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[parent] tangent plane of quadratic surface (Result)

The common equation of all quadratic surfaces in the rectangular $ (x,\,y,\,z)$-coordinate system is

$\displaystyle Ax^2+By^2+Cz^2+2A'yz+2B'zx+2C'xy+2A''x+2B''y+2C''z+D = 0$ (1)

where $ A,\,B,\,C,\,A',\,B',\,C',\,A'',\,B'',\,C'',\,D$ are constants and at least one of the six first is distinct from zero. The equation of the tangent plane of the surface, with $ (x_0,\,y_0,\,z_0)$ as the point of tangency, is
$\displaystyle Ax_0x+By_0y+Cz_0z+A'(z_0y+y_0z)+B'(x_0z+z_0x)+C'(y_0x+x_0y)+ A''(x+x_0)+B''(y+y_0)+C''(z+z_0)+D = 0.$
This is said to be obtained from (1) by polarizing it.

Example. The tangent plane of the elliptic paraboloid $ 4x^2+9y^2 = 2z$ set in the point $ (x_0,\,y_0,\,z_0)$ of the surface is $ 4x_0x+9y_0y = z+z_0$, and especially in the point $ (\frac{1}{2},\,\frac{1}{3},\,1)$ it is $ 2x+3y -z-1 = 0$.



"tangent plane of quadratic surface" is owned by pahio.
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See Also: tangent of conic section, quadratic surfaces


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Cross-references: elliptic paraboloid, polarizing, point, surface, tangent plane, quadratic surfaces, equation
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This is version 4 of tangent plane of quadratic surface, born on 2005-01-29, modified 2006-11-01.
Object id is 6681, canonical name is TangentPlaneOfQuadraticSurface.
Accessed 1855 times total.

Classification:
AMS MSC: 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
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