Re: Point on the curve - easier in implicit form?



Tony wrote:
>
> In my recent studies I often find the statement that point on
> the curve determination is simpler if curve is in implicit,
> rather than in parametric form.
>
> However, in the case of quadratic and cubic curves I fail to
> see any obvious advantage of implicit form in determining if
> point lies on the curve. In both cases I end up calculating
> the value of one polynomial and solving one quadratic/cube
> root equation.
>
> What am I missing?

Surely if your point is specified by values of each coordinate,
either numeric or parametric (the latter defining some other
curve or variety) its easier to plug it into an implicit form,
say f(x, y) = 0, and see if the result gives "0 = 0".

.

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