The Interaction of Curves and Surfaces

Simon Salamon

Politecnico di Torino

The presentation will focus on the interaction between curves and surfaces in space, developing examples from Alfred Gray' s book The Modern Differential Geometry of Curves and Surfaces with Mathematica (third edition by Elsa Abbena and the speaker, CRC 2006). We visualize the tangent developable surface associated to a twisted cubic curve, and this motivates the study of the discriminant of a cubic surface in complex projective    3-space relative to the twistor projection, a current topic of pure mathematical research. This study begs the question of how best to visualize an object in higher dimensions using Mathematica, in this instance a surface with singularities in 4-space. Successful implicit plotting and animation of families of algebraic curves and surfaces relies on the power and versatility of graphics commands in versions 6 and 7.

Ingots with assigned parameters

Consider a cuboid of length a, width b, and depth c:

s(0):={Orange,Cuboid[{0,0,0},{3,2,1}]}
S(i_):=Graphics3D[s[i]]
Show[S(0),Boxed->False]

Its associated parameters are:
    perimeter = 4 (a + b + c) = 4 p
    area = 2 (bc + ca + ab) = 2 q
    volume = abc = r
They are essentially the coefficients of the cubic polynomial with roots a,b,c:

f(x_)=-p (x^2)+q x-r+(x^3)
su:={p->a+b+c,q->a b+a c+b c,r->a b c}
g(x_)=Factor[f(x)/.su]

Question: What are the conditions on three numbers 4p, 2q, r that guarantee the existence of an ingot with this perimeter, area and volume? Two necessary conditions follow from well-known inequalities:

(p^2)>3 q
(p^3)>27 r

Equivalently, when does a cubic polynomial with assigned coefficents p,q,r > 0 admit three positive roots a,b,c? To answer this, first consider the condition that at least two of the roots are equal:

F=Discriminant[f(x),x]
Factor[F/.su]

Answer: A necessary and sufficient condition is the non-negativity of the discriminant:

F≥0

Ruled surfaces

The region in (p,q,r)-space for which there is an ingot with the specified parameters is the region "touching or inside" the quartic surface F=0:

S(1):=ContourPlot3D[F==0,{p,.1,5},{q,.1,5},{r,.1,2},Mesh->None,
                                          ContourStyle->Opacity[.5],PlotPoints->20]
pqr:=Module[{},a=RandomReal[5];b=RandomReal[5];c=RandomReal[5];
                          {PointSize[.015],Point[{p,q,r}/.su]}]
s(2)=Table[pqr,{n,500}];
Show[Array[S,2]]

The singular locus of the discriminant surface F=0 is obtained by setting the roots all equal:

a:=t
b:=t
c:=t
γ(t_)={p,q,r}/.su

It is a twisted cubic curve γ representing the "fold" in the surface:

S(3)=ContourPlot3D[F==0,{p,-3,3},{q,-3,3},{r,-3,3},Mesh->None,
                                      PlotPoints->50,Boxed->False,Axes->False];
S(4):=ParametricPlot3D[γ(t),{t,-3,3},PlotStyle->{Blue,Thickness[.015]}]
Show[Array[S,2,3]]

This surface is in fact the tangent developable of γ, the union of its tangent lines. Any such surface has Gaussian curvature K=0.

v(i_):=Module[{a,b},a:=γ((i/10));b:=γ'((i/10));{Red,Line[{a-b,a+b}]}]
s(5):=Table[v(i),{i,-20,20}]
Show[Array[S,3,3]]

A Möbius band in space is a simple example of a ruled surface with K≠0, and contains infinitely many lines:

m(u_,v_):={v cos((u/2)) cos(u)+cos(u),v sin(u) cos((u/2))+sin(u),v sin((u/2))}
ParametricPlot3D[m(u,v),{u,0,4 π},{v,1,2},Mesh->None,PlotStyle->Yellow,
                                     BoundaryStyle->Red,Boxed->False,Axes->None]

Cubic surfaces

By contrast, non-singular cubic surfaces in space possess only a finite number of lines (27 over the field ℂ of complex numbers). In the simplest example, none of these lines are real and so they are all invisibile:

W(j_):=(∑_(i=0)^3)((w(i))^j)
fermat=W(3)/.{w(0)->1,w(1)->x,w(2)->y,w(3)->z}
ContourPlot3D[fermat==0,{x,-2,2},{y,-2,2},{z,-2,2},Axes->False]

The so-called Cayley cubic is not non-singular. It has four singular points forming the vertices of a curved tetrahedron, the edges of which give rise to six real lines on the cubic surface:

σ:=(Sqrt[2])
sv:={w(0)->-x σ-z+1,w(1)->x σ-z+1,w(2)->-y σ+z+1,w(3)->y σ+z+1}
cayley=Expand[(1/48) (4 W(3)-((W(1))^3))/.sv]
ContourPlot3D[cayley==0,{x,-4,4},{y,-4,4},{z,-4,4},Axes->False]

The last two pictures represent real slices of objects that really exist in complex projective 3-space ((ℂℙ)^3), a manifold of 6 real dimensions, which is the natural environment for the study of such surfaces.

Another discriminant

There is a twistor projection ((ℂℙ)^3)-> (𝕊^4) to the 4-sphere, locally a mapping  π : (ℂ^3) -> (ℝ^4) from complex 3-space to real 4-space. It was used by R. Penrose to encode the field equations of physics into a complex setting to help reconcile gravity and quantum theory. Consider the Fermat (first) cubic again, but using coordinates in which ζ parametrizes the fibres of π, and x,y,z,t represent (ℝ^4):

w(0):=1
w(1):=ζ
w(2):=-ζ (z-I t)+x+I y
w(3):=I t+ζ (x-I y)+z
fermat2=Collect[W(3),ζ]

In joint work with M. Povero, we were interested in mapping this cubic surface to (ℝ^4) using π. The latter will be 3 to 1 except on certain exceptional points of (ℝ^4) over which there are 1, 2 or ∞ points of the cubic. These exceptional points form a 2-dimensional surface Σ, which is characterized by the vanishing of the associated discriminant d. The real real part of d is especially complicated:

d(x_,y_,z_,t_):=Discriminant[fermat2,ζ]
cx(X_):=ComplexExpand[X]
X(x_,y_,z_,t_)=cx(Re(d(x,y,z,t)))
Y(x_,y_,z_,t_)=cx(Im(d(x,y,z,t)));

The surface Σ has a total of 9 singular points in (ℝ^4). Here we verify one of them:

sw:={x->(1/2),y->((Sqrt[3])/2),z->0,t->0}
{x,y,z}/.sw
{X(x,y,z,t),((∂X(x,y,z,t))/(∂x)),((∂X(x,y,z,t))/(∂y)),((∂X(x,y,z,t))/(∂z)),((∂Y(x,y,z,t))/(∂t))}/.sw

The zero set of the real part of the discriminant in one plane displays 6-fold symmetry:

ρ:=2.1
ContourPlot3D[X[.5,x,y,z]==0,{x,-ρ,ρ},{y,-ρ,ρ},{z,-ρ,ρ},Mesh->None,
                               Boxed->False,Axes->False,PlotPoints->20,PlotRangePadding->-.7,
                               ContourStyle->{Blue,Opacity[.8],Specularity[White,30]}]

Here are slices of the zero set of the real (red) and imaginary part (green) of the discriminant for t=1/2:

duo(t_):=Module[{r,p},
r=1.5;
p(1):=ContourPlot3D[X(x,y,z,t)==0,{x,-r,r},{y,-r,r},{z,-r,r},
                                          Mesh->None,Boxed->False,Axes->False,PlotPoints->5,             
                                          ContourStyle->{Red,Opacity[.8],Specularity[White,30]}];
p(2):=ContourPlot3D[Y(x,y,z,t)==0,{x,-r,r},{y,-r,r},{z,-r,r},
                                         Mesh->None,Boxed->False,Axes->False,PlotPoints->5,
                                         ContourStyle->{Green,Opacity[.8],Specularity[White,30]}];
p(3):=Graphics3D[{PointSize[.03],Point[{0,0,1}]}];Show[Array[p,2]]]
duo((1/2))

Varying t, we build a table for a short video:

t1:= Table[duo[l/100],{l,-100,100}]
video1:=Export["disc1.avi", t1]

Boundary strings

It is hard to ascertain the intersection of the real and imaginary surfaces, but in a key breakthrough we used RegionFunction to determine the intersection curves. Consider first the problem of trying in one dimension lower; trying to determine a surface by its planar slices:

tor1:=((((x^2)+(y^2)+(z^2)+3))^2)-16 ((x^2)+(y^2))
tor2:=tor1/.x->x-6
tor=Expand[tor1 tor2-25 (x^2)];
r:=3.2
cp(o_,a_):=ContourPlot3D[Evaluate[tor==0],{x,-r,3 r},{y,-r,r},{z,-.4 r,.4 r},Mesh->None,Boxed->True,Axes->False,PlotPoints->10,ContourStyle->{Green,Opacity[o],Specularity[White,30]},BoxRatios->{2,1,.4},RegionFunction->({x,y,z}|->x<a),BoundaryStyle->{Thick,Red},ImageSize->600]
Show[Table[cp(0,a),{a,-3,9}]]

Show[cp(.2,10)]

In our problem, the curves are in 3-dimensional space. The single frame below exhibits a slice that is the so-called alternating knot (Subscript[9, 35]):

sl(t_):=Module[{r},
e=X(x,y,z,t);
f=Y(x,y,z,t);
r=1;
 ContourPlot3D[Evaluate[f==0],{x,-r,r},{y,-r,r},{z,-r,r},Mesh->None,Boxed->False,Axes->False, PlotPoints->20,ViewPoint->{1,0,1},ContourStyle->{Green,Opacity[.1],Specularity[White,30]},RegionFunction->({x,y,z}|->e>0),BoundaryStyle->{Thick,Red},ImageSize->600]]
sl(.3)

Again, varying t, we build a table for a short video:

t2:= Table[sl[l/100],{l,-100,100}]
video2:= Export["disc2.avi", t2]

A knotty conclusion

Oriented surfaces without singularities in space are classified topologically by their genus g and Euler number χ=2-2g. The case g=0 or χ=2 corresponds to a sphere, and χ=0 to a torus. In joint work with J. Viaclovsky, the discriminant for a quadric surface is known to be a torus with 0,1 or 2 pinch points. In the last case, χ=2 but the surface (illustrated) is not homeomorphic to a sphere.

g(u_,v_):={cos(u) (4 cos(v)+3),sin(u) (4 cos(v)+3),4 sin(v)}
ParametricPlot3D[g(u,v),{u,0,2 π},{v,0,2 π},PlotRangePadding->-1,
                                   Axes->False,Boxed->False,PlotRange->(({{-8, 8}, {-4, 8}, {-8, 8}})),ImageSize->500]

Another type of "singular torus" (the so-called Bohemian dome) can be formed by projecting a very natural torus (the product of two circles) in (ℝ^4) to (ℝ^3) by means of a linear projection. Its points of self-intersection form curves rather than isolated points:

h[u_, v_] = {Cos[u]/2, Sin[u]/2, Cos[v], Sin[v]}
m[1] := {1, 0, 0, 0}
m[2] := {0, 1, 1, 0}/σ
m[3] := {0, 0, 0, 1}
hh[u_, v_] := Table[h[u, v] . m[i], {i, 3}]
ParametricPlot3D[hh[u, v], {u, 0, 2π}, {v, 0, 2π}, PlotRange -> {{-.5, .5}, {-1.1, 1.1}, {-1, .98}}, 
                                           Boxed -> False, Axes -> None, Mesh -> None, PlotStyle -> {Magenta, Opacity[0.8], Specularity[White, 30]}]

The surface Σ defined by the discriminant of a cubic in ((ℂℙ)^3)projecting to  (𝕊^4) is MUCH more complicated. However, the animations we have constructed allow us to determine the topology of Σ.

Show[GraphicsRow[{sl[1.01],sl[1],sl[4/5],sl[1/2],sl[1/5],sl[0]}], ImageSize->1200]

The key frames in the animation show that the surface has Euler number

3        +        (-3+1)      +        0         +       (-6+2)       all times two,     +    (-6+3),

which totals -9 as predicted!

References

M. Povero: Tesi di dottorato, Politecnico di Torino, XXI ciclo, 2009

R. Penrose: The Road to Reality: A Complete Guide to the Laws of the Universe, Random House, 2005

S. Salamon and J. Viaclovsky, Orthogonal complex structures in (R^4), Mathematische Annalen 343, 2009

Oriented surfaces without singularities in space are classified topologically by their genus g and Euler number χ=2-2g. The case g=0 or χ=2 corresponds to a sphere, and χ=0 to a torus. In joint work with J. Viaclovsky, the discriminant for a quadric surface is known to be a torus with 0,1 or 2 pinch points. In the last case, χ=2 but the surface (illustrated) is not homeomorphic to a sphere.

g(u_,v_):={cos(u) (4 cos(v)+3),sin(u) (4 cos(v)+3),4 sin(v)}
ParametricPlot3D[g(u,v),{u,0,2 π},{v,0,2 π},PlotRangePadding->-1,
                                   Axes->False,Boxed->False,PlotRange->(({{-8, 8}, {-4, 8}, {-8, 8}}))]

Another type of "singular torus" (the so-called Bohemian dome) can be formed by projecting a very natural torus (the product of two circles) in (ℝ^4) to (ℝ^3) by means of a linear projection. Its points of self-intersection form curves rather than isolated points:

h[u_, v_] = {Cos[u]/2, Sin[u]/2, Cos[v], Sin[v]}
m[1] := {1, 0, 0, 0}
m[2] := {0, 1, 1, 0}/σ
m[3] := {0, 0, 0, 1}
hh[u_, v_] := Table[h[u, v] . m[i], {i, 3}]
ParametricPlot3D[hh[u, v], {u, 0, 2π}, {v, 0, 2π}, PlotRange -> {{-.5, .5}, {-1.1, 1.1}, {-1, .98}}, 
                                           Boxed -> False, Axes -> None, Mesh -> None, PlotStyle -> {Magenta, Opacity[0.8], Specularity[White, 30]}]

The surface Σ defined by the discriminant of a cubic in ((ℂℙ)^3)projecting to  (𝕊^4) is MUCH more complicated. However, the animations we have constructed will allow us to determine the topology of Σ and confirm that χ=-9. Details will be explained in the talk, time permitting.

References

M. Povero: Tesi di dottorato, Politecnico di Torino, XXI ciclo, 2009

R. Penrose: The Road to Reality: A Complete Guide to the Laws of the Universe, Random House, 2005

S. Salamon and J. Viaclovsky, Orthogonal complex structures in (R^4), Mathematische Annalen 343, 2009