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 J = λ 1 λ 2 ≠ 0 we can distinguish various cases:<br />
1. If det A ≠ 0 (this is, b 03 ≠ 0) The reduced equation of the affine quadric<br />
is<br />
0 = 2b 03 x 3 + λ 1 x 2 1 + λ 2 x 2 2<br />
and we have:<br />
a) If sign(λ 1 ) = sign(λ 2 ), this is J > 0, the reduced equation of the affine<br />
quadric is of the form<br />
0 = dx 3 + a 2 x 2 1 + b 2 x 2 2<br />
which is the equation of an elliptic paraboloid.<br />
b) If sign(λ 1 ) ≠ sign(λ 2 ), this is J < 0, the reduced equation of the affine<br />
quadric is of the form<br />
0 = dx 3 + a 2 x 2 1 − b 2 x 2 2<br />
which is the equation of an hyperbolic paraboloid.
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