that-are-not-aligned-with-the-epipolar-line_fig3_251875719 . On the finite sum of powers of cosinces shows non geometric meanings of several powers of Cosines which is not intuitions provoking concept http://pefmath.etf.rs/accepted/1632-Fonseca-f2.pdf https://www.researchgate.net/publication/234499089_Closed-form_summations_of_Dowker's_and_related_trigonometric_sums . I have gone through several of books on Trigonometric sums and series which are not shown graphically.