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March 15th, 2015, 04:31 PM   #1
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infinite straight line to circle transformation

Consider the line x = 1/2,
can I algebrically transform this line into a unit circle centered around some point .
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March 16th, 2015, 03:50 AM   #2
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Yes. The centre may be at $(0, -\frac12)$. The line passing through $(0, -\frac12)$ - the point on the circle farthest from the line - and $(x, \frac12)$ intersects the circle at one point, which is the image of $(x,y)$.

You can do this with a circle above the line instead.
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March 16th, 2015, 06:56 AM   #3
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Did you mean the point nearest the line x = 1/2 where the line through (0, -1/2) and (1/2, y) intersects the circle is the image of (1/2, y)?

I don't see how this can work, as there will be one point on the circle that is not in the image of the line.
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March 16th, 2015, 07:00 AM   #4
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The projective line x = 1/2 (or any other projective line!) is homeomorphic to any given circle on the affine plane. But the affine line x = 1/2 is only homeomorphic to a circle minus a point.
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March 16th, 2015, 12:21 PM   #5
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Here's a diagram.

Our point $C(a,y)$ is to be projected onto the unit circle centred at $A(a-1,b)$. It's image is at $E(x',y')$.

Now $|\vec{BC}| = y-b$ and $|\vec{DB}| = 2$. These are the opposite and adjacent (respectively) of the angle $\alpha$ in the right-angled triangle $\triangle{BCD}$. Thus we have $$\tan \alpha = {y-b \over 2} = t$$

Now, the triangle $\triangle{ADE}$ is isosceles since $|\vec{AD}| = |\vec{AE}| = 1$ so $\angle{AED} = \alpha$ and thus $\angle{DAE} = \pi - 2\alpha$ and $\angle{BAE} = 2\alpha$.

Thus we have $$x' = a - 1 + \cos 2\alpha \qquad y' = b + \sin 2\alpha$$
Now we notice that $$\cos 2A = \cos^2 A - \sin^2 A = {1 - \tan^2 A \over sec^2 A} = {1 - \tan^2 A \over 1+\tan^2 A} \\ \sin 2A = 2\sin A \cos A = { 2\tan A \over sec^2 A} = {2 \tan A \over 1+\tan^2 A}$$
And thus we have
$$x' = a - 1 + {1 - t^2 \over 1+t^2} \qquad y' = b + { 2t \over 1+t^2}$$
or
$$x' = a - 1 + {4 - (y-b)^2 \over 4 + (y-b)^2} \qquad y' = b + {2(y-b) \over 4 + (y-b)^2}$$

Note that $b$ is arbitrary in terms of the validity of the projection, so we would normally choose $b=0$.
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March 16th, 2015, 02:15 PM   #6
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$$y' = {4(y-b) \over 4 + (y-b)^2}$$
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