Let ℍ be the higher half airplane, the set of advanced actual numbers with constructive imaginary half. Once we measure distances the best way we’ve mentioned within the final couple posts, the geometry of ℍ is hyperbolic.
What’s a circle of radius r in ℍ? The identical as a circle in any geometry: it’s the set of factors a hard and fast distance r from a middle. However whenever you draw a circle utilizing one metric, it could look very completely different when considered from the attitude of one other metric.
Suppose we placed on glasses that gave us a hyperbolic perspective on ℍ, draw a circle of radius r centered at i, then take off the hyperbolic glasses and placed on Euclidean glasses. What would our drawing appear like?
Within the earlier submit we gave a number of equal expressions for the hyperbolic metric. We’ll use the primary one right here.
Right here the Fraktur letter ℑ stands for imaginary half. So the set of factors in a circle of radius r centered at i is
Divide the expression for d(x + iy, i) by 2, apply sinh, and sq.. This provides us
which is an equation for a Euclidean circle. If we multiply either side by 4y and full the sq., we discover that the middle of the circle is (0, cosh(r)) and the radius is sinh(r).
Abstract to date
So to recap, if we placed on our hyperbolic glasses and draw a circle, then change out these glasses for Euclidean glasses, the determine we drew once more seems to be like a circle.
To place it one other approach, a hyperbolic viewer and a Euclidean viewer would agree {that a} circle has been draw. Nonetheless, the 2 viewers would disagree the place the middle of the circle is situated, and they might disagree on the radius.
Each would agree that the middle is on the imaginary axis, however the hyperbolic viewer would say the imaginary a part of the middle is 1 and the Euclidean viewer would say it’s cosh(r). The hyperbolic observer would say the circle has radius r, however the Euclidean observer would say it has radius sinh(r).
Small circles
For small r, the hyperbolic and Euclidean viewpoints practically agree as a result of
cosh(r) = 1 + O(r²)
and
sinh(r) = r + O(r³)
Huge circles
Be aware that in case you requested a Euclidean observer to attract a circle of radius 100, centered at (0, 1), he would say that the circle will lengthen exterior of the half airplane. A hyperbolic observer would disagree. From his perspective, the true axis is infinitely far-off and so he can draw a circle of any radius centered at any level and keep inside the half airplane.
Shifting circles
Now what if we checked out circles centered some other place? The hyperbolic metric is invariant underneath Möbius transformations, and so particularly it’s invariant underneath
z ↦ x0 + y0 z.
This takes a circle with hyperbolic heart i to a circle centered at x0 + i y0 with out altering the hyperbolic radius. The Euclidean heart strikes from cosh(r) to y0 cosh(r) and the radius modifications from sinh(r) to y0 sinh(r).
