Abstrakt
Recently, we have studied four-dimensional synthetic constructions of real regular quadric sections of four-dimensional cones with an ellipsoidal (for unruled quadrics) and a one-sheet hyperboloidal (for ruled quadrics) directrix in the affine classification. The four-dimensional space is visualized in the double orthogonal projection onto two mutually perpendicular 3-spaces, in which a four-dimensional object is represented by its two conjugated three-dimensional images in one modeling 3-space.
This way, tools of the classical descriptive geometry are generalized and conveniently used with interactive computer graphics for synthetic constructions in the four-dimensional space. In this contribution, synthetic constructions of all the real singular quadrics in the double orthogonal projection are carried out.
Each singular three-dimensional quadric is ruled, and hence for finding the most of real cases, we choose hypercones containing a one-sheet hyperboloid. Spatial sections of a one-sheet hyperboloidal hypercone through its singular point (vertex) are three dimensional real cones or two real planes intersecting in a line.
Considering a hypercone with an improper singular point (i.e. four-dimensional hypercylinder) with a one-sheet hyperboloidal directrix, the following spatial sections: an elliptic, parabolic, and hyperbolic cylinder, or two parallel planes; can be derived. Furthermore, to obtain a double plane, or a proper and improper planes, as spatial sections, we choose a singular four-dimensional quadric with at least a singular line.
We visualize hyperquadrics with their spatial sections in the double orthogonal projection and support the constructions with their analytic derivations in the projective extension of the real space. All visualizations are supplemented with interactive 3D models with step-by-step constructions.
The purpose of the presented work is to show how a generalization of descriptive geometry methods of Monge's projection is applied for a deeper understanding and investigation of the properties of four-dimensional hyperquadrics.