The books I have state this redult as obvious from the definitions, but it is not clear to me why this is so.
2026-04-01 15:58:00.1775059080
Why 9 points determine a quadric
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A generic quadric surface has equation: $$ ax^2+by^2+cz^2+dxy+cxz+eyz+fx+gy+hz+k=0 $$ You can alway divide for a not null coefficient so you have $9$ independent coefficients that you can determine giving $9$ points.