As it is well known, Isaac Newton had to develop the differential calculus, (jointly with Gottfried Leibniz), with particular reference to the historical definition of velocities as the time derivative of the coordinates, $v = dr/dt$, in order to write his celebrated equation $m a = F(t, r, v)$, where $a = dv/dt$ is the acceleration and $F(t, r, v)$ is the Newtonian force acting on the mass $m$. Being local, the differential calculus solely admitted the characterization of massive points. The differential calculus and the notion of massive points were adopted by Galileo Galilei and Albert Einstein for the formulation of their relativity, thus acquiring a fundamental role in 20th century sciences.
In his Ph. D. thesis of 1966 at the University of Turin, Italy, the Italian-American scientist Ruggero Maria Santilli pointed out that Newtonian forces are the most widely known in dynamics, including action-at-a-distance forces derivable derivable from a potential, thus representable with a Hamiltonian, and other forces that are not derivable from a potential or a Hamiltonian, since they are contact dissipative and non-conservative forces caused by the motion of the mass $m$ within a physical medium. Santilli pointed out that, due to their lack of dimensions, massive points can solely experience action-at-a-distance Hamiltonian forces.
On this ground, Santilli initiated a long scientific journey for the generalization of Newton's equation into a form permitting the representation of the actual extended character of massive bodies whenever moving within physical media, as a condition to admit non-Hamiltonian forces. Being a theoretical physicist, Santilli had a number of severe physical conditions for the needed representation. One of them was the need for a representation of extended bodies and their non-Hamiltonian forces to be invariant over time as a condition to predict the same numerical values under the same conditions but at different times.
The resulting new calculus, today known as Santilli IsoDifferential Calculus, or IDC for short, stimulated a further layer of studies that finally signaled the achievement of mathematical and physical maturity. In particular, we note: the isotopies of Euclidean, Minkowskian, Riemannian and symplectic geometries; the isotopies of classical Hamiltonian mechanics, today known as the Hamilton-Santilli isomechanics, and the isotopies of quantum mechanics, today known as the isotopic branch of Hadronic mechanics.
The main purpose in this lecture is to represent some recent researches of Santilli iso-mathematics in the area of the plane geometry. This lecture is devoted to the iso-plane geometry. It summarizes the most recent contributions in this area.
Straight iso-lines are introduced. Iso-angle between two iso-vectors is defined. They are introduced iso-lines and they are deducted the main equations of iso-lines. They are given criteria for iso-perpendicularity and iso-parallel of iso-lines. Iso-reflections, iso-rotations, iso-translations and iso-glide iso-reflections are introduced. We define iso-circles and they are given the iso- parametric iso-representations of the iso-circles. We introduce iso-ellipse, iso-parabola and iso-hyperbola and they are given some of their basic properties. The lecture is provided with suitable examples.
[1] bibitem{9} R. M. Santilli, Embedding of Lie-algebras into Lie-admissible algebras, {Nuovo Cimento} { 51}, 570 (1967). \ http://www.santilli-foundation.org/docs/Santilli-54.pdf bibitem{10} R. M. Santilli, An introduction to Lie-admissible algebras, {Suppl. Nuovo Cimento}, { 6}, 1225 (1968). bibitem{11} R. M. Santilli, Lie-admissible mechanics for irreversible systems. {Meccanica}, { 1}, 3 (1969). bibitem{13} R. M. Santilli, On a possible Lie-admissible covering of Galilei's relativity in {Newtonian mechanics for nonconservative and Galilei form-noninvariant systems}, {1}, 223-423 (1978). \ http://www.santilli-foundation.org/docs/Santilli-58.pdf bibitem{14} R. M. Santilli, Need of subjecting to an experimental verification the validity within a hadron of Einstein special relativity and Pauli exclusion principle, {Hadronic J.} {1}, 574-901 (1978). \ http://www.santilli-foundation.org/docs/Santilli-73.pdf bibitem{15} R. M. Santilli, { Lie-admissible Approach to the Hadronic Structure,} Vols. I and II, Hadronic Press (1978).\ http://www.santilli-foundation.org/docs/santilli-71.pdf\ http://www.santilli-foundation.org/docs/santilli-72.pdf bibitem{16} R. M. Santilli, { Foundation of Theoretical Mechanics,} Springer Verlag. Heidelberg, Germany, Volume I (1978). { The Inverse Problem in newtonian mechanics.}\ http://www.santilli-foundation.org/docs/Santilli-209.pdf\ Volume II, { Birkhoffian generalization lof hamiltonian mechanics,} (1982),\ http://www.santilli-foundation.org/docs/santilli-69.pdf bibitem{25} R. M. Santilli, A possible Lie-admissible time-asymmetric model of open nuclear reactions, {Lettere Nuovo Cimento} { 37}, 337-344 (1983).\ http://www.santilli-foundation.org/docs/Santilli-53.pdf bibitem{28} R. M. Santilli, Invariant Lie-admissible formulation of quantum deformations, {Found. Phys.} { 27}, 1159- 1177 (1997).\ http://www.santilli-foundation.org/docs/Santilli-06.pdf bibitem{29} R. M. Santilli, Lie-admissible invariant representation of irreversibility for matter and antimatter at the classical and operator levels, {Nuovo Cimento B} { 121}, 443 (2006).\ http://www.santilli-foundation.org/docs//Lie-admiss-NCB-I.pdf bibitem{33} R. M. Santilli and T. Vougiouklis, Lie-admissible hyperalgebras, {Italian Journal of Pure and Applied Mathematics}, (2013).\ http://www.santilli-foundation.org/Lie-adm-hyperstr.pdf bibitem{34} R. M. Santilli, { Elements of Hadronic Mechanics,} Volumes I and II, Ukraine Academy of Sciences, Kiev, second edition 1995.\ http://www.santilli-foundation.org/docs/Santilli-300.pdf\ http://www.santilli-foundation.org/docs/Santilli-301.pdf bibitem{35} R. M. Santilli, { Hadronic Mathematics, Mechanics and Chemistry,}, Vol. I [18a], II [18b], III [18c], IV [18d] and [18e], International Academioc Press, (2008). \ http://www.i-b-r.org/Hadronic-Mechanics.htm bibitem{39} R. M. Santilli, Lie-isotopic Lifting of Special Relativity for Extended Deformable Particles, {Lettere Nuovo Cimento} 37, 545 (1983).\ http://www.santilli-foundation.org/docs/Santilli-50.pdf bibitem{40} R. M. Santilli, { Isotopic Generalizations of Galilei and Einstein Relativities,} Volumes I and II, International Academic Press (1991).\ http://www.santilli-foundation.org/docs/Santilli-01.pdf\ http://www.santilli-foundation.org/docs/Santilli-61.pdf bibitem{42} R. M. Santilli, Origin, problematic aspects and invariant formulation of q-, k- and other deformations, {Intern. J. Modern Phys.} 14, 3157 (1999). \ http://www.santilli-foundation.org/docs/Santilli-104.pdf bibitem{43} R. M. Santilli, Isonumbers and Genonumbers of Dimensions 1, 2, 4, 8, their Isoduals and Pseudoduals, and Hidden Numbers of Dimension 3, 5, 6, 7, {Algebras, Groups and Geometries} Vol. 10, 273 (1993).\ http://www.santilli-foundation.org/docs/Santilli-34.pdf bibitem{52} R. M. Santilli, Nonlocal-Integral Isotopies of Differential Calculus, Mechanics and Geometries, in {Isotopies of Contemporary Mathematical Structures}, P. Vetro Editor, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 7-82 (1996).\ http://www.santilli-foundation.org/docs/Santilli-37.pdf bibitem{57} R. M. Santilli, Iso, Geno- Hypermathematics for matter and their isoduals for antimatter, {Journal of Dynamical Systems and Gerometric theories} {2}, 121-194 (2003). bibitem{59} R. M. Santilli, {Acta Applicandae Mathematicae} { 50}, 177 (1998). \ http://www.santilli-foundation.org/docs/Santilli-19.pdf bibitem{60} R. M. Santilli, Isotopies of Lie symmetries, Parts I and II, {Hadronic J.} { 8}, 36 - 85 (1985). \ http://www.santilli-foundation.org/docs/santilli-65.pdf bibitem{61} R. M. Santilli, {JINR rapid Comm., } {6}, 24-38 (1993). \ http://www.santilli-foundation.org/docs/Santilli-19.pdf bibitem{62} R. M. Santilli, Apparent consistency of Rutherford's hypothesis on the neutron as a compressed hydrogen atom, {Hadronic J.} 13, 513 (1990).\ http://www.santilli-foundation.org/docs/Santilli-21.pdf bibitem{63} R. M. Santilli, Apparent consistency of Rutherford's hypothesis on the neutron structure via the hadronic generalization of quantum mechanics - I: Nonrelativistic treatment, {ICTP communication} IC/91/47 (1992).\ http://www.santilli-foundation.org/docs/Santilli-150.pdf bibitem{64} R. M. Santilli, Recent theoretical and experimental evidence on the apparent synthesis of neutrons from protons and electrons, {Communication of the Joint Institute for Nuclear Research}, Dubna, Russia, number JINR-E4-93-352 (1993). bibitem{65} R.M. Santilli, Recent theoretical and experimental evidence on the apparent synthesis of neutrons from protons and electrons, {Chinese J. System Engineering and Electronics} Vol. 6, 177-199 (1995).\ http://www.santilli-foundation.org/docs/Santilli-18.pdf bibitem{70} R. M. Santilli, { Isodual Theory of Antimatter with Applications to Antigravity, Grand Unification and Cosmology,} Springer (2006). bibitem{71} R. M. Santilli, A new cosmological conception of the universe based on the isominkowskian geometry and its isodual, Part I pages 539-612 and Part II pages, Contributed paper in {Analysis, Geometry and Groups, A Riemann Legacy Volume}, Volume II, pp. 539-612 H.M. Srivastava, Editor, International Academic Press (1993). bibitem{72} R. M. Santilli, Representation of antiparticles via isodual numbers, spaces and geometries, {Comm. Theor. Phys.} 1994, 3, 153-181.\ http://www.santilli-foundation.org/docs/Santilli-112.pdfAntigravity bibitem{73} R. M. Santilli, Antigravity, {Hadronic J.} 1994 17, 257-284.\ http://www.santilli-foundation.org/docs/Santilli-113.pdfAntigravity bibitem{76} R. M. Santilli, Isotopic relativity for matter and its isodual for antimatter, {Gravitation} 1997, 3, 2. bibitem{77} R. M. Santilli, Does antimatter emit a new light? Invited paper for the {Proceedings of the International Conference on Antimatter}, held in Sepino, Italy, on May 1996, published in Hyperfine Interactions 1997, 109, 63-81.\ http://www.santilli-foundation.org/docs/Santilli-28.pdf bibitem{78} R. M. Santilli, Isominkowskian Geometry for the Gravitational Treatment of Matter and its Isodual for Antimatter, {Intern. J. Modern Phys}. 1998, D 7, 351.\ http://www.santilli-foundation.org/docs/Santilli-35.pdfR. bibitem{79} R. M. Santilli, Classical isodual theory of antimatter and its prediction of antigravity, {Intern. J. Modern Phys.} 1999, A 14, 2205-2238.\ http://www.santilli-foundation.org/docs/Santilli-09.pdf bibitem{80} R. M. Santilli, Lie-admissible invariant representation of irreversibility for matter and antimatter at the classical and operator levels, {Nuovo Cimento} B, Vol. 121, 443 (2006). \ http://www.santilli-foundation.org/docs/Lie-admiss-NCB-I.pdf bibitem{81} R. M. Santilli, The Mystery of Detecting Antimatter Asteroids, Stars and Galaxies, American Institute of Physics, {Proceed.} 2012, 1479, 1028-1032 (2012).\ http://www.santilli-foundation.org/docs/antimatter-asteroids.pdf bibitem{} S. Georgiev, Foundations of Iso-Differential Calculus,Vol. I-VI. Nova Science Publisher, 2014.
