Analysis of mapping of general II degree surfaces in collinear spaces

Abstract

Mapping of projective creations which includes the II degree surfaces in the projective, general collinear spaces is complex. In order to simplify it, firstly the characteristic parameters must be constructively determined: vanishing planes, axes and centers of spaces. All II degree surfaces are mapped using the common elements of absolute conic and infinitely distant conic of quadrics in the infinitely distant plane of space, which provide the determination of parameters of any surface of II degree. The common elements of their associated pair of conics in vanishing plane of space are used. The paper analyzed the conditions of choice of general surface of II degree in the first space to be mapped into the respective general surface of II degree in the second collinear space. The mapping is biunivocal. A sphere is chosen in the first space, and it was analyzed how it should be placed in respect to the characteristic parameters of the space, so that it would be mapped in rotating or triaxial general surfaces of II degree in the second space.

Keywords

• collinear spaces
• general surface of II degree
• characteristic parameters of space
• rotating surfaces
• triaxial surfaces