Torus is the formal name for the surface often described as that of a
doughnut
; but since quite a log of doughnuts are in fact simpler lumps (with
holes in them), that should really be talking about the specific subtype that's
called a ring doughnut
, and even that involves ignoring the fact that it
may well have a cavity within it, that might otherwise be counted as
contributing to the boundary of the object. However, as the
qualifier ring
hints, there is another familiar object we can use to
describe this shape: the surface of a ring. Perhaps better to visualise (given
that rings are usually much thinner in one direction than the other) would be a
car-tyre (of course, any tyre will do, but those of cars and trucks are fatter
in proportion to their circumference than those of bicycles, for example), with
the valve smoothed away
of course.
In general the n-dimensional Torus, T(n), is simply the cartesian product of n copies of the one-torus, T(1), better known as a circle; but what's usually meant by Torus, when the dimension isn't specified, is the two-torus, T(2), visualised by an embedding of it in three-dimensional space. One obvious way to do such an embedding is to have a circle of (big) radius R and, at each point of it, construct a circle, of (smaller) radius r, about that point in the plane perpendicular to it at that point. If we take the plane of the first circle as our first two co-ordinates, that results in a parameterisation of the torus as:
The parts of this closest to and furthest from the origin (centre) are the circles with typical points (R −r).[Cos(u), Sin(u)] and (R +r).[Cos(u), Sin(u)]; if we make the inner one of these equal to the cross-section's redius, r, we get R = 2.r and can write our typically point as simply r.[(Cos(v) +2).Cos(u), (Cos(v) +2).Sin(u), Sin(v)].
Just as puncturing a sphere and projecting the remainder outwards can map all but one point of a sphere to the two-dimensional plane, we can puncture a torus and map the neighbourhood of the puncture to the outer reaches of that two-dimensional plane, with the topology of the rest of the torus then being localised to a region in which we'll naturally want to pick the centre of our co-ordinates. This region looks somewhat like an idealized version of the land-surface where a bridge crosses a stream, or a stream running through a tunnel under a road. So next I want to find a parameterisation of a surface of this form, that ideally has some nice symmetries.
Written by Eddy.