George Adams was interested in asymptotic lines as possible
interfaces between physical and ethereal forces. In terms of counterspace this might be equivalent to a linkage
between space and counterspace.
An asymptotic line is a kind of boundary between positive and negative curvature on a surface. For example, consider a ruled hyperboloid:
The red plane intersects it in a circle, a curve which has positive curvature, while the blue plane intersects it in a hyperbola, which
has negative curvature. If we rotate the plane from red to blue, at one position it meets the surface in two straight lines called
rulers, which have infinite (or no) curvature. Those lines are asymptotic lines because they mark the transition between
cross sections with positive and negative curvature. There are many asymptotic lines on a surface, and the rulers are the asymptotic
lines in this case.
It is clear that no such argument can be applied to an ellipsoid as all intersecting planes meet it in ellipses, which have positive curvature. Surfaces such as the ruled hyperboloid are said to have negative curvature because planes can meet them in curves with either positive or negative curvature, and only such surfaces can have asymptotic lines.
Another way of expressing all this is to say that curves with positive curvature have their centres of curvature inside the surface, while those with negative curvature have their centres of curvature outside. The asymptotic curves are a transition between these two cases. For a circle the centre of curvature is obviously its centre, while for other curves it varies and at a given point it is the centre of the tangential circle in the osculating plane which has the same curvature as the curve at that point.
For more complex surfaces there may exist points such that all the curves through them have their centres of curvature on only one side
of the surface, known as elliptical points, and hyperbolic points with centres of curvature on both sides for the various
curves passing through it. A surface must possess hyperbolic points for it to contain asymptotic lines.
The spiralling curves on the vortex below are its asymptotic lines:
For such surfaces we have to go to the infinitesimal and consider the asymptotic direction at a point P on the surface.
If we take all possible planes in that point each meets the surface in a curve, and the asymptotic direction is that tangent at P
which separates tangents to curves with positive curvature from those with negative curvature. In general there exist two
asymptotic directions through P, tangential to two curves such that the tangent at every point of them is an asymptotic direction,
and hence those two curves are asymptotic lines of the surface.
Furthermore, the asymptotic lines are curves whose osculating planes coincide with the tangent planes at each point of the curve. Now an osculating plane at a point P on a curve is that plane in which the tangent at P is momentarily turning i.e. the curve momentarily lies in that plane. In the diagram below the red plane represents a tangent plane and the three tangents illustrate what is meant, although they should of course be 'consecutive' tangents as the curve only lies in the plane at the indicated point of tangency:
If that plane is also tangential to the surface then the curve is neither turning towards the inside of the surface nor towards its outside,
and hence demarcates those curves with their centres of curvature on one side of the surface from those with them on the other.
Hence the tangent is an asymptotic direction.
Returning to the vortex shown above, it is a surface defined by path curves with lambda equal to -1.618. There can be many sets of path curves on one such surface, and in fact its asymptotic curves are also path curves. This is not a trivial result, and a little manipulation is required to prove it. Given a path curve specified by its lambda and epsilon, a unique surface exists for which it is an asymptotic line, and all other such asymptotic lines on that surface with the same sense have the same parameters. The parameters of the surface are mu and beta, where mu is the 'lambda' value for its vertical profiles (always negative) and beta is the cotangent of the angle defining the horizontal logarithmic spiral cross sections. Thus given such a spiral together with a given point O below the centre of the spiral, the surface may be envisaged as composed of all the vertical path curves specified by mu which start from O and intersect the spiral.
If beta = 0 then the spirals degenerate to circles, which is the case for the previous vortex. In that case the asymptotic lines in the opposite sense have the same lambda but their epsilon is reversed in sign i.e. they are essentially the same path curves but winding round the surface in the opposite sense. mu equals lambda in that case, and it is epsilon that singles out the asymptotic path curves from all the others. The previous vortex has epsilon = 0.2429. For more general surfaces such as above the second set of asymptotic path curves has a different lambda from the first. Another way of thinking of the surface is to take a fixed vertical path curve and the axis, and rotate a horizontal logarithmic spiral such that its centre remains on the axis and it always intersects the path curve; its plane will move upwards or downwards parallel to itself. The diagram below shows a set of logarithmic spirals for such a surface seen from the top, with an example of each type of asymptotic line:
The practical formulae for calculating the asymptotic path curves derived by the author are:
It is clear that although a unique surface is defined by choice of lambda and epsilon for the asymptotic line, a given surface has in
general two possible values each of epsilon and lambda, corresponding to the two sets of asymptotic lines winding in opposite senses
(note that epsilon is the same for both apart from sign, but lambda is distinct). Also, setting beta = 0 makes lambda unique and
equal to mu, as stated above.
The late Dr. Georg Unger first analysed asymptotic path curves and derived the following formula for George Adams:
The great mathematician Gauss studied curves in surfaces when commissioned to make a survey of Germany, and derived the equations for such curves and their curvatures. This is the subject matter of differential geometry, and several types of curvature are defined. A very beautiful theorem is that of Meusnier which states that the circles of curvature of all plane sections through the same line element of a surface lie on a sphere. There are two principal radii of curvature at a point, R1 and R2, obtained by solving the equation for coincidence of the two directions of curvature for a normal section. The tangents for R1 and R2 define the principal directions, and in general two curves through a point are such that all their tangents are principal directions. Such curves are called lines of curvature. The Gaussian curvature is defined as K=1/R1R2, and the average curvature H by 2H=1/R1+1/R2. The angles between two asymptotic lines through a point are bisected by the lines of curvature through that point. Texts on vector algebra or differential geometry derive these results.