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Steiner Surfaces
by Adam Coffman. Images rendered by POV-Ray 3.1. |
| Background | Equations and Graphics | Animation | Links |
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Steiner Surfaces
by Adam Coffman. Images rendered by POV-Ray 3.1. |
| Background | Equations and Graphics | Animation | Links |
Background
The "real projective plane" is the set of lines through the origin in
real 3-dimensional space; when each of these lines is represented by
one point, the resulting set is, in an abstract way, a smooth,
two-dimensional surface. There are many ways to represent this
surface as a two-dimensional subset of three dimensional space, and
some of these representations were studied by J. Steiner. Here's some
background on the geometry of Steiner's images of the real projective
plane.
A convenient way to assign coordinates to the real projective plane is to pick a basis of 3-space, and denote by [u0 : u1 : u2] the line equal to the set of all scalar multiples of the non-zero ordered triple (u0,u1,u2). So [1 : 2 : 3] denotes the same line as [2 : 4 : 6], and another example is [0 : 1 : 0], which is the u1-axis. This mathematical description of the projective plane is called a "homogeneous coordinate system," and it generalizes to any dimension: real projective n-space is the set of lines through the origin in n+1 dimensions.
The starting point for our approach to Steiner surfaces is the Veronese variety, a smooth, 2-dimensional surface, given by embedding the projective plane into projective 5-space by the homogeneous parametric equations
A natural way to classify Steiner surfaces is to say that two are equivalent if they are related by linear transformations of the domain (the projective plane, [u0 : u1 : u2]) and the range (projective 3-space, coordinates [x0 : x1 : x2 : x3]). A classification of Steiner surfaces was known in the XIX century in the case where the coordinates and projective transformations are allowed to be complex. A classification in the real case has been studied in our recent paper :
A. Coffman, A. Schwartz, and C. Stanton, The algebra and geometry of Steiner and other quadratically parametrizable surfaces, Computer Aided Geometric Design (3) 13 (April 1996), 257-286.
Updated comments on the paper: 
A parametrization of a Steiner surface patch in a 3-dimensional affine neighborhood by quadratic rational functions can be obtained from the quadratic polynomials by "dehomogenizing:" using two domain variables u=u1/u0 and v=u2/u0, and three range variables, so that dividing by the first homogeneous coordinate transforms, for example,
The following list of Steiner surfaces represents each equivalence class from the classification theorem, and gives several different ways to describe each object. Starting with a quadratic homogeneous parametrization in [u0 : u1 : u2], a homogeneous implicit equation can be derived. This procedure is called "elimination of parameters," and I used the algebra program Macaulay. Then, the inhomogeneous implicit equation is given, in the affine coordinates x=x1/x0, y= x2/x0, z=x3/x0. There are also parametrizations by trigonometric or other functions, which are more efficient for parametrizing the whole surface, instead of just a patch.
Some of the parametric surfaces have self-intersections along line segments, called double lines. When the parameters are eliminated to get the implicit equation, there are some "extra solutions": points which satisfy the implicit equation, but which do not appear as images of the parametric maps. These extra solutions extend the line segments to infinite lines, and appear as "whiskers" sticking out of the surface. (In general, if a line meets a quartic at more than four points, then the line must be contained completely inside the quartic.) In the POV code, some of the whiskers have been clipped off, and some are already invisible because the rays that reflect off the surface miss these lines entirely.
I've posted some code, which at least has worked for me, using a Gateway 2000 E-3100 computer. The Maple code worked with Maple V Release 4 Version 4.00b. The command with(plots); invokes the Maple graphics package, and then the plot3d, display3d and animate3d can be displayed in a new Maple window where one can rotate, color, and otherwise manipulate the surfaces in space.
The POV-Ray implementation was based on similar web sites, and was developed using POV-Ray™ Version 3.1a.watcom.win32. Click on the small picture to get a larger picture. The code for the larger picture is available as a .txt file. The POV program uses the affine implicit equation of the surface, instead of the parametric plot used by Maple.
I would be glad to hear about any use you find for this page, including pictures you make and post to the WWW. I will also entertain questions and suggestions. My email is CoffmanA (at) ipfw.edu; you can acknowledge this resource by linking to this URL: http://www.ipfw.edu/math/Coffman/steinersurface.html.
1. |  |
Steiner's Roman Surface. Three double lines, six pinch
points, and a triple point. For a few more images, see my graphics
gallery.
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2. |  |
This surface can be transformed into Steiner's Roman Surface,
by a complex change of coordinates (transforming x to ix and z
to iz interchanges the Type 1 and 2 implicit equations, where
i2=-1). It can also be transformed into the Type 3
Cross-cap by a complex transformation, but Types 1, 2, and 3 are not
related by any real transformation. It has two real pinch points and
three double lines meeting at a triple point, and, unlike the Roman or
Cross Cap, is not compact in any affine neighborhood.
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The linear transformation which interchanges the x0
and x2 coordinates gives another view of the Type 2
surface, where two of the double lines are on the plane at infinity.
The double line connecting the two real pinch points is still visible
in this affine neighborhood. Planes containing this double line
intersect the surface along parabolas, so this representation has been
called a "parabolic Steiner surface."
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3. |  |
Steiner's Cross Cap. (equations from F. Apéry's book)
One real double line is visible, but there are also two complex double
lines, which would only be visible after a complex coordinate change,
resulting in a Type 1 or 2 surface. There are two real pinch points,
which curve in different directions (see the patches in the larger
picture). It's also known as the Cross-Cap or Crosscap, and the pinch
points are also called cross-cap singularities or Whitney
singularities.
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4. |  |
Two of the three double lines in the Type 2 surface here
coincide, forming a "tacnodal" line, along which two neighborhoods of
this noncompact surface appear to be tangent. This line contains one
of the two pinch points, and the remaining double line connects the
two pinch points.
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5. |  |
Two of the three double lines in Steiner's Roman Surface here
coincide, forming a tacnodal line meeting the other double line in a
"T" shape, and four singularities at the segment endpoints. It is
related to the previous surface by complex but not real
transformations.
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6. |  |
The three double lines of Steiner's Roman Surface coincide,
forming an "oscnodal" line. I call this type the "Cross Cup," since
it resembles the Cross Cap, but with the double line tangent to the
surface.
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7. |  |
Quadratically parametrized surfaces can also have
cubic implicit equations. The surfaces turn out to be ruled by
lines. There are trigonometric parametrizations, but also convenient
algebraic parametrizations. This cubic surface has a double line and
no pinch points. It is sometimes called Zindler's conoid.
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8. | This ruled cubic has a double line connecting two pinch points. Here are two representatives of Type 8. | |
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In this affine neighborhood, only one of the two pinch points
is visible. The other is "at infinity." This affine variety is the
classic Whitney's
Umbrella. There is another image on my graphics
gallery page.
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The linear transformation which changes the x0
coordinate to x0-x3 shows both pinch points in
the x0=1 affine neighborhood. The surface is called Plücker's
Conoid.
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9. |  |
Cayley's
ruled cubic has a double line and a singularity called a "unode,"
which is not a pinch point.
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The linear transformation which changes the x0
coordinate to x0+x1 gives another view of
Cayley's ruled surface.
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10.Quadric Cases | 
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Some quadric varieties can also be parametrized by homogeneous
quadratic polynomials. For example, the sphere :
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POV include file: a .txt file summarizing the implicit equations used in the above graphics, in POV format.
square to
load an animated gif
Animations of Steiner Surfaces | ||
 916KB
gif
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The Steiner Roman Surface, rotating around one of its double lines. | |
 215KB
gif
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The Steiner Cross-Cap Surface. The linear transformation
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 278KB
gif
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Several different Steiner surfaces. Unlike the above two, which
were different views of the same object, in this animation, the object
is deforming from one type of Steiner surface to another. The idea is
that Types 1, 2, and 3 are stable under perturbation of the parametric
equations, and that the Types 4-10 are unstable intermediate cases.
In the following equations, the time-dependent quantities are
p=cos(t)/sqrt(3) and q=sin(t), 0<=t<=2Pi. The variety is a sphere
when p+q=0 (two solutions for t, so two frames), a Type 5 when p=q or
p=0 (four frames), and a Roman or Cross-Cap surface for all other
values of t. Note the similarity of the homogeneous parametrization
with the previous section's equations for Types 5 and 10.
Here's some Maple code, showing some of the frames in this loop, parametrized by real numbers e instead of the periodic functions p(t), q(t). Varying the parameter e gives a Cross Cap for e < 0, the sphere for e = 0, the Cross Cap again for e between 0 and 2, surface #5 when e = 2, and the Roman Surface for e > 2.
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 822KB
gif
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Another deformation of Steiner surfaces. As in the previous
animation, the coefficients of the parametric equations depend on
another parameter t, 0<=t<=2Pi, and most of the surfaces are either
the Type 2, which is not compact in any affine neighborhood, or the
Cross Cap, which in this affine neighborhood appears as two components
that could be separated by a flat plane. The quantities
p=cos(t)/sqrt(3) and q=sin(t) are used again in the following
equations. When p=0 (for two frames), the equations define a
hyperboloid of two sheets, and when p+q=0 or p-q=0 (for four frames),
the surface is a Type 4.
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The Boy Surface | ||
 557KB
gif
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| The Boy Surface is named after Werner Boy, who discovered this immersion of the real projective plane in three dimensions. It can be realized as a real algebraic variety of degree six, unlike a Steiner Surface which has degree at most four. It has a triple point, like the Roman Surface, but no pinch points. The photo shows the sculpture at Oberwolfach. The animation demonstrates F. Apéry's homotopy, where the equations of the surfaces depend on t, 0<=t<=1. The parametric equations are fourth degree, except that at t=0, the (u02+u12) quantity can be canceled from each of the four components, so the t=0 surface is a quadratically parametrized Steiner surface (Type 1). The implicit equations are sixth degree, but again at t=0, the polynomial factors into a Steiner quartic and z2. The z2 factor defines the xy-plane, but it is not in the image of the parametric equations --- it's a locus of extra solutions from the implicitization, just like the whiskers in the still pictures, and it is not included in the animation sequence. The image of Apéry's quartic parametrization of Boy's surface is the t=1 sextic variety, without any extra implicit solutions. |
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Links related to Steiner surfaces :
Software: Books:
Academic papers, abstracts, etc.:
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Links to pictures of Steiner surfaces:
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