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Welcome to the GeometrifyingTrigonometry wiki! We need to follow GEOMETRIFYING TRIGONOMETRY IS ALGEBRA OVER LOCKED SET(FOR TRIGONOMETRY 2D IT IS HEP ARRANGEMENT AS DEFINED) (https://geometrifyingtrigonometry.quora.com/Algebraic-Structure-of-Geometrifying-Trigonometry)

Algebraic Structure of Geometrifying Trigonometry

A RIGHT ANGLED TRIANGLE IS (BY DEFINITION OF GEOMETRIFYING TRIGONOMETRY SANJOY NATH(C) ) THE SIMPLEST LOCKED SET OF LINE SEGMENTS(OTHER LINE SEGMENTS ARE DEFINABLE , OR ADDEBLE AS PER PROBLEMS SOLVING NEEDS)

Before starting the Precised Definition of Geometrifying Trigonometry

While working on the more precised definition of Pivoting action of * operation for Geometrifying Trigonometry we have seen that ∞ (Pivoting with Aligning With Scale To Fit ) done between second HEP Arrangement(Locked Set) on superimposing with First HEP arrangement(Locked Set) need some more specific understanding. We have seen that there are 7 possible configurations can occur due to multiple usages of / and \ operation on second HEP arrangement(Locked Set) Is Sin(Θ) x Cos(Θ) equal Cos(Θ) x Sin(Θ) when interpreted or drawn geometrically?

Another Example where the Division is also shown . Division is * operation with reversed

Every multiplication in Trigonometric expression convertsinto * operation when it comes to Geometrifying Trigonometry Domain. Geometrifying Trigonometry precisely says we can replace * operation with any of these 7 operations ∞ , ∞ / , ∞ / \ , ∞ / \ / , ∞ ,∞ \ /,∞ \ / \ and after writing several conditions through Autocad programming Simulations

we have found that Sanjoy Nath(c) Conjecture that ∞ is independent , ∞ / has same effect as ∞ \ / \ , ∞ \ has same effect as ∞ / \ / And ∞ / \ has same effect as ∞ \ / So ultimately if we categorize the output configurations then we see there are 4 types of unique symmetric cases generated due to 7 different types of orientations or flips this gives insight that we can proceed with only these four conditions of * operation

After doing extensive geometric simulations through Autocad programming , we have seen that Locked sets are defined as block of line segments where we will never scale these blocks with different proportions.If we have to scale these blocks , then we will do the scale_x=scale_y=Scale_z every time and we will never explode these locked sets

We have seen that if Trigonometric expressions have + before that then doing ∞/ (Same as ∞ \ / ) gives better result and if - is there before the trigonometric expression then ∞\ (Same as ∞ / \ /) gives better results.

We prefer ∞ / or ∞ \ operations normally due to above reasons since these will give us unique outputs and unique inverses in closures of Line Segment space

Trigonometry is treated as simple Algebra when we try to solve some equations to get some values which are simple scalars and these scalar answers are Trivial for practical geometric needs which causes dilemma for Geometric programmars who writes program for professional Engineering applications .So we try to preserve the Geometry and sometimes we try to inductively auto generate all possible pictures from the given Trigonometric expressions . This shows how ambiguous it becomes in terms of possible geometric outputs

Now natural Question arise is "What is the definition of best result for * operations?" We clarify this with the concept that Suppose we have to do HP*HP then the better result of * is that which gives output line segment exactly overlaps on the line segment H (Since H is the first line segment of Geometrifying Trigonometry Duplets)

For every computable resolutions of any topic , we need to provide a good well defined Structure

GT = { L , σ , D , Σ , * , . , +,-,x ,÷ , = , µ ,■ ,τ}

L ={Set of Locked Arrangement Sets}

Σ={Set of Line Segments free line segments or Line Segments well defined and lockedly arranged conformally inside the well defined Locked Sets}

D =Doublets sets defined in the Given Locked Sets{Locked set of line segments) . Each Doublet represent the trigonometric ratios in terms of (name of denominator) .( name of numerator) forms Example Sin(Θ) => H.P , Cos(Θ) = > H.B , tan(Θ) => B.P etc... If we need additional line segments σ , then we define these accordingly for given problems

. is the Operator which works inside given Locked Set and it cannot work outside any given Locked Set Arrangement since it is the mapping of Denominator Line Segment to Numerator Line Segment in the Trigonometric Ratios

L is set of Locked Sets used in the given Trigonometric Expressions . To represent the whole Trigonometric Expression (Given) in Geometrifying Trigonometry form we need to identify and understand number of Different Angles (Example Θ , δ,Φ,α...) are present in the Trigonometric Expression.

If somehow we can know the number of different angles involved in the Trigonometric Expression , then we can say the numbers of Locked Set Arrangement necessary to formulate the GT String(Geometrifying Trigonometry Stringology representations)

Next comes the question of numbers of Doublets involved. If in a given Locked Set Arrangement , there are n Line Segments pre arranged and locked then we need (nC2) x (2!) numbers of Doublets .Reciprocal of Trigonometric Ratios are defined as Reversed Doublets . Every Alphabets in the String Doublets represent a pre arranged and locked line segment in Definitions of Geometrifying Trigonometry

DRAG of HEP Locked Arrangement [we are working to define this more precisely]

µ (alt 230) MOVE or DRAG (Representing FREE MOVEMENT OR FREE DRAG operation on whole Locked Set independent of * operations)[NORMALLY WE DONT ALLOW FREE DRAG , FREE ROTATION OF HEP ARRANGEMENTS]

τ (ALT 231) TORQUE or ROTATE (Free Rotation of the whole Locked Set independent of * operation) [We will define these more precisely here after some more workouts]

■ (alt 254) FREE SCALING IS DEFINED ON FREE LOCKED SET AS SCALAR ALGEBRAIC MULTIPLICATION ON WHOLE TRIGONOMETRIC EXPRESSION THAT DEFINES SOME LINE SEGMENTS MAPPING.

THIS CASE WHOLE LOCKED SET(ARRANGEMENT) SCALES PROPORTIONATELY ALONG ALL DIMENSIONS

= Equal to Sign in Our normal Daily use Trigonometry simply evaluates and compares Length value (Scalar value) of two calculated data for given angles BUT in reality when we try to think the fact geometrically , then we see that Geometry can compare things with pure ruler and compass.If we think in this way of comparizations then we see that EQUALITY is not simple comparing of lengths values as scalar things

We see that STRONG EXACT EQUALITY means the line segments are exactly overlapping on one another where as GOOD EQUALITY is defined as the line segments are COLLINIER but not overlapped and equal in length , GOOD Equality is Parallel and equal in length , ORDINARY EQUALITY means the broken lines not on single collinier line are summed to make equal length where sum is allowed only when these LINE segments are not parallel.HERE Geometrifying Trigonometry is STRICTLY DIFFERENT from vector algebra because

in case of GEOMETRIFYING TRIGONOMETRY we cannot sum or substract the Line Segments if they are not parallel .Sum means + which is start-end - start - end chain of the line segment(This is like Toe - Tip - To - Tip chain... of Vector sum) and the substraction is also possible if the line segments are parallel to each other. If Two line Segments are parallel to each other then we can Translate the start point of one line segment on the end point of another parallel line segment (DRAG operation which we are defining more precisely in some other theorems)

We have checked that many of such operations are not completely cross verified yet in CAD programming.We are working with checking of Associativity for all 7 types of * operations and also for Other operations.We are trying to search best algorithms or redefinitions regarding PIVOT , ALIGN,SCALE TO FIT of ∞ operation where as we have seen through simulations that ∞/ and ∞\ works best

[we are working to define this more precisely] this is refined when we continue to write programs for 3d softwares.We are defining here with more refinements because we are not only mathematics lovers but we do it for practical purpose.We have practical purpose at hand directly used for clients needs.

in GEOMETRIFYING TRIGONOMETRY LOCKED SET IS CONSIDERED AS GEOMETRICALLY TRIGONOMETRICALLY ARRANGED PREDEEFINED LOCKED SET WHICH RETAINS PROPORTIONS AND ANGLES BETWEEN EACH OTHER ON ANY KIND OF OPERATIONS DONE ON THAT

x (Simple x symbol) and ÷ (ALT 246) are the scaler multiplications on LOCKED SET . The multiplication with scalar are not allowed directly on Free Line Segments in Geometrifying Trigonometry . When we scale up or scale down , we actually do that on any Locked Set (GEOMETRICALLY TRIGONOMETRICALLY ARRANGED PREDEEFINED LOCKED SET is SIMPLE TO UNDERSTAND as AUTOCADs BLOCK object having some predefined LINE SEGMENTS PROPERLY ARRANGED THERE)

scalar multiplication (x operation is nothing but doing Scale /scaling of Locked set) or scalar division (This is also multiplication with reciprocal of the value ÷ ) So obviously we cannot multiply or divide any scalar number with LOCKED SET in GEOMETRIFYING TRIGONOMETRY . SO to make the definitions precise and simpler we consider that all possible types of x and ÷ are done on right side of trigonometric expressions

While writing programs for Geometrifying Trigonometry (As we are writing for Autocad, For Tekla , For Revit , For Dynamo , For Grasshopper,For Rhino ,For Octave ,For MatLAB,For Blender...) we will push all left side scalar type number objects to right side of Trigonometric sub expressions.Sub expressions are the Trigonometric strings seperated with + or - symbols

• Additions have 8 types of Arrangements and also - Substractions have 8 types of arrangements. We cannot Add nor Substract any scalar number object in Geometrifying Trigonometry.(+) Additions or ( - ) Substraction always work on some LOCKED SET when particular LINE SEGMENT OBJECT INSIDE that locked set is PARALLEL OR COLLINIER to OTHER TRIGONOMETRIC SUBEXPRESSION s LINE SEGMENT

Actually Additions or Substractions can work on line segments but line segments cannot move freely on space (As per GEOMETRIFYING TRIGONOMETRY) , so obviously any line segment can travel anywhere in 2D space (We will extend this to nD on some next time) AS AN INTEGRATED LOCKED PART OF ITS LOCKED SET (Think it As Autocads Block object )

TOO MUCH PRECISE DEFINITION OF ∞ (ALT 236) OPERATOR We know we have already defined 37 common line segments used in Trigonometric Expressions from several exercise books 37 Line Segments and 10 Points Identified to Symbol set of Geometrifying Trigonometric Formal System

We know ∞ does three things on second HEP LOCKED SET OBJECT to Superimpose one line segment of second HEP Arrangement(LOCKED SET) on the other line segment of First HEP Arrangement (LOCKED SET) where THE MOST CRITICAL OPERATION IS ROTATION

ALIGNMENT of one object with other is simply PIVOTING + ROTATING + SCALE TO FIT , where we consider ROTATION as first task that ∞ does.MINIMUM ENERGY ROTATION is the key factor here for ALIGNMENT . We choose pivoting later but we choose the direction of rotation first which will align the line segment of second HEP (Locked set) to overlap on the line segment on First HEP (Locked Set)

Suppose as per definition of Geometrifying Trigonometry Stringology Conventions , we convert the whole Trigonometric expressions into STRINGOLOGY forms examp,e (Sin(Θ))^2 [Read Sin Squared Theta as Sin(Θ) x Sin (Θ ) as usual notations] is turned into H.P_ΘH.P_Θ (here we have intoduced _ (Underscore operator to identify the GOVERNING ANGLE OF PARTICULAR LOCKED SET ARRANGEMENT), suppose a given trigonometric expression is with only one angle (GOVERNING ANGLE say Θ) considered , then the GEOMETRIFYING TRIGONOMETRY STRING simplifies to H.PH.P or more simply HP*HP

here in the expression we see that HPHP has a substring of PH sub part . With this we simply understand that the multiplication in Trigonometric expression is turned into * operation in GEOMETRIFYING TRIGONOMETRY which has 7 types of varieties as discussed earlier but we are concerned with ∞/ or ∞\ only for normal school level geometry cases.For proffessional programming cases we consider the other types of operation also NOW LETS FOCUS ONLY ON ∞

never confuse this ROTATION with τ (Free rotation of LOCKED SET ARRANGEMENT OPERATION)

This ROTATION is inharent inside ∞ operation which has two types of cases possible one is MINIMUM ENERGY ROTATION and other is MAXIMUM ENERGY ROTATION . WE HAVE USED THE MINIMUM ENEERGY ROTATIONS for ∞ operator which means more angle covered in rotations means MORE ENERGY USED to rotate and if we use less angular travel to align line segments , then there is LESS ENERGY USED . In this terms we have planned the rotation to do LEAST ENERGY SPENDING of angular rotation of one LOCKED SET to superimpose LINE SEGMENT on ANOTHER

Example of [Sin(Θ)]^2 means H.PH.P where PH is there means Right Side line segment H has to rotate (90-Θ) degrees to become vertical (Aligned on Perpendicular) which is anti clock rotation.If we choose clock wise rotation , then We have to Cover angular movement of (90+Θ) degrees which means For Sin(Θ) Case anticlockwise rotation gives LESS ENERGY ROTATION so we will rotate the Second HEP Arrangement (LOCKED SET Simplest form of LOCKED SET HEP ARRANGEMENT IS A RIGHT ANGLED TRIANGLE Having H , P, B Line Segments only)

From this example it is clear now that we are chosing Rotation operation intrinsic inside ∞ operation with specific calculations which we have put into our AFFINE TRANSFORMATIONS MECHANISM OF ALGORITHM FOR GEOMETRIFYING TRIGONOMETRY) then we do scale to fit Hypotenuse of second (Right Hand Side GT String Locked Set) Locked Set on the Perpendicular of Left Hand Side Locked Set Since PH substring is visible in the string [[[ RECALL H.PH.P where P*H is explained in detailin last line here]]] Our algorithm searches the Pivot points accordingly [check hyperlink above]

Now we want to explain the scalar multiplications in GT STRINGS .Here we have designed our scalar multiplication as BODMAS typr of OPERATOR PRECEDING MECHANISM .Strict working rule that we have followed here is that we will do all kind of scaling ( scalar x or ÷ from right side that is When all geometric operations and Affine transformations on Trigonometric expressions * operations like[whichever is best or all possible actions ∞ , ∞/ ,∞/\ , ... done] NEVER DO ANY SCALING OPERATION UNTIL THESE ACTIVITIES ARE DONE

SIMILARLY , NEVER DO ANY + , - ACTIVITY UNTIL ALL SUB EXPRESSIONS (WITH BRACKETS , *) ARE DONE . WE HAVE TO KEEP IN OUR MIND THAT WE HAVE TO CONVERT ALL DENOMINATORS PART OF SUB PARTS OF WHOLE TRIGONOMETRIC EXPRESSIONS INTO NUMERATORS BY CHOOSING RIGHT LINE SEGMENT OR THROUGH REVERSING THE DOUPLETS OF GT STRINGS. THEN WE NEED TO COMPLETE ALL * TASKS(OPERATIONS) THEN ONLY WE CAN DO + - TASKS[OPERATIONS] GEOMETRICALLY OTHERWISE WE WILL DO WRONG INTERPRETATIONS OF GEOMETRY