• Introduction
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• Duality map
• Dual parametrization
• Dual equation
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The equations of the Cayley surface and of the dual surface:
 
    The approach used here is based on what I have learned from Sevín Recillas. 

    We start with a version of the implicit equation of the Cayley surface which is valid at points of the Cayley surface where all four coordinates are nonzero.  The idea is to divide the implict equation by  wxyz,  so that we work with the implicit equation  G(w,x,y,z) = 0,  where  G(w,x,y,z) = ¹⁄_w + ¹⁄_x + ¹⁄_y + ¹⁄_z.  This leads to the following version of the duality map:
 
      (w:x:y:z) ---> (^∂G⁄_∂w: ^∂G⁄_∂x: ^∂G⁄_∂y: ^∂G⁄_∂z) = (¹⁄_w² : ¹⁄_x² : ¹⁄_y² : ¹⁄_z² ).
 
(A minus sign has been cancelled in the last homogeneous coordinate vector.) 

    Since  ¹⁄_w² = (¹⁄_w)²  (and similarly for the other variables), our current version of the implicit equation of the Cayley surface leads to the following version of an implicit equation of the dual surface:

	_    _    _    _
	√w + √x + √y + √z = 0.

We can transpose two terms to the opposite side of the equation, and then square both sides:

	_    _     _    _
	√w + √x = -√y - √z ;

	__             __
	w + 2√wx + x = y + 2√yz + z.

We can isolate the two square roots on one side of the equation, and square both sides again. Doing this and a little bit of simplifying, we obtain the following:

	__     __
	2√wx - 2√yz = -w - x + y + z;

	____
	- 8√wxyz = w² + x² + y² + z²
	- 2wx - 2wy - 2wz               - 2xy - 2xz - 2yz.

    If we square both sides of this equation, we clearly obtain the following result:
 
      64wxyz = (w² + x² + y² + z² - 2wx - 2wy - 2wz - 2xy - 2xz - 2yz)².
 
This equation is symmetric in the variables w,x,y,z. 

    Another nice feature is that this equation clearly exhibits the fact that each of the coordinate planes intersects the dual surface in a plane conic curve;  more specifically, each coordinate plane is tangent to the dual surface at every point of the corresponding curve.  In other words, this intersection occurs with multiplicity = 2.  For instance, the equation of the intersection of the dual surface with the plane  w = 0  (obtained by substituting  w = 0  into the implicit equation) is:
 
      (x² + y² + z² - 2xy - 2xz - 2yz)² = 0.

    On the other hand, using this version of the implicit equation of the dual surface to study the singular locus of the dual surface seems (to me, anyway) to be excessively complicated.  On a separate page I will present a change of coordinates that yields a version of the implicit equation which exhibits the singular locus much more directly.