• Cayley surface:
     • With the lines
     • Without lines
     • Duality
     • Equation
• Steiner surface:
     • Introduction
  (Not yet available)
     • The triple point
  (Not yet available)
     •

• Cayley surface:
     • Introduction
     • Duality
     • Equation
     • View without    the lines
• Steiner surface:
     • Introduction
     • The triple point

• Surface thumbnails
• JGV homepage

Projective duality:
 
    The figure shows part of the Cayley surface.  It is a surface of degree 3, with 4 singular points.  These singular points are often called nodes.  There are 9 lines on the Cayley surface.  Six of them join pairs of nodes.  Thus, we can view the nodes as vertices of a tetrahedron, and these six lines are the edges of the tetrahedron.  The other three lines lie in the trigangent plane, which will be discussed below.

    The Cayley surface is the dual variety of the Steiner surface.  This means that the points of the Cayley surface correspond bijectively to the tangent planes of the Steiner surface (except that the correspondence is not bijective along finitely many subvarieties).  Here are some specific features of this correspondence:

• Each of the 6 lines that joins a pair of nodes on the Cayley surface is the locus of tangency of a single plane.  Therefore each of these lines corresponds to a single point on the Steiner surface -- actually one of the pinch points.
• Each of the 4 plane conics on the Steiner surface is the locus of tangency of a single plane.  Therefore, each of the plane conics on the Steiner surface corresponds to a single point on the Cayley surface -- actually one of the nodes(!!).  Click here to see the a drawing of the portion of the Steiner surface around the triple point and these 4 plane conics.
• At the triple point of the Steiner surface there are three distinct tangent planes.  This imlies the existence of a plane which is tangent to the Cayley surface at three distinct points.
  • Sevín Recillas, who got me interested in the Cayley surface, called this plane the tritangent plane.
  • The intersection of the tritangent plane with the Cayley surface is the union of 3 lines which intersect pairwise in 3 distinct points.  These lines correspond in a natural way to the 3 lines on the Steiner surface.  {The lines on the Steiner surface go through the triple point.}