Achieved non calculus non imaginary number spline grids with PURE RULER COMPASS

[dbhunia]

Dear Experts,

Questions first:

1. How can we get splines polynomial equations?

2. How can we get exponential coefficients of splines fit in list commands to cross verify locus?

3. How can we snap normal and tangent on any spline through some jig of Autocad? Is there any ready made system for that?

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Theory of Geometrifying Trigonometry(C) started with endeavor to represent every Trigonometric Expressions with pure ruler compass method and to generate picture forms (ALL POSSIBLE) hidden inside any Trigonometric expressions. This helped us to achieve line segment unique addressing , fractals nomenclatures , spline grids addressing , Automated Theorem Proving through pure Autocad Picture Proof , Dynamic Geometry etc...

All of these types of grid lines placements and Anchor bolts placements without any iterations are possible through direct deterministic method through Theorems of Geometrifying Trigonometry(C)
These can help the cnc cut lines generations and also pre deterministic length measurements framework for cutlines and weld lines of cnc tool paths for many such shapes without using complex numbers and without using calculus.

Many Theorems of Geometrifying Trigonometry(C) directly address some particular splines and Grid lines , Tangents to these Grid lines at particular point , Normals at particular points. None of these formula are derived through use of calculus. These Theorems are generated through pure Euclidean Axioms and Trigonometric Axioms/Rules/Definitions.

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One out of 32^n numbers of possible splines in (  1  +  tan(Θ)   ) ^ n  is demonstrated here.

Theory , Algorithms , Video Demonstrations , pdf files are available in GIT HUB

Theory and algorithms

https://github.com/SanjoyNath/SanjoyNathGeometrifyingTrigonometry/wiki

Videos , pictures

https://github.com/dbhunia/GeometrifyingTrigonometryNARRAYS/wiki

https://github.com/dbhunia/GeometrifyingTrigonometryNARRAYS/wiki/Automated-GeometrifyingTrigonometry....

Other Mathematical references

https://file.scirp.org/pdf/OJDM20110200009_48842543.pdf we have used algorithms of pure RULER and COMPASS and originates with Trigonometric expressions
https://sun.iwu.edu/~the/CAM%208547_TXHe_FinalVersion.pdf
https://sun.iwu.edu/~the/frame.html
https://en.wikipedia.org/wiki/Perfect_spline
https://en.wikipedia.org/wiki/Complex_analysis
https://en.wikipedia.org/wiki/Real_analysis
https://en.wikipedia.org/wiki/Mandelbrot_set
https://en.wikipedia.org/wiki/Gilbreath_shuffle#Gilbreath_permutations
https://link.springer.com/article/10.1007/s00440-016-0698-0 we have achieved these without complex numbers

Our mathematical model is completely different from this nor purely of other types as below
https://www.cambridge.org/vi/academic/subjects/mathematics/historical-mathematical-texts/theory-imag...
https://arxiv.org/pdf/1811.02040.pdf