CONSTRUCTIVE REPRESENTATION OF FOURTH-ORDER SPACE CURVES

Abstract

There are several possibilities of constructing and examining fourth-order space curves, such as analytically for metric requirements, synthetically in position geometrical analyses, in projective or constructive geometry. Computer graphics expands possibilities and widens survey. There are several possibilities for the constructive geometrical classification of fourth-order space curves. Two main classes are first and second-kind fourth-order space curves, to be classified e. g. according to their relation to a plane in the infinity. Classification may be according to the appearance of the common polar tetrahedron, various involutions, decompositions. singular points, smoothness or bifurcation of fourth-order space curves (whether being pair or odd). Cases of constructing fourth-order space curves of the first kind are considered, answering different classifications, also concerning symmetry conditions. Practical applications mainly involve involutions of fourth-order space curves. Junction curves of shell surfaces are often decompositional, symmetrical fourth-order space curves of the first kind. All these will be directly illustrated, without aiming at completeness.