Dual curve: Difference between revisions

Content deleted Content added
Reverted 4 edits by 66.30.119.105 (talk): This was correct. (TW)
Line 12:
:<math>X=\lambda \frac{\partial f}{\partial x}(p, q, r),\,Y=\lambda \frac{\partial f}{\partial y}(p, q, r),\,Z=\lambda \frac{\partial f}{\partial z}(p, q, r).</math>
 
Eliminating ''p'', ''q'', ''r'', and λ from these equations, along with ''f(p,q,r)=0Xp''+''Yq''+''Zr''=0, gives the equation in ''X'', ''Y'' and ''Z'' of the dual curve.
 
For example, let ''C'' be the [[Conic section|conic]] ''ax''<sup>2</sup>+''by''<sup>2</sup>+''cz''<sup>2</sup>=0. Then dual is found by eliminating ''p'', ''q'', ''r'', and λ from the equations
:<math>X= 2\lambda ap,\,Y=2\lambda bq,\,Z=2\lambda cr,\,ap^2Xp+bq^2Yq+cr^2Zr=0.</math>
The first three equations are easily solved for ''p'', ''q'', ''r'', and substituting in the last equation produces
:<math>\frac{X^2}{2\lambda a}+\frac{Y^2}{2\lambda b}+\frac{Z^2}{2\lambda c}=0.</math>