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Thread: Doubts about the meaning of "canonical"

  1. #1
    Senior Member
    May 2010

    Doubts about the meaning of "canonical"

    Hi there. I have a doubt about what we are talking about when we ask, in linear algebra, for the canonical form of a conic.

    The doubt is basically, if for example, the canonical form of a circle includes a circle translated at any point, for example:

    $\displaystyle (x-2)^2+(y-3)^2=3^2$

    Or if it only reefers to the circle when it is centered at the origin, and then if I have that equation I must make a translation to get:

    $\displaystyle x'^2+y'^2=3^2$

    Essentially its the same, but I'm not sure when it is in the canonical form, and what the canonical forms of a conic are, and what isn't.

    Thats all.

    Thanks for posting and over :P
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  2. #2
    MHF Contributor Bruno J.'s Avatar
    Jun 2009
    In a way, it's only a matter of preference. A good "canonical" form must be such that there is exactly one such form for any given conic. The form you mention has that property, and it is easy to read off the conic's parameters from it.

    The two equations you give are of the same form. They only represent the same circle for as long as $\displaystyle (x,y)$ and $\displaystyle (x', y')$ are related by the translation you speak of.

    It's really a matter of preference. I could also choose to write a conic as $\displaystyle ax^2+bxy+cy^2=1$, and this form is also unique.
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