Abstract
The article is devoted to the consideration of a number of theoretical questions of projective geometry related to specifying and displaying imaginary objects, especially, conics. The lack of development of appropriate constructive schemes is a significant obstacle to the study of quadratic images in three-dimensional space and spaces of higher order. The relationship between the two circles, established by the inversion operation with respect to the other two circles, in particular, one of which is imaginary, allows obtain a simple and effective method for indirect setting of imaginary circles in a planar drawing. The application of the collinear transformation to circles with an imaginary radius also makes it possible to obtain unified algorithms for specifying and controlling imaginary conics along with usual real second-order curves. As a result, it allows eliminate exceptional situations that arise while solving problems with quadratic images in spaces of second and higher order.
Highlights
Nowadays the attention of many researchers working in the field of constructive geometry is directed to solving problems related to modeling quadrics
The well-known, but almost forgotten constructive schemes for constructing quadrics given with nine points located in three-dimensional space could not be implemented practically due to the instrumental complexity of such schemes in the planar drawing [1]
The development of automation tools for geometric constructions based on the theoretical principles of projective geometry, the beginning of introducing imaginary objects into constructive geometry make it possible to remove from the agenda the issue of instrumental limitations of the geometric method [2]
Summary
Nowadays the attention of many researchers working in the field of constructive geometry is directed to solving problems related to modeling quadrics. Since the choice of the plane position in the general case can be arbitrary, the probability of the absence of an explicit (real) intersection of a straight line with the projections of conical sections is very high. This situation, should not led to the algorithm failure in general problem solving, because it may have real result. Corresponding effective project algorithms, developed on effective structural geometric schemes, should present These considerations explain the relevance of the study, some of obtained results will be discussed below
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