The pencils of curves of the third and fourth order obtained by mapping the pencils of conics

The principle of relativistic geometry is used to compare the inversion as a quadratic transformation in the plane with its spatial presentation in the form of two stereographic projections: from a `plane` onto a sphere and from the sphere back onto the same `plane` (from the antipodal centre of projection). Both the classical and the relativistic interpretations are presented. Different types of conic symmetry are analysed, conformal (axial and point symmetry) and non-conformal. Harmonic symmetry, which is a completely bijective and conformal central inversion, is used to present the transformation of all types of pencils of conics and their specific features into the pencils of curves of the 3rd and 4th order and their mapped specific features and vice versa. The basic transformation is inversion which is interpreted in two ways: as a quadratic transformation in the classical projective geometry and as a pure symmetry in the relativistic geometry, with constant comparison of the two geometric systems and their opportunities for explications and generalizations. A pencil of curves of the 4th order obtained by inverting the pencil of quadric cut by a `plane` into a pencil of equivalent surfaces of the 4th order cut by the same `plane` is further presented. The recognition of the equivalence of inversion with the classical axial symmetry (non-conformal symmetry) has created numerous possibilities for mapping curves and surfaces and obtaining new forms which can be of great use both in the theory of geometry and in the practice of architecture. The research in the field of expanded symmetry will greatly widen the scope of further discoveries about the plane and surface forms. The projection model was used to create the Lisp routine and the AutoCAD software was used for the purposes of computer drawing the pencils of conics, pencils of curves of the 3rd and 4th order and generating and presenting the spatial models of the surfaces. Korišćenjem principa relativističke geometrije, inverzija kao kvadratna transformaciju u ravni upoređena je sa njenim prostornim izvođenjem u vidu dva stereografska projektovanja, sa "ravni" na sferu i sa sfere na istu "ravan" (iz antipodnog centra projiciranja). Dato je klasično i relativističko tumačenje. Analizirani su takođe različiti tipovi simetrija konike, konformna (osna i centralna) i nekonformna. Harmonijskom simetrijom, kao potpuno bijektivnom i konformnom centralnom inverzijom, prikazana je transformacija svih šest tipo