Graphic representation of a triaxial ellipsoid by means of a sphere in general collinear spaces

For graphic representation of the projective creations, such as the quadrics (II degree surfaces) in projective, general collinear spaces, it is necessary to firstly determine the characteristic parameters, such as: vanishing planes, axes and centers of space. An absolute conic of a space is an imaginary conic, residing in the infinitely distant plane of that space. The common elements of the absolute conic and infinitely distant conic of a quadric in the infinitely distant plane of that space are the autopolar triangle and two double straight lines which are always real and it is necessary to use the common elements of their associated pair of conics in the vanishing plane of the associated space. The quadric axes are passing through the apices of the autopolar triangle, and they are important for graphic representation of the quadrics. In order to map a sphere in the first space into the triaxial ellipsoid in the second space, it is necessary to select a sphere so that its center is not on the axis of that space and that it intersects the vanishing plane of the second space along the imaginary circumference, which is in general position with the figure of the absolute conic of the second space (the associated pair of conics in the vanishing plane).

Analysis of mapping of general II degree surfaces in collinear spaces

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.

Constructive procedure for determination of absolute conic figure in general collinear spaces

When they are collinear, projective spaces set with five pairs of biuni-vocally associated points are general. In order to map quadrics (II degree surfaces), in these spaces, the absolute conic was used. Geometrical position of all the absolute points in the infinitely distant plane of one space, i.e. an absolute conic of space cannot be graphically represented. To the infinitely distant planes are associated by the vanishing planes, and the absolute conics are associated by the conic in the vanishing planes, that is, figures of the absolute conics. Prior to mapping the quadrics, it is necessary to constructively determine the characteristics parameters such as the vanishing planes, axes and centers of space, and then the figures of the absolute conics, in the vanishing planes of both spaces. In order to constructively determine the figure of the absolute conic in the second space, a sphere in the first space was used, which maps into a rotating ellipsoid in the second space. The center of the sphere is on the axis of the first space, and the infinitely distant plane intersects it along the absolute conic. The associated rotational ellipsoid, whose center is on the axis of the seconds space is intersected by the vanishing plane of the first space along the imaginary circumference aI, whose real representative is circumference az. The circumference aI is the figure of the absolute conic of the first space. General collinear spaces are presented in a pair of Monge's projections.