, and approaches -
as v->0 and
+ as v->. Nothing new is gotten from other
such maximal intervals.
(Nov 1, 2001)
Contents:
1. Discussion
2. Definition and Curvature
3. The Standard Picture
4. Symmetries
5. Maximal Regular Domains
6. The Singularity Curves
7. The Kuen Domain Map
8. Reparametrization
9. Assembling the Saucer
10.The A Surface
11.The B Surface
12. Global Properties
13. Phantom Spheres
The original meaning of the word surface was as the 2-dimensional boundary of a 3-dimensional object, such as "the surface of the earth" or "the surface of a cube." In this sense, Kuen's surface--the artifact shown here-- is indeed a surface. However, its creases and edges show that it is not a smooth surface, that is, a 2-dimensional differentiable manifold. That would require each point to have a neighborhood like those in the Euclidean plane.
Our goal is to cut Kuen's surface into its largest smooth pieces, and to examine
their geometry and how they fit together. This dissection is possible because the
surface is described as the image of a certain differentiable function, say F, defined
on an open domain in R2. Thus it is easy to remove all the singular
points of F, leaving only the regular points, that is, the points p at which
the derivative vectors dF/du and dF/dv are linearly independent. Then for a
region D consisting solely of regular points, F carries small Euclidean neighborhoods
of each point of D to smooth neighborhoods in the image F(D), which is thus a
smooth surface.
Kuen's surface is famous for a rare property: it has constant negative Gaussian curvature. Plenty of surfaces have negative curvature--constancy is what is rare (except for a few surfaces of revolution).
Of course, Kuen's surface does not have constant Gaussian curvature everywhere. There is no way to define curvature at singular points, since there is no smooth neighborhood on which the necessary calculus can operate.
By definition, Kuen's surface--with parameter a--is the image of the mapping given in Mathematica as
kuen[a_][u_,v_] := (2a/(1+u^2*Sin[v]^2))*
{(Cos[u]+u*Sin[u])*Sin[v],(Sin[u]-u*Cos[u])*Sin[v],Cos[v]}+
{0,0,a*Log[Tan[v/2]]}
We note at once that the misfit term Log[Tan[v/2]] is well defined on the interval
0<v<, and approaches - as v->0 and
+ as v->. Nothing new is gotten from other
such maximal intervals.
To compute the curvature, we use the command superK.m built at the end of the notes on Mathematica. Assuming this command has been saved in a "dot-m" notebook, as described there, it is installed now by
<<superK.m
Then
superK[kuen[a]][u,v]
yields -1/a2. Here Mathematica gives the common-sense result, ignoring the fact that curvature is undefined on the creases.
Since the entire function kuen is a scalar multiple of a, the geometry for arbitrary a follows immediately from that for a=1. (See the analogous situation with Kummer's surface.) From now on, we use only a=1, defining
kuen1[u_,v_]:=kuen[1][u,v]
The central region of Kuen's surface, shown above, is plotted by
stan =
ParametricPlot3D[kuen1[u,v]//Evaluate,
{u,-4.5,4.5}, {v,0.05,Pi-0.05},
PlotPoints->{40,50}, PlotRange->All]
Here "//Evaluate" keeps Mathematica from complaining, and "PlotRange" blocks Mathematica's otherwise valuable tendency to clip pictures to achieve a nicely proportioned result.
We imagine this standard picture as showing a robot gripping a saucer in the tips of its arms. Views from different angles (and somewhat lower) can be produced by the command
spin[q_] := Show[stan, Boxed->False,
Axes->False,
ViewPoint->{2Cos[q],2Sin[q],0.5}]
The term "Kuen's surface" is elastic and is commonly applied to any sizeable portion of the total image of the function kuen1. It will turn out that there is a great deal more to Kuen's surface than is shown in the standard figure above. But no matter how large a region is pictured, the fact that Gaussian curvature is negative means that every small neighborhood is saddle-shaped.
Pictures as above suggest that Kuen's surface is symmetric about the planes y=0 and z=0. This is readily proved. First, evaluating kuen1[u,v] - kuen1[-u,v] gives a result of the form {0,f[u,v],0}, and the y-coordinate of kuen1 evidently changes sign when u changes sign.
Thus the u<0 half of Kuen's surface is the mirror image, in the plane y=0, of the u>0 half.
Second, evaluating kuen1[u,v] - kuen1[u,-v]
gives a result of the form {0,0,g[u,v]}. We claim that the z-coordinate of kuen1
changes sign when v is replaced by -v. In fact, Mathematica
evaluates the z-coordinate of kuen1[u,v] + kuen1[u,-v] as
Log[Cot[v/2]] + Log[Tan[v/2]] = Log[Cot[v/2]·[Tan[v/2]] = Log[1] = 0
Thus the v<Pi/2 half of Kuen's surface is the mirror image,
in the plane z=0, of the u>/2 half.
These symmetries mean that information from u>0, v>/2
translates automatically to the entire domain of kuen1.
Recall that kuen1 is defined only for v in the interval (0,), so its domain is reduced from R2 to D = R x (0,).
To find the singular points of kuen1, we use the metric components E, F, G, whose commands ee, ff, gg are given toward the end of the notes on Mathematica. By a well-known criterion, a point (u,v) of D is singular if and only if the function EG-F2 is zero at (u,v).
Now ff[kuen1][u,v] turns out to be identically zero, so there will be two possibilites for singular points, E=0 and G=0. Entering
ee[kuen1][u,v]
yields 16u2Sin[v]2/
(2+u2-u2 Cos[2v])2.
Every factor here is positive on D except u2, so we have found that
(0,v) is a singular point of kuen1 for all 0<v<. In short,
the v-axis in D consists of singular points.
Finally,
gg[kuen1][u,v]
yields N/D, where
N = (2-u2+u2Cos[2v])2 Csc[v/2]2Sec[v/2]2 and D = 4(2+u2- u2Cos[2v])2
In this case, the new singular points are given by
2-u2+u2 Cos[2v] = 0
These points lie on two curves, which can be described as the graphs of the functions u= ±uu[v] on the v-axis, where
uu[v_]:=
Sqrt[2]/Sqrt[1-Cos[2v]]
These curves, together with the v-axis, separate the domain D of kuen1 into four regions:
A: 0< u<uu[v], B: uu[v]<u,
A': -uu[v]<u<0, B': u<-uu[v].
These are the maximal regions of regularity for kuen1, so
the restricting the mapping to them gives four basic surfaces that constitute
the non-singular part of the total Kuen surface.
These surfaces are respectable geometric objects, with well-defined Gaussian curvature, geodesics, shape operators, principal curves and curvature, intrinsic distance, and so on. None of these features are well-defined at the singular points of the total Kuen surface.
If (u,v) is a singular point of the mapping kuen1, we call its image kuen1 [u,v] a singularityof Kuen's surface. As we just saw, the singular points make up three curves, and the images of these curves under kuen1 are the singularity curves of Kuen's surface:
The central singularity curve:
cs[v]:= kuen1[0,v]
The positiveand negative singularity curves:
ps[v_]:= kuen1[uu[v],v], ns[v_]:= kuen1[-uu[v],v],
where uu[v] is as defined in the preceding section.
The three singularity curves slice the total Kuen surface into the four basic surfaces described above, so these curves give a kind of framework for the total Kuen surface. We now examine them, beginning with the central curve cs.
Entering cs[v] in Mathematica gives its formula:
{2Sin[v], 0, 2Cos[v]+Log[Tan[v/2]]}
It follows at once that cs lies in the y=0 plane, asymptotically approaches
the z-axis as v->0 and v->, and that its x-coordinate goes
from 0 to 2 and back as v goes from 0 to . Because cs is a
plane curve, the plot given by
ParametricPlot3D[cs[v]//Evaluate,
{v,0.01,Pi-0.01},
PlotPoints->100, PlotRange->All]
proves that cs makes a loop. Let c1 and c2 be the v-parameter values at
which cs cuts across itself. Symmetry implies that the crossing lies in the z=0 plane.
By plotting the z-component cs[v][[3]] of cs and using Mathematica's
FindRoot command, we find the approximate values c1=0.2926,
c2=2.8490. These values are spaced evenly around v=/2, where cs[/2]={2,0,0} is the extreme point of the
loop.
It is not hard to recognize cs in the standard picture: If we imagine it as robot
gripping a saucer, then cs traces the rim of the saucer and runs along the crease
that is evident above the saucer--and by symmetry, below it, as well).
But for two exceptional points, the Kuen singularities all have the same
character--like two pages of a book opening out from a singularity curve.
This figure shows strips of the A and A' surfaces along the segment
2.5 < v < 3.0 of the central
singularity curve. The flaring surfaces are the same throughout, but above the
crossing point of cs they form the crease just mentioned, and below
it they form the
knife-edge rim of the saucer. (What happens exactly at the crossing will be
concealed by other parts of the Kuen surface.)
The other two singularity curves, ps and ns, are symmetrical about the plane y=0. They can be plotted simultaneously by
kuenpsns =
ParametricPlot3D[{ps[v],ns[v]}//Evaluate,
{v,0.21,Pi-0.21}, PlotPoints->400]
The coordinate expressions of ps and ns are too complicated to be generally
useful. However, taking ps for concreteness (ns is its mirror image) it is evident that
the tip of this arm occurs at the point where ps crosses the z=0 plane. A plot of the
z-component of ps[v] shows that it has only one zero, at v=/2. The
tip here is not a corner but a needle point, as can be seen by asking
Mathematica for a close-up view. These needle point singularities-- one each on ps
and ns--are uniquely characterized as the only points at which singularity curves are
themselves singular, that is, have vanishing velocity vectors.
The three singularity curves are plotted together by
Show[{kuencs,kuenpsns}//Evaluate, Boxed->False, Axes->False]
The singularity curves are easy to spot in the standard picture above, and
we see that the tips of ps and ns are indeed the points at which the saucer is gripped.
Thus the singularity of ps at /2, and its mirror image for ns, are truly
exceptional points. Entering Simplify[ps[Pi/2]] in Mathematica
and then finding approximate numerical values by N[%] gives
ps[Pi/2] = kuen[1,Pi/2] =
{Sin[1]+Cos[1],Sin[1]-Cos[1],0}
= {1.382.., 0.301..,0}
For ns[/2], reverse the sign of the y-components.
Our job now is to fill in the patches of surface spanned by the singularity curves.
This map shows the essential features of the domain of kuen1:
) that map to the tips of ps
and ns, where the saucer is gripped.
/2), which maps to the nose of the saucer
(point with largest x-coordinate).
The image of the region A under the mapping kuen1 will be called the A surface, and similarly for the other three regular regions. In the next section we show how these surface combine to create the saucer.
Since Mathematica only plots from rectangular domains, it cannot deal with A
directly, so we will reparametrize A to a convenient rectangle, (0,4) x (0,). A natural way to do this is proportionally: sending each point
(u,v) in this rectangle to the point (u',v) such that u/4 = u'/uu[v], for uu[v]
as in Section 5.
This gives the change of variable mapping
chv[u_,v_]:= {u*uu[v]/4,v]
Following this map with kuen1 gives a plottable function. Since the values of chv are 2-vectors we must alter kuen1 from a function of two variables to a function on 2-vectors, as follows:
kuenv[{u_,v_}]:= kuen1[u,v]
Then the required reparametrization of kuen1 is
ffA = Composition[kuenv,chv]
Reparametrization of kuen1 on the B and B' is much simpler.
For B, the map (u,v)->(u+uu[v],v) shifts the infinite rectangle u>0, 0<v<, onto B, so kuen1[u+uu[v],v] is defined on that rectangle.
For B' the reparametrization is by (u,v)->(u-uu[v],v).
The most remarkable feature of the Kuen surface is surely the saucer. To see how it is constructed, we will assemble pieces of the four basic surfaces, using parametrizations from the preceding section.
The essential piece is the part of the A surface that appears in the standard picture. Here we use the PlotRange option to restrict the x,y,and z coordinates to just the region needed.
ParametricPlot3D[ffA[u,v]//Evaluate,
{u,0,4}, {v,0.2,Pi-0.2}, PlotPoints->{30,50}],
PlotRange->{{0,2},{-0.1,1},{-1,1}}]
Recall that the rim of the saucer is given by the central singularity curve cs. Here we imagine that (with v decreasing) the A surface is descending from z=1 at upper left, guided by the rim. It then swings around a pivot point and--following the loop in cs--cuts back through itself.
If the lower left of this picture is unclear, recall that the z<0 part of the A-surface is simply the mirror image of the z>0 part.
Note that these "pivot points" are exactly the tips of the singularity curve ps and ns, which makes them the most exceptional points in the Kuen surface.
Now we want to add the other side of the saucer, which is given symmetrically by
the A' surface.
.
A check of signs in the change of variables used for the A surface shows that the mapping ffA works for A' (where u<0) as well as it does for A (where u>0). Thus to add the A' half of the saucer to the figure above, we need only change the range of u to {-4,4} and in the PlotRange replace {-0.1,1} by {-1,1}.
Comparison of this figure above with stan makes it clear that the two remaining
basic surfaces -- from B (where y<0) and B' (where y>0) -- must provided the arms
that hold the saucer. (See also the paired figures in Section 6.)
The right arm is plotted by the command
rightarm=ParametricPlot3D[
kuen1[u-uu[v],v]//Evaluate,
{u,-3,0},{v,Pi/2-0.8,Pi/2+0.8},
PlotPoints->{40,20},
Boxed->False, Axes->False]
Then the command Show[saucer, rightarm] attaches the arm to the preceding picture, giving a crude but descriptive version of the standard picture.
(The left arm can be neglected since it would be hidden in this picture.)
The standard image of Kuen's surface lies in in a rather small region of
R3 with Ð2<z<2 . But the Log(Tan) term in the z-component of
kuen1 shows that the total surface stretches from z=-
to z=+. So the question is: What are the basic surfaces like for
regions beyond the standard picture?
This picture is produced by the command
ParametricPlot3D[
ffA[u,v]//Evaluate,
{u,0,4}, {v,0.15,Pi-0.15},
PlotPoints->{40,40},
PlotRange->All].
It shows that as the A surface extends beyond the part in stan, it rather suddenly begins to spiral..
This spiralling increases violently as v approaches or 0 --
too violently for Mathematica to produce reasonable pictures. Furthermore, the
regions of the A surface with large u cover up those with smaller u. So we adopt
a different strategy, describing the A surface by plotting a large number of its
u-parameter curves.
The Domain Map shows that u-parameter curve for v const is defined on the u- interval (0,uu[v]) and hence runs from the central singularity curve to the positive singularity curve, since
kuen1[0,v]=cs[v] to kuen1[uu[v],v]=ps[v].
A machine for plotting them is given by
upar[v_]:=
ParametricPlot3D[kuen1[u,v]//Evaluate,
{u,cs[v],ps[v]},PlotRange->All,
PlotPoints->200]
We first build a u-parameter picture of the part of the A surface in the top half of
stan (z</2) by drawing several u-parameter curves and then
attaching sections of cs and ps. The result is the bottom part of the figure on the left
below. The low end of cs is the lower half of the rim of the saucer. The highest
u-parameter curves run from v=/2 up to v=2.92, which is about
where the spiralling of the A-surface becomes serious.
In plotting u-parameter curves for v past 2.92, too many curves can be confusing,
so in the figure we have added only those for v=3.08 and v=3.12. These should make
it clear how violent the spiralling becomes as v increases toward
On the right is a strip of the A surface whose top curve is a slightly enlarged copy of the start of the v=3.12 curve on the left. Note that the winding is ever tighter as v increases and that the strip rises as it goes around -- as already shown in the u-parameter picture.
The violent behavior of the A surface might have been anticipated since these u-parameter curves all end at ps, and just how wildly ps itself is spiralling can be shown by the plot
ParametricPlot3D[
ps[v]//Evaluate,{v,0.02,Pi-0.02},
PlotRange->All, PlotPoints->10000]
This plot will show only a segment of cs from z=-3 to z=+3, nevertheless the
accuracy demanded by PlotPoints->10000 is not enough to produce a smooth image.
The remaining small interval 3.135<v< has u-parameter
spirals whose heights z rise unboundedly toward .
Note that these spirals end very close in the unit cylinder
x2+y2=1, since they end in ps, which we know,
approaches that cylinder.
The A' surface can be neglected , since it is just a mirror image of the A surface.
It is not easy to predict what the B and B' surfaces will be like when they are extended beyond the "arms" used in Section 9 to build the saucer. This figure, which extends the left arm, is produced by the command
ParametricPlot3D[
kuen1[u+uu[v]]//Evaluate,
{u,0,5},{v,0.15,Pi-0.15},
PlotPoints->{30,50},PlotRange->All]
Though not very elegant, it shows that, like the A surface, the B surface is going to spiral. For the interval of v that produces the left arm, this is not noticeable, but at about v= 2.9 (and symmetrically, near -2.9) the spiralling suddenly becomes rapid.
But there are significant differences between this picture and the comparable
picture of the A surface. There the figure plotted all points of the surface between the
limits on v. Here the u-parameter runs only to u=5. However, this is enough to show
that a curtain is being wrapped around the figure--not evenly, but carried farther for
values of v farther from v=/2.
To see what happens as the wrapping continues, we draw a few u-parameter
curves.
The Domain Map shows that the u-parameter curve for v const is defined for all u>uu[v]. So these curves all start on the positive singularity curve ps.
For any fixed v, the formula for kuen1 shows at once that the spiralling u-parameter curve is headed for limit {0,0,Log[Tan[v/2]]} on the z-axis. As v increases, it takes longer for the curves to get near the z-axis, so the upper limit of u must be increased. Here is a machine that allows for this.
upar[v_,b_]:=
ParametricPlot3D[kuen1[u,v]//Evaluate,
{u,uu[v],b},PlotPoints->1000,PlotRange->All,
Boxed->False, Axes->False]
For v large, the u-parameters curves of the A surface start near the z-axis and run to ps. By contrast, the u-parameter curves of the B surface start from ps and run toward a limit on the z-axis.
The figure shows four such curves peeling off from ps. We draw ps starting at a, but end it prematurely (dots at upper left), since otherwise its many spirals on the way to d would obscure the figure.
For v=/2, the u-parameter curve (starting at a in the
figure) lies in the z=0 plane and runs from the tip of the left arm to a tight spiral around
the origin {0,0,0).
The v=2.5 curve, starting at b, is much the same, but has begun to climb. For v=3.0, starting at c, the rate of climb has increased.
At v=3.12, the spiral seems to be climbing a sphere, starting at d near its equator.
For large values of v an immense run of the u parameter is needed for the curves
to get close to their limits in the z-axis.
The four basic surfaces that derive from A, B, A', B' are each differentiably equivalent to the plane R2, and thus they share its topological properties Geometrically, all four are incomplete, that is, each has geodesics that cannot be extended over the entire real line, with constant speed.
Furthermore, all four are inextendible, that is, none can be a part of a larger isometrically immersed surface in R3 (proved by showing that normal curvatures approach infinity at singular points).
I suspect that the map kuen1 is one-to-one on B. If so, the A surface is a genuine surface in R3, that is, a 2-dimensional submanifold.
Similarly, if the A surface fails to be one-to-one only in the self-intersection involved in the construction of the saucer, then it would be an immersed surface on a par with the standard immersion of the Klein bottle in R3.
Intersections between the A and B surfaces seem to be rare, but by contrast, the A and A' surface, being symmetrical about the plane y=0, meet on every turn of their spirals, and the same is true for the B and B' surfaces.
Beyond its central part, stan, the full Kuen surface is not just a tangle of spirals. Significant structure is represented by the phantom spheres, whose existence is suggested by the u-parameter pictures of the A and B surface in the two preceding sections. The highest drawn u-parameter curve for A suggests a (southern) hemisphere, that of B a northern hemisphere. The two can be combined by drawing u-parameter curves that run from cs in the A surface on through ps well into the B surface. Here is a general command for these plots:
PlotRange->All,PlotPoints->1000]
The sphere is already evident in this plot uAB[3.1,150], and values of v
nearer It is easy to verify that for large v, the u-parameter curve, v constant, lies very
close to a sphere S[v] whose radius is nearly 1. Such a curve starts at
cs[v]=
{2Sin[v],0,2Cos[v]+Log[Tan[v/2]]
and approaches the limit {0,0,Log[Tan[v/2]} as v-> To determine how close a u-parameter curve is to its sphere, consider the
square distance from the curve to the center:
dsq[v_]:=
(kuen1[u,v]-{0,0,Cos[v]+Log[Tan[v/2]]}).
(kuen1[u,v]-{0,0,Cos[v]+Log[Tan[v/2]]}) (Here the dot between the two expressions denotes Mathematica's dot product.)
Plot[dsq[3.1],{u,0,100},
PlotRange->All]
shows that the u-parameter curve v=3.1 is very close to its sphere. Larger values
of v give increasingly accurate fits. Of course, these spheres S[v], which have
Gaussian curvatures approaching K=+1, are not part of the Kuen's surface.
We can use the above information to show that the entire Kuen surface lies
strictly inside the cylinder x2+y2=4 of radius 2, with the sole exception of the nose of the saucer: cs[¹/2]={2,0,0}.
In fact, Mathematica computes the derivative D[dsq[v],u] to be strictly
negative for u>0, so the largest value of dsq[v]is at u=0. Its value there is
Cos[v]2+4Sin[v]2. This function has a unique
maximum at v= By symmetry this argument holds for u<0 also. Actually, plots for dsq
will show that most of the Kuen surface, say the part with |z|>3, does not stray far
outside the unit cylinder.
Back to Graphics- Assisted Geometry
do even better.
.
We want the z-axis to run through the poles of S[v], so its center will be at
{0,0,Cos[v]+Log[Tan[v/2]]} and its diameter is -2Cos[v]. Thus as v approaches , the radius of S[v] approaches 1.
Taking v=3.1, for example, the plot produced by
/2, where its value is 4. Thus every point
kuen1[u,v] has distance less than 2 from some point on the z-axis--
except for points on u -> kuen1[u,Pi/2]. Consequently, every point
kuen1[u,v] has x-coordinate smaller than 2, except for
kuen1[0,Pi/2]={2,0,0}, which is indeed the nose of the saucer.