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From: stolfi@jumbo.UUCP
Newsgroups: comp.graphics
Subject: Re: Can You Transform Quadric Surfaces With 4x4 Matrices?
Message-ID: <686@jumbo.dec.com>
Date: Thu, 29-Jan-87 15:30:37 EST
Article-I.D.: jumbo.686
Posted: Thu Jan 29 15:30:37 1987
Date-Received: Sat, 31-Jan-87 04:42:26 EST
References: <255@onion.cs.reading.ac.uk>
Reply-To: stolfi@jumbo.UUCP (Jorge Stolfi)
Followup-To: comp.graphics
Organization: DEC Systems Research Center
Lines: 69
Keywords: quadratic surface transformation matrix homogeneous projective map
Summary: Sure: Q --> M~ Q M~'
X-Edited: last by stolfi on Thu Jan 29 12:30:12 1987 PST

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Stephen Marsh asked:

>   If I have a surface defined by 
>   
>    Q(x,y,z) = ax^2 + by^2 + cz^2 + dxy + eyz + fxz + gx + hy + iz + k 
>    
>   (ie sphere, cone etc) does anyone know if there is a
>   straightforward way of transforming this surface using the
>   usual 4x4 scale, rotate and translate matrices?  

Let the coordinates of a point p be written as a 4-element row vector
[x y z 1] (or [X Y Z W] in homogeneous coordinates, where x=X/W, y=Y/W,
z=Z/W).  

A projective transformation (one that preserves straight lines) of R^3
is determined by a 4x4 matrix M, such that the image of a point p is
simply the product p M of the 4-vector p by the matrix M.
Linear, Euclidean, and affine transformations or R^3 (rotations,
translations, scalings, reflections, etc.) are all special cases of
projective maps.  For example, a translation by (dx, dy, dz) is given
by the matrix 

    1   0   0   0
    
    0   1   0   0
    
    0   0   1   0
    
    dx  dy  dz  1
    
A quadratic surface can be represented by a symmetric 4x4 matrix Q, such
that a point p lies on the surface if and only if the matrix product
p Q p' = 0, where p' is the transpose of p, i.e.  the 4-element COLUMN
vector <x y z 1>.  

Specifically, the quadratic surface with equation 

   ax^2 + by^2 + cz^2 + dxy + eyz + fxz + gx + hy + iz + k = 0
   
is given by the matrix

      a    d/2   f/2   g/2
      
     d/2    b    e/2   h/2
     
     f/2   e/2    c    i/2
     
     g/2   h/2   i/2    k
     
So, how do we map a quadratic surface Q by a map M?  We want a surface K
such that p is on Q if and only if (p M) is on K; that is, we want a matrix
K such that 

  p Q p' = 0  iff   (p M) K (p M)' = 0
  
Since (p M)' = M' p', we conclude that 

  K = M~ Q  M~'
  
will do the trick, where M~ is the inverse of M, and M~' its transpose.

j.

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