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Projective Parametrization

GR242

New member
Mar 20, 2018
2
I'm trying to parametrize some conics by the projective line, giving the answer in homogeneous co-ordinates.

I get how you do this in the affine case - if I have some ellipse, say:

$$\frac{x}{a^2} + \frac{y}{b^2} = c^2$$

then I can pick a point $(0, c/b)$ and run some line $y = tx+ c/b$ through it and work out algebraically the point of intersection to get expressions for $x$ and $y$ in terms of $t$ (or, I guess, alternatively, make a change of co-ordinates $y \mapsto y + c/b$ and use the line $y = tx$ - obviously all the algebra works out the same and gives me $x=0$ and $x = f(t) y= tf(t)$ as required.

I've got a bit of a block, though, in terms of convincing myself that this works out the same for projective space. I can't see how it accounts for the 'point at infinity' that is my projection point.

Later on I'm after parametrising some sphere over the complex projective line, which I guess is related to stereographic projection but again I feel like I have to do something different that I can't quite see to make it projective. Any help gratefully received - I'm really operating against some kind of mad mental block here...
 

GR242

New member
Mar 20, 2018
2
sorry, obviously meant
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = c^2$$ and for $c/b$ read $cb$