CHAPTER III: CONICS AND QUADRICS - OCW UPM
CHAPTER III: CONICS AND QUADRICS - OCW UPM
CHAPTER III: CONICS AND QUADRICS - OCW UPM
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AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda<br />
If two of the eigenvalues of A 00 vanish (suppose λ 2 = λ 3 = 0), hence:<br />
det A = 0, det A 00 = 0, J = 0 and tr A 00 = λ 1 .<br />
In certain coordinate system the matrix of the quadric is<br />
⎛<br />
⎞<br />
b 00 0 0 0<br />
0 λ 1 0 0<br />
⎜<br />
⎟<br />
⎝ 0 0 0 0 ⎠<br />
0 0 0 0<br />
and the reduced equation of the quadric is<br />
1. If b 00 ≠ 0 we have<br />
0 = b 00 + λ 1 x 2 1.<br />
a) If sign(b 00 ) = sign(λ 1 ), the reduced equation of the affine quadric is of<br />
the form<br />
0 = p 2 + a 2 x 2 1<br />
and the affine quadric is a pair of imaginary parallel planes .
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