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`βPythagorean TriplesβGeometric Approachβπ¦-interceptβConic Section Example π₯2+2π¦2=1βKey Ingredients for Binary Operation: π₯2+ππ¦2=1ββGeneralization to degree 3 with three variablesββGeneralization to twists: ππ₯2+ππ¦2=πβπ₯2+ππ¦2=πβKey Ingredients for Parameterisation: π₯2+ππ¦2=1βSource and Reference`

Pythagorean Triples

Pythagorean triples: π₯2+π¦2=1 with rational solutions. The rational solution (π₯,π¦) can also be reference to another fixed rational point i.e. (1,0)

Geometric Approach

By sweeping a line about a fixed rational point (1,0), the slope of the line jointing the rational point (π₯,π¦) can be used to represent this rational point. And the slope is equal to π¦π₯β1. Since the rational solutions of π₯2+π¦2=1 always preserves the rationality of slope. In other words, the rational point (π₯,π¦) is mapped to a rational slope. ```π0={(π₯,π¦):π₯2+π¦2=1,π₯β 1} π:π0ββ given by π(π)=π¦π₯β1 ``` Similarly, the map π can be defined for more general curves. And the properties of π are
• π is 1-to-1
• π is surjective.
• πβ1:ββπ0 is given by rational functions
These propertiesβa parameterisation

π¦-intercept

Similar to Riemann stereographic projection, the slope of the sweeping line can be projected as the π¦-intercept on the π¦-axis. The parameterisation of the 2-degee equation with one fixed point gives one unique association with the remaining point. Let πβ1(π‘)=(π₯,π¦); π₯2+π¦2=1 βπ(π)=π‘βπ¦π₯β1=π‘ βπ¦=π‘(π₯β1) and π₯2+π¦2=1 Let π§=π₯β1 βπ¦=π‘π§ and (π§+1)2+π¦2=1 β(π§+1)2+(π‘π§)2=1 βπ§2+2π§+π‘2π§2=0 β(1+π‘2)π§2+2π§=0 If π§β 0 β(1+π‘2)π§+π§=0 βπ§=β21+π‘2βπ₯β1=β21+π‘2 βπ₯=π‘2β11+π‘2ββ βπ¦=π‘π§=π‘β21+π‘2=β2π‘1+π‘2 βπβ1(π‘)=(π₯,π¦)=π‘2β11+π‘2,β2π‘1+π‘2

Conic Section Example π₯2+2π¦2=1

Similar to other conic section. Binary Operation of π₯2+2π¦2=1
Let π(π₯,π¦),π(π€,π§) be the solution of the equation. β(π₯+π¦β2π)(π₯βπ¦β2π)=1 and (π€+π§β2π)(π€βπ§β2π)=1 β(π₯π§β2π¦π€)2+2(π₯π€+π¦π§)2=1 So πβπ=π where π=(π₯π§β2π¦π€,π₯π€+π¦π§) map π:π0ββ given by π(π)=π¦π₯β1 where π0={(π₯,π¦):π₯2+2π¦2=1, π₯β 1} πβ1(π‘)=(π₯,π¦), where π¦=π‘(π₯β1) Let π§=π₯β1β(π§+1)2+2π‘2π§2=1βπ§((1+2π‘2)π§+2)=0 βπ§=β2/(1+2π‘2) βπ₯=2π‘2β11+2π‘2 βπ¦=β2π‘21+2π‘2 βπβ1(π‘)=2π‘2β11+2π‘2,β2π‘21+2π‘2 πβ1:ββπ0

Key Ingredients for Binary Operation: π₯2+ππ¦2=1

π₯2+ππ¦2=1
• RHS being 1
• (π₯+π¦βπ)π=π₯'+π¦'βπβπ₯2π+ππ¦2π=1

Generalization to degree 3 with three variables

π₯3+2π¦3β6π₯π¦π§+4π§3=1βmanipulate β2

Key Ingredients for Parameterisation: π₯2+ππ¦2=1

• A rational point (1,0)
• The degree being 2βπβ1:ββπ0

Source and Reference

ID: 201100016 Last Updated: 16/11/2020 Revision: 0 Ref:

References

1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science &amp; Engineering
2. Wheatstone, C., 1854, On the Formation of Powers from Arithmetical Progressions
3. Stroud, K.A., 2001, Engineering Mathematics
4. Coolidge, J.L., 1949, The Story of The Binomial Theorem

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