Other curves; hypercycles
| Consider the circles tangent to a fixed line
at the point P on the line with centre M. As you move M out towards the
boundary the circle approaches a limiting curve, one that touches the boundary
at a point. This curve is called a horocycle through P.
Such curves (they can be loosely thought of as d-circles with infinite radius, if you think of the similar set up in the Euclidean plane) are not d-circles in the space as they have a point on the boundary and no centre point in the non E sense. Move the point M. The d-circle approaches one of the two limiting horocycle curves through P, shown in red. These curves were important in the work of Bolyai and Lobachevsky |
Centres of Circles?Other curves; hypercycles
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