• 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

The implicit equation and lines on the surface:
 
    The Cayley surface is most naturally defined as a surface in  P³  [projective 3-space].  Our pictures can only show part of a particular affine representative of the surface, i.e. where  R³  is identified with the complement of some plane in  P³.  There are several versions of the implicit equation of the Cayley surface, depending on a choice of coordinate system in  P³.  The simplest version of the implicit equation, given in terms of homogeneous coordinates  w,x,y,z,  is:
 
            wxy + wxz + wyz + xyz = 0.
 
We will call this version the standard equation.

    We find six lines on the Cayley surface by studying its intersections with the coordinate planes.  For instance, the intersection of the Cayley surface with the plane  w = 0  satisfies the equations  w = xyz = 0.  Therefore, this intersection is the union of the following three lines:
 
    L_wx (w = x = 0);    L_wy (w = y = 0);    L_wz (w = z = 0);
 
Similarly, if we study the intersections of the Cayley surface with the other coordinate planes, we find the previous lines again, along with the following additional lines:
 
    L_xy (x = y = 0);     L_xz (x = z = 0);     L_yz (y = z = 0).
 
These six lines can be viewed as the edges of the tetrahedron whose vertices are  (1:0:0:0), (0:1:0:0), (0:0:1:0)  and  (0:0:0:1);  this is sometimes called the "coordinate tetrahedron" in  P³.  {We will see later that these four vertices are the four nodes of the Cayley surface.}

    We find three additional lines on the Cayley surface by studying the intersection of the surface with the plane  w + x + y + z = 0.  Choosing two variables, for instance  w and x,  we can re-write the standard equation as follows:
 
            wx(y + z) + (w + x)yz = 0.
 
This makes it clear that the intersection of the Cayley surface with the plane  w + x + y + z = 0  contains the following line:
 
    L₀₁ (w + x = y + z = 0).
 
(We could have plausibly called this line  L₂₃  by choosing the variables  y and z  instead of  w and x.)  By choosing other pairs of variables, we find the following additional lines:
 
    L₀₂ (w + y = x + z = 0);     L₀₃ (w + z = x + y = 0).
 
Since the Cayley surface has degree = 3, these three lines must account for its entire intersection with the   w + x + y + z = 0. 

    We have now found 9 lines on the Cayley surface.  Click here to see an explanation of why there are no other lines on this surface.