Re:Programmer’s guide to homogeneous coordinates
by contravariant on Sunday May 14, 2017 @06:52PM
(#14337968)
> All the surfaces described by an equation of degree n are the same.
Hang on a minute, let's not get ahead of ourselves. First of all the degenerate surfaces (e.g. (x - w)^n = 0) are clearly different from the other surfaces. Second of all, while you can use simple linear transformations to transform most quadrics into one another, there are two cases where this won't work if you use real numbers. Up to linear isomorphism these are:
X^2 + Y^2 + Z^2 - W^2 = 0
X^2 + Y^2 - Z^2 - W^2 = 0
(to get all degenerate or 'trivial' cases you also need to include equations where some of the coefficients are 0, or all coefficients are the same, see Sylvester's law of inertia for more information).
Admittedly in 2D there is only one interesting case, X^2 + Y^2 - W^2 = 0 (note that this is a cone, hence why all 2D quadrics can be generated by slicing a cone).
And that's just the 2nd order, in higher orders things get complicated quick, although I suppose that in the complex projective plane all surfaces (or rather curves) of the same degree actually are equivalent, albeit only topologically.
Hang on a minute, let's not get ahead of ourselves. First of all the degenerate surfaces (e.g. (x - w)^n = 0) are clearly different from the other surfaces. Second of all, while you can use simple linear transformations to transform most quadrics into one another, there are two cases where this won't work if you use real numbers. Up to linear isomorphism these are:
X^2 + Y^2 + Z^2 - W^2 = 0
X^2 + Y^2 - Z^2 - W^2 = 0
(to get all degenerate or 'trivial' cases you also need to include equations where some of the coefficients are 0, or all coefficients are the same, see Sylvester's law of inertia for more information).
Admittedly in 2D there is only one interesting case, X^2 + Y^2 - W^2 = 0 (note that this is a cone, hence why all 2D quadrics can be generated by slicing a cone).
And that's just the 2nd order, in higher orders things get complicated quick, although I suppose that in the complex projective plane all surfaces (or rather curves) of the same degree actually are equivalent, albeit only topologically.