Considered projective algorithms for constructing curves of the second order de-fined by a mixed set of real and imaginary elements. Composed graphical algorithms can be used to Dene a metric conic section, the incident not only valid, but also pairwise complex conjugate imaginary elements (imaginary point and an imaginary tangent) defined elliptic involutions on the line or in the direct beam. The problem of constructing a conic section given five points, two of which are imaginary. Also considered the problem of constructing conic curve defined by a point and four imaginary tangents. To solve problems using geometrically exact graph algorithms, eliminating the need to perform any algebraic calculations. The metric of the desired curve is determined using a specialized computer program. In the software algorithm uses only two graphics primitive is a straight line and a circle, so the accuracy of determination of a metric is equal to the accuracy of the representation of simple geometric objects in the base graphics package.

Keywords:Elliptic involution, the harmonic homology, the polar line, the beam of conic sections, autopolarity triangle.