Light travels in a straight line and so you would have to see the tube vanish to a point before you would see it curve upwards. I’m assuming “down” is the outside wall of the ring, like your picture.
Angular resolution of the naked eye is about one arcminute, about 0.0017 degrees. Let’s assume you won’t notice a rise going just over your head, let’s say 2 metres, before your vanishing point.
Some geometry and some trig suggests this ultimately comes out that it would be about 7,000 metres x (the width of the tube in metres) to your vanishing point.
More geometry and trig, and if my calculations are right, the radius to the centre of the torus, i.e. the point around which it would revolve, is about ( (the square of the width of the tube in metres) / 15 ) x 100 million metres.
Big in other words.
If your tube is 500 metres across, like your picture perhaps, that would be 35km to the vanishing point, and 2 and 2/3 billion km across to the other side of the ring.
Centred on the sun, the ring would be a ways outside the orbit of Saturn. Cold.
My calculations might be wildly wrong. Why not check on https://stackexchange.com ?
https://worldbuilding.stackexchange.com
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