The pencil of the 4th and 3rd order surfaces obtained as a harmonic equivalent of the pencil of quadrics through a 4th order space curve of the 1st category

Abstract

This paper shows the process of inverting the 4th ordered space curve of the first category with a self-intersecting point (with two planes of symmetry) and determining its harmonic equivalent. There are harmonic equivalents for five groups of surfaces obtained through the 4th order space curve of the 1st category. Mapping was done through a system of circular cross-sections. Both classical and relativistic geometry interpretations are presented. We also designed spatial models - a spatial model of the pencil of quadrics and a spatial model of the pencil of equivalent quadrics. Besides the boundary surfaces, one surface of the 3rd order, which is an equivalent to a triaxial ellipsoid, passes through this pencil of surface of the 4th order. The center of inversion is located on the contour of the ellipsoid. The parabolic cylinder is mapped into its equivalent, by mapping the contour parabola of the cylinder, in the frontal projection, in relation to the center and the sphere of inversion into a contour curve of the 4th order surface. The generating lines of the parabolic cylinder, which are in a projecting position and pass through the antipode, are mapped into circles (also in a projecting position) whose diameters are from the center of inversion to the contour line. The application of the 4th order surfaces in architectural practice is also presented.

U radu je inverzijom preslikana prostorna kriva 4. reda prve vrste sa samopresečnom tačkom (sa dve ravni simetrije) i određen je njen harmonijski ekvivalent. Prikazani su harmonijski ekvivalenti za pet grupa površi koje su dobijene kroz prostornu krivu 4 reda 1 vrste. Preslikavanje je rađeno preko sistema kružnih preseka. Dato je klasično i tumačenje u relativističkooj geometriji. Takođe su urađeni i prostorni modeli - prostorni model pramena kvadrika i pramena ekvivalentnih kvadrika. Kroz ovaj pramen površi 4. reda, osim graničnih površi, prolazi i jedna površ 3. reda koja je ekvivalent troosnom elipsoidu. Centar inverzije nalazi se na konturi elipsoida. Parabolički cilindar se preslikava u svoj ekvivalent, tako što se konturna parabola cilindra, za drugu projekciju, preslika u odnosu na centar i sferu inverzije u konturnu krivu površi 4. reda. Izvodnice paraboličkog cilindra, koje su u projicirajućem položaju i prolaze kroz antipod, preslikavaju se u krugove (takođe u projicirajućem položaju) čiji su prečnici od centra inverzije do konturne linije. Prikazana je i primena površi 4. reda u arhitektonskoj praksi.