1. Smooth Curves and Surfaces
Granada 2003
Diff. Geom. Class, 2017
Smooth Curves and Surfaces
and their Application
Title
Carlo H. Séquin
University of California, Berkeley

2. Granada 2003
Applications ?
What can you do with the math of curves and surfaces that you are learning in this class ?
The theme of my lecture is: …

3. Based on Curves Sculptural shapes dominated by smooth curves
Granada 2003
Based on Curves
Sculptural shapes dominated by smooth curves
Let’s look at some sculptural shape defined or dominated by smooth curves.

4. Figure-8 Knot Bronze, Dec. 2007 Carlo Séquin
Florida 1999
Figure-8 Knot Bronze, Dec Carlo Séquin
This sculpture celebrates the Figure-8 knot. Mathematical knots are always closed curves. Here the curve happens to be very smooth and symmetric. This little sculpture won the silver medal at the AMS Exhibit in 2009.
2nd Prize, AMS Exhibit 2009

5. “Pax Mundi” Team effort: Brent Collins, Steve Reinmuth, Carlo Séquin
Granada 2003
“Pax Mundi”
Here is a sculpture conceived by artist Brent Collins, and realized in the bronze foundry of Steve Reinmuth. I was doing the detailed design work.
Team effort: Brent Collins, Steve Reinmuth, Carlo Séquin

6. Brent Collins (1997) “Hyperbolic Hexagon II”
Granada 2003
Brent Collins (1997)
Brent is a wood sculptor, living on Gower MO, out in the nowhere, about half an hour north of Kansas City.
Here you can see him holding up “Hyperbolic Hexagon II” – our very first collaborative piece.
This came out of a special purpose program I wrote to make such Saddle Toroids, which I called “Sculpture Generator 1.”
“Hyperbolic Hexagon II”

7. Steve Reinmuth, Bronze Studio, Eugene OR
Granada 2003
Steve Reinmuth, Bronze Studio, Eugene OR
Steve Reinmuth, the key man in the realization of Pax Mundi. He runs a bronze foundry in Eugene, Oregon.
Here he is tightening the bolts during the installation.
Reinmuth Bronze Studio, Inc Meadow Lane Eugene, OR, 97402

8. Ribbon Sculptures Altamont Collins: Pax Mundi (1994) Stelvio
Granada 2003
Ribbon Sculptures
Altamont
This original wood sculpture by Brent Collins was a key inspiration to me, and it inspired me to write computer programs to create similar sculptures, all based on smooth curves.
Collins: Pax Mundi (1994)
Stelvio

9. Dominated by Surfaces Sculptural shapes dominated by smooth surfaces
Granada 2003
Dominated by Surfaces
Sculptural shapes dominated by smooth surfaces
Next let’s look at sculptural shapes dominated by smooth surfaces.

10. “Volution_2” ( 2 tunnels )
Granada 2003
“Volution_2” ( 2 tunnels )
This shape is based on a minimal surface (It has zero mean-curvature everywhere).
It was built on a 3D printer, and then cast and patina’ed by Steve Reinmuth.
Patina by Steve Reinmuth

11. Saddle Toroids Twisted Hexagon Monkey Trefoil
Granada 2003
Saddle Toroids
Twisted Hexagon
Here is the sculpture that prompted me to start a collaboration with Brent Collins.
On the right you see again some derived shapes.
Monkey Trefoil
Brent Collins (1993) Hyperbolic Hexagon

12. Granada 2003
Eva Hild
More inspirational surface come from Eva Hild is another one of my artistic heroes.

13. A large collection of ceramic creations & metal sculptures
Granada 2003
Eva Hild
A large collection of ceramic creations & metal sculptures
Eva Hild has a very rich portfolio of 2-manifold sculptures. Most of them are in ceramic, some of them in metal.

14. A rich collection of “topological” sculptures!
Granada 2003
Charles O. Perry
A rich collection of “topological” sculptures!
Charles Perry is yet another one of my heroes.
Here is a large collection of 2-manifold sculptures by Charles Perry, most of them in metal. He calls them “topological sculptures”. He had an active interest in this branch of mathematics.
Many of his sculptures are primarily defined by the smooth curves that define the edges of these sculptures.
He physically starts with steel cables that he suspends with strings in his studio and pulls into the desired 3D shape. Then he forms a smooth surface between those defined border curves. So there is a close interaction between curves and surfaces.

15. Brent Collins’ Pax Mundi 1997: Wood, 30”diam.
2006: Commission from
H&R Block, Kansas City
to make a 70”diameter version in bronze.
My task: Define the master geometry.
CAD tools play important role!
Let’s start by gaining a better understanding of this simpler sculpture. Brent Collins made this wood model in Then in 2006, he received a commission from H&R Block in Kansas City to make a larger version in bronze for the atrium of their headquarters.
-- and I received a phone call: “Carlo, can you help?” My task was to make a computer description of this shape and scale it up to the desired size.
Key question: How do you model such a shape?

16. How to Model Pax Mundi ... Already addressed that issue in 1998:
Pax Mundi could not be done with Sculpture Generator I
Needed a more general program !
Used the Berkeley SLIDE environment.
First: Needed to find the basic paradigm   
Fortunately I had already addressed that issue in It was clear that my “SG1” could not produce such shapes! For this sculpture I needed a more general modeling program. But first I had to figure out: What is the conceptual model behind Pax Mundi. Brent had told me that this ribbon was contained in the surface of a sphere. == But how do you define such a curve in the computer ?

17. B-Splines (Basis-Spline)
Granada 2003
How to specify an arbitrary, smooth curve ?
{{ So here is a quick exercise: How do you define this 2D, kidney-shaped curve in 2D – so you can enter it into a computer for further processing?? }}
??? { let students answer … -> Blackboard work … }
>>
Computer graphics people use something called a B-spline…
B-Splines (Basis-Spline)

18. Granada 2003
Bspline
Bspline
This is a polynomial function that has minimal support with respect to a given degree of smoothness. For a cubic B-spline the influence of a given control point can be felt as far as two neighbors away. Its influence smoothly increase from zero, max’es out in the middle, and then decays again to zero.
At an arbitrary point along the curve, the influence of four control points is felt.

19. Granada 2003
Cubic B-Spline Curves
One segment depending on the 4 control points A,B,C,D, is given by the polynomial:
Q = A(1 -3t +3t2 -t3)/ B(4 -6t2 +3t3)/ C(1 +3t +3t2 -3t3)/6 + D(t3)/6
The point at t=0 is given by: A/6 + 4B/6 + C/6,
The point at t=1 is given by: B/6 + 4C/6 + D/6.
Here you can see how the influence of the 4 nearest control points is calculated. Each hump is composed of four cubic polynomial terms. Note that these 4 coefficients sum up to unity!
This is enough to guarantee that the resulting curve is tangent continuous and curvature continuous.
-- How does such a formulation help to specify the curve of interest?

20. Closed B-spline Closed loop
Granada 2003
Closed B-spline
It allows on to draw a crude control polygon with just a few control points and they then define a continuous, smooth curve! By moving some of the control points around, we can fine-tune the geometry of the curve. If you want to see more wiggles in the curve, then you need to introduce more control points.
This also allows you to make curves that smoothly close into a loop – You just need to repeat a few control vertices from the beginning of the sequence, to make sure that the last part of the curve sees the same conditions as the beginning.
Closed loop

21. Sculptures by Naum Gabo
The ribbon curve in Brent’s sculpture also reminded me of sculptures by Naum Gabo, which I had seen as a student in a visit to Paris.
The edges in these two sculptures form an undulating pathways on a sphere.
I call these types of curves “Gabo curves”.
Pathway on a sphere:
Edge of surface is like seam of tennis- or base-ball;
 “2-period Gabo curve.”

22. 2-period “Gabo Curve”
Here is how I define them in my computer program: The bluish rectangle on the right is a Mercator projection of the surface of the Earth. The equator is the horizontal line in the middle; North pole is on top, South pole at the bottom.
In this domain I defined an undulating curve that crosses the equator a specified number of times. Here we have two full waves. -- The red curve is a B-spline. Because of its overall symmetry, I only needed 3 parameters (shown by the small blue arrows) to control the shape of the curve, -- primarily its amplitude, and the width (or bulginess) of its lobes.
Approximation with quartic B-spline with 8 control points per period, but only 3 DOF are used (symmetry!).

23. 4-period “Gabo Curve” Same construction as for as for 2-period curve
And if we cram four complete periods around the equator, the result would look like this.
Same construction as for as for 2-period curve

24. Pax Mundi Revisited
Can be seen as: “Amplitude modulated, 4-period Gabo curve”
Now, with this new view of things, I could characterize Pax Mundi as an “amplitude-modulated, 4-period Gabo curve.”
Here the globe lies on its side, the two poles are at the left and right extrema.

25. Progressive Sweeps Sculpture is not just a mathematical curve.
Granada 2003
Progressive Sweeps
Sculpture is not just a mathematical curve.
There is some substance; it has volume.
Define shape by sweeping a cross section along a given 3D space curve.
But this sculpture is not just a line … There is some substance; it has some volume.
-- A convenient way to describe this, is by sweeping a crescent-like profile along the given Gabo curve.

26. 2-period Gabo Sculpture
Tennis ball – or baseball – seam used as
sweep curve.
Here is a very simple form produced by sweeping that same crescent-like cross section along a simple 2-period Gabo curve, which has the shape of the seam of a baseball or tennis ball.

27. Viae Globi Family (Roads on a Sphere)
Here I have increased the number of periods in the Gabo curve from 2 to 5.
periods

28. SLIDE-GUI for “Pax Mundi” Shapes
Florida 1999
Good combination of interactive 3D graphics and parameterizable procedural constructs.
All this control was then captured in a modular program, constructed within the Berkeley SLIDE environment. This is a graphics program that my graduate students built in the 1990s. It provides a good combination of interactive 3D graphics and parametrizable procedural constructs. On display you see my final model of Pax Mundi.
This new generator program has three columns of sliders controlling respectively: the sweep curve – the cross-sectional shape – and the application of it along the sweep.

29. Modularity of Gabo Sweep Generator
Sweep Curve Generator:
Gabo Curves as B-splines:
Cross Section Fine Tuner:
Paramererized shapes:
Sweep / Twist Controller:
How is cross section applied?
Here is what the 3 sets of controls do:
-- The first one defines the sweep curve.
-- The 2nd one fine-tunes a selected parametrized cross section. The middle one was used for Pax Mundi.
-- The 3rd bank of sliders controls how it is applied to the sweep curve. And here is where it starts to tie in with something you have learned about curves.

30. Intrinsic Sweep Mode Keep cross section perpendicular to tangent.
Granada 2003
Intrinsic Sweep Mode
Keep cross section perpendicular to tangent.
Place cross section into the x-y-plane of the Frenet frame.
Keep orientation / rotation as it was in the defining x-y-coordinate system.
Add any additional azimuth angle as a rotation around the z-axis (tangent).
This is the simplest sweep model:

31. “Natural” orientation with Frenet frame
Granada 2003
Intrinsic Sweep Mode
Problems at inflection points (in plane):
 Pinched-off “hour-glass” shapes.
This simple model may cause problems: At inflection points (in the middle of an S-shape), the Frenet frame flips through 180 degrees, yielding the hour-glass shape you see in the middle.
“Natural” orientation with Frenet frame

32. Minimum-Torsion (-Rotation) Sweep
Granada 2003
Minimum-Torsion (-Rotation) Sweep
Project orientation of cross section forward, from one vertex of the sweep polyline to the next.
Neutralize rotation of Frenet frame
We can avoid this, if we compensate for the intrinsic rotation of the Frenet frame and rotate the cross section by the same amount in the opposite direction. -- This is the same as saying that from one incremental sampling point on the curve to the next, we simply forward-project the cross section from one x-y-plane of the Frenet frame to the next one. This then results in a garden hose that has no visible twisting. This is called a minimum-torsion or minimum-rotation sweep. Often this is a better basis to start modeling a sweep along an arbitrary space curve.
intrinsic minimum torsion

33. Azimuth / Twist Control
Granada 2003
Azimuth / Twist Control
Starting with torsion/rotation minimization:
On top of this we can add two important parameters that affect the sweep overall:
Azimuth is a constant additional rotation applied to all the cross sections at all the sample points. It would make a toroidal structure turn through itself like a smoke-ring.
Twist is an additional component in the azimuth that starts at zero and builds up incrementally from one vertex to the next, until it reaches the full value specified at the last vertex.
azimuth = 0, azimuth = 90, azimuth = 90, twist = 0; twist = 0; twist =180;

34. Local Azimuth Control = “Warp”
Granada 2003
Local Azimuth Control = “Warp”
Starting with torsion/rotation minimization:
Sometimes we want additional local control: “Warp” works on an individual control point.
If sweep is specified explicitly along a poly-line, then we see a momentary change just at the one location where warp was added.
For smooth curves, like a B-spline, it gets interpolated like any other geometric component of that spline (e.g., x,y,z). We then see a smooth back and forth twisting of the ribbon.
We can also apply a local scaling of the cross section, and that will also get smoothly interpolated.
azimuth = 0, azimuth = 0, twist = 0; warp = -90;

35. Azimuth / Twist Control
Granada 2003
Azimuth / Twist Control
Controls applied to the 2-period Gabo curve:
These are some of the changes we can effect with these controls on the sweep along the 4-period Gabo curve of Pax Mundi.
For Pax Mundi, the natural orientation was working quite nicely; I just needed a little warp control to soften the torsional transitions between two subsequent hair-pins.
If we started with a minimum-torsion sweep, then we could still adjust the azimuth to have the ribbon be mostly tangential to the sphere, or perpendicular to it. And on top of that we can add any amount of twist.
Natural orientation with Frenet frame
Torsion Minimization: Azimuth: tangential / normal
900° of twist added.

36. Extension: Free-form Curve on a Sphere
Granada 2003
Extension: Free-form Curve on a Sphere
Spherical Spline Path Editor (Jane Yen)
Starting with a program to capture Pax Mundi in a CAD model, I gradually expanded the generality of this sweep construct.
A first step was to not just have Gabo curves on the sphere, but arbitrary undulating pathways, defined by just a few control points.
Jane Yen wrote a nice program based on “Circle Splines” that made sure that the curve through these points would be always smooth and exhibit a smoothly varying amount of curvature.
Nice smooth interpolating curves through sparse data points

37. Many Different Viae Globi Models
Granada 2003
Many Different Viae Globi Models
This then allowed me to make more “curvy” Viae Globi models (i.e “roads on a sphere”).
In the middle is Maloya, inspired by a swiss mountain road.
On the left is Stelvio, a pass route in Italy with 40+ hair-pin curves.
At right is Altamont pass, a high-way pass into the Central Valley in CA with multiple parallel lanes.

38. Granada 2003
Maloja
Here is a small metal cast of Maloya.

39. Extending the Paradigm: Aurora-M
Granada 2003
Extending the Paradigm: Aurora-M
Simple path on sphere,
but more play with the swept cross section.
This is a Möbius band.
It is morphed from a concave shape at the bottom to a flat ribbon at the top of the flower.
 “Sweep-Morph”
Here is a further extension of the sweep paradigm. This ribbon still follows a simple path on a sphere, but it is a Moebius band.
At the bottom, the ribbon has the crescent profile of all the sculptures you have seen so far. But when the ribbon finally meets itself at the top, there is a problem: the ribbon curvatures point in opposite directions! In order to be able to close the ribbon into a loop, we have to straighten out this lateral curvature. This means that the cross sectional profile has to gradually change. We call this a sweep-morph.
This paradigm extension allows to specify a different cross section at every control point of the sweep curve.

40. Paradigm Extension: Sweep Path is no longer confined to a sphere!
Granada 2003
Paradigm Extension: Sweep Path is no longer confined to a sphere!
The next extension was to allow the path to deviate from the sphere surface. Now we can make paths the correspond to mathematical knots.
Chinese Button Knot

41. Florida 1999
Chinese Button Knot (Knot 940) Bronze, Dec Carlo Séquin cast & patina by Steve Reinmuth
Here is a medium-complexity, 9-crossing knot – known as the Chinese button knot.
This shape was also first printed in ABS plastic on a FDM machine and then converted to bronze with an investment casting process.
This was done by Steve Reinmuth in his bronze foundry in Eugene, OR.

42. Design Requires Paradigm Extension
Granada 2003
Design Requires Paradigm Extension
A more recent design by Brent Collins required this paradigm extension.
It is called “Music of the Spheres”, because …
Music of the Spheres (Brent Collins)

43. Definition of Sweep Path (hugging 4 different spheres)
Granada 2003
Definition of Sweep Path (hugging 4 different spheres)
The sweep path is defined as indicated on the right hand side: It deviates 3 times from the outer sphere and loops around 3 smaller inner spheres indicated by black lines, and in the process forms a topological trefoil knot.
-- Brent Collins got a commission to do this in bronze at a large scale, and he asked me to do the detailed CAD model as for Pax Mundi. For each unique piece of geometry, we then had to make a mold to cast positive replicas in wax, which could then be used in an investment-casting process.
Music of the Spheres (Brent Collins)

44. Partitioning; Joint Design
Granada 2003
Partitioning; Joint Design
So I wanted to maximally exploit the symmetry of this sculpture to minimize the number of the master molds needed. Here, the minimal geometry that needs to be defined comprises 1/3 of the whole sculpture, shown in white. But this piece is too large to be handled. Even 1/9th of the sculpture was still too large to fit in Steve Reinmuth’s kiln. So I divided the white piece into 6 smaller pieces as shown on the right. I also provided some alignment stubs to make assembly of these pieces easier.
Alignment stubs
1/3 = unique
geometry
18 pieces:
fit in kiln!

45. Assembly of Music of the Spheres
Granada 2003
Assembly of Music of the Spheres
Close to final assembly of “Music of the Spheres”. Photo conveys a sense of the physical labor involved in building such a sculpture.

46. Installation at MWSU, Feb. 2013
Granada 2003
Installation at MWSU, Feb. 2013
Just finishing the installation at MWSU
Steve Reinmuth Brent Collins

47. Illuminated Music of the Spheres
Granada 2003
Illuminated Music of the Spheres
And it looks even more spectacular at night!
Photo by Phillip Geller

48. Modeling “Tetra” by Charles Perry (3)
Granada 2003
Modeling “Tetra” by Charles Perry (3)
As mentioned before, Perry’s sculptures are defined by smooth curves: the edges of his sculptures. In this one, the edges are just 4 equal-sized circles.
Left: his Tetra sculpture suitably annotated. Right: my CAD model in the computer. Middle another view of his sculpture with less perspective distortion. After the border curves have been put in place, Perry then defines the surface that is suspended between them with additional short curves, also formed by thinner steel cables.
Annotated sculpture image
CAD model of Perry’s “Tetra”
Perry’s “Tetra”

49. “Tetra_6M” Modification of Perry’s Tetra Sculpture
Granada 2003
“Tetra_6M” Modification of Perry’s Tetra Sculpture
All SIX tetra-edges are twisted through 180.
This also makes it single-sided!
This one has 3 identical borders, forming a Borromean link !
Here is another Perry-type sculpture with just three boundary curves, interlinked in a Borromean configuration. Ideally – what kind of surface should be suspended between those borders? .

50. Granada 2003
Soap Films
Perhaps we can just use the shape of a soap film. Because of surface tension, such a film will always try to minimize the overall surface area and result in a minimal surface.

51. Granada 2003
Minimal Surfaces
Here are some classical minimal surfaces. At all points of these surfaces, we have some nicely balanced saddles, where the maximal and minimal principal are of equal and opposite magnitude. -- Minimal surfaces can take on rather complex shapes consisting of multiple adjoining saddles nicely blended together; and they can even form infinitely large periodic lattices with infinitely many saddles.
Tp right: Costa surface; bottom left triply periodic minimal surface.
The two principal curvatures (maximal and minimal) are of equal and opposite magnitude at every point of the surface!

52. “Tetra” by Charles Perry
Granada 2003
“Tetra” by Charles Perry
Perry’s “Tetra” is clearly NOT a minimal surface:
Region of positive Gaussian curvature.
It turns out, Perry’s “Tetra” is NOT a minimal surface.
The region of positive Gaussian curvature would have to be much more like a flat ribbon. Not as interesting a sculpture.

53. “Atomic Flower II” by Brent Collins
ISAMA 2004
“Atomic Flower II” by Brent Collins
Here is a sculpture by Collins who also has such curved ribbons but of more negative Gaussian curvature.
Minimizing this curvature to zero makes the sculpture less pleasing.
Collins’ sense of aesthetics wins!
Minimal surface in smooth edge (captured by John Sullivan)

54. 3 Monkey Saddles with 180º Twist
ISAMA 2004
3 Monkey Saddles with 180º Twist
This verdict becomes even more convincing, when looking at this model of sculpture that I did with Brent Collins.
On the left is a Scherk-Collins Toroid coming out of my Sculpture Generator. It is composed of three monkey saddles and a twist of 180 degrees.
On the right are the very same rims spanned by a true minimal surface, that is: a soap film.
As you can see in the center things are happening that are not very pretty. Also the ribbons have become too flat and too bland.
Again, the Scherk-Collins geometry clearly wins.
Maquette made with Sculpture Generator I
Minimal surface spanning three (2,1) torus knots

55. Triply Periodic Minimal Surfaces
Granada 2003
Triply Periodic Minimal Surfaces
But in some cases, minimal surfaces are a very good solution, as in these cells that one may find as part of a triply periodic surface. In both cases, there is a cubic cell, with a border that is composed of 12 quarter circles, -- two each on every face of a cube.
Schoen’s F-RD Surface Brakke’s Pseudo Batwing modules
Surface embedded in a cubic cell,
12 “quarter-circle” boundaries on cube faces

56. A Loop of 12 Quarter-Circles Simplest Spanning Surface: A Disk
Granada 2003
A Loop of 12 Quarter-Circles Simplest Spanning Surface: A Disk
Here is probably the simplest such element. Its boundary is a single closed chain of 12 quarter circles. The minimal surface inside is topologically equivalent to a simple disk; and a soap film could actually take on this shape.
Minimal surface formed under those constraints

57. Higher-Genus Surfaces
Granada 2003
Higher-Genus Surfaces
Enhancing simple surfaces with extra tunnels / handles
On the left again is this same element, seen from a different angle. Going towards the right, I kept the boundary rim the same, but made the soap film surface more complicated by poking some holes into it, which will then blend into nice smooth tunnels and create a minimal surface of higher genus.
“Volution_0” “Volution_2” “Volution_4”
warped disk tunnels tunnels

58. Ken Brakke’s Surface Evolver
Granada 2003
Ken Brakke’s Surface Evolver
For creating constrained, optimized shapes
The program that allows me to find these surfaces is Ken Brakke’s “Surface Evolver.” In the example here, I make a surface with two tunnels.
I start with a rough polyhedral surface that outlines the desired shape. This surface is triangulated and gradually refined, and is tied to ever more finely defined quarter-circle rims. The inner vertices of this mesh are balanced against their neighbors to locally minimize the surface area.
Start with a crude
polyhedral object
Subdivide triangles
Optimize vertices
Repeat the process

59. Optimization Step To minimize “Surface Area”:
Granada 2003
Optimization Step
To minimize “Surface Area”:
move every vertex towards the equilibrium point where the area of nearest neighbor triangles (Av ) is minimal, i.e.:
move along logarithmic gradient of area:
To achieve this, every vertex is wiggled by a small amount to find the position that minimizes the sum of the areas of the surrounding triangles. We take only small steps towards this preferred position and do this for all vertices simultaneously in several rounds of optimization.

60. “Volution” Surfaces (Séquin, 2003)
CAD 2004, Thailand
“Volution” Surfaces (Séquin, 2003)
And these are the kind of surfaces that result, -- with no tunnels – and with 5 tunnels, respectively.
“Volution 0” “Volution 5” Minimal surfaces of different genus; same boundary.

61. “Volution’s Evolution”
Granada 2003
“Volution’s Evolution”
This shows the topological evolution of these shapes. This group is now in an art museum in Valencia in Spain.
At this point some of you may interject that you have never seen a soap film take on a shape like the one on the right – or even the one in the middle with just two tunnels.
And you would be right! This is for the same reason that you may never see a configuration like this…
0 tunnels tunnels tunnels

62. An Unstable Equilibrium
Granada 2003
An Unstable Equilibrium
While this balancing act is theoretically possible – it is not going to last long in nature!
… will not last long!

63. Fighting Tunnels The two side-by-side tunnels are not a stable state.
Granada 2003
Fighting Tunnels
The two side-by-side tunnels are not a stable state.
If one gets slightly smaller, the pull of its higher curvature will get stronger, and it will tug even more strongly on the larger tunnel.
It will collapse to a zero-diameter and pinch off.
But in a computer we can add a constraint that keeps the two tunnels the same size!
Specifically two soap-film tunnels right next to one another are NOT stable. If one gets slightly smaller, it assumes higher curvature, and it starts to exert an even stronger pull on the wall that it shares with its neighbor. It will shrink more and more, and quickly collapse to a zero-diameter, and then pinch off, and disappear.
But in a computer, we can add a constraint that keeps the two tunnels the same size during the whole refinement and optimization process!

64. Stable vs. Unstable Equilibria
Granada 2003
Stable vs. Unstable Equilibria
Stable equilibrium is immune to small disturbances.
Unstable equilibrium will run away when disturbed.
Computer can help to keep a design perfectly balanced.
The soap films you are going to see in nature are in a stable equilibrium. But some of the shapes I have created are in an un-stable equilibrium.
But I can force my computer programs to keep this precarious balance by maintaining a specified symmetry at all times.

65. Limitations of “Minimal Surfaces”
Granada 2003
Limitations of “Minimal Surfaces”
“Minimal Surface” - functional works well for large-area, edge-bounded surfaces.
But what should we do for closed manifolds ?
Spheres, tori, higher genus manifolds … cannot be modeled by minimal surfaces.
 We need another functional !
There are some limitations of what shapes a minimal surface can assume. If we would like to discover the most beautiful egg-shape – how could minimal surfaces help us? Closed surfaces cannot be modeled by ordinary minimal surfaces.

66. Closed Soap-film Surfaces
Granada 2003
Closed Soap-film Surfaces
So how does nature do this?
Soap films can take on spherical shapes, rather than saddle shapes, when the air-pressures inside and outside are different.
Fortunately, there is another energy-functional that is quite suitable to optimize closed shapes…
Pressure differences:  Spherical shapes

67. Surface Bending Energy
Granada 2003
Surface Bending Energy
It is based on bending energy. Bending a thin (metal) plate increases its energy; it stores energy like a taught spring. The color diagram (right) shows the local energy density in a flat ribbon that was bent into a Möbius band.
By integrating this stored energy over the whole surface, we can obtain another measure for surface quality. We want to make this value as low as possible. This leads to the notion of Minimal Energy Surfaces (MES).
Bending a thin (metal) plate increases it energy.
Integrating the total energy stored over the whole surface can serve as another measure for optimization:
 Minimal Energy Surfaces (MES)

68. Surface Bending Energy
Granada 2003
Surface Bending Energy
Cost (“penalty”) is based on local curvature.
Minimum Energy Surfaces (MES): Bending Energy =  (k1)2 + (k2)2 dA  Only planes are “ideal.”
This is a frequently used cost functional.
This is a classical functional used to “optimize” surfaces or make them more “fair”..

69. Minimum Energy Surfaces (MES)
Granada 2003
Minimum Energy Surfaces (MES)
sphere, cones, cyclides, Clifford torus
Lawson’s genus-5 surfaces:
A sphere is the “best” (cheapest) closed surface. It has a total bending Energy of 4π – regardless of radius. Other shapes that result from this functional are cones, cyclides, and tori.
Among all tori, the one with the absolute lowest bending energy is the Clifford torus where the ratio between the two radii is root(2).
For higher genus surfaces, the energy minimizing shapes are not so nice, they typically contain a "perforation" zone consisting of narrow tunnels and thin pillars.
“Perforation zone”

70. Lawson Surfaces of Minimal Energy
Granada 2003
Lawson Surfaces of Minimal Energy
12 little legs
Genus Genus Genus 11
The genus-3 ME-surface has 4 reasonably sized pillars. For genus-5, the 6 pillars are already quite a bit smaller; and by genus 12, they have become truly tiny. These are not satisfactory, beautiful shapes!
Also the perfect genus-0 shape is the sphere – but it still has a cost (or penalty) of 4π. -- Shouldn’t this be ZERO ?
That is why, in the 1990 time frame, Henry Moreton and I tried to find a better functional.
Shapes get worse for MES as we go to higher genus …
[ … see models ! ]

71. A Better Optimization Functional?
Granada 2003
A Better Optimization Functional?
Penalize change in curvature !
Minimum Variation Surfaces (MVS):  (dk1/de1)2 + (dk2/de2)2 dA  Spheres, Cones, Various Tori, Cyclides …
The Sphere now has a cost/penalty of zero!
We came up with the idea to integrate the CHANGE of curvature over the whole surface.
On a sphere we can move in all directions, and the curvature is always the same – so there is no change. Thus the sphere now has a cost/penalty of zero!
This functional also produces Cones, Various Tori, and Cyclides,
For higher genus surfaces, we get more pleasing results with a better balance between the tunnels and the handles surrounding them.

72. Minimum-Variation Surfaces (MVS)
Granada 2003
Minimum-Variation Surfaces (MVS)
Here are some of the resulting shapes of lowest MVS costs.
Genus 3
D4h
Genus 5 Oh
The most pleasing smooth surfaces…
Constrained only by topology, symmetry, size.

73. Comparison: MES   MVS (genus 4 surfaces)
Granada 2003 
Comparison: MES   MVS (genus 4 surfaces)
Here is a direct comparison with the MES surfaces (shown on the left). These are all cost-minimizing shapes.
The MVS surfaces on the right exhibit a much better balance between the tunnels and the toroidal arms. 

74. Things get worse for MES as we go to higher genus:
Granada 2003
Comparison MES  MVS
Things get worse for MES as we go to higher genus:
pinch off
Here is a comparison of genus 5 surfaces. The difference gets much more pronounced the higher we go in genus.
===
Conclusion: depending on what we define as our energy functional, we can get different kinds of optimal surfaces.
In some cases this minimum-energy shape coincides with an aesthetic optimum.
In general the MVS functional seems to do a better job than the MES functional.
I just wanted to make you aware of this alternative …
3 holes
Genus-5 MES
MVS keep nice toroidal arms

75. Granada 2003
Papers on MVS etc
H. P. Moreton and C.H. Séquin: "Functional Optimization for Fair Surface Design,'' Proc. ACM SIGGRAPH'92, Chicago, July 1992, and Computer Graphics, Vol 26, No 2, pp
P. Joshi and C. H. Séquin, "Energy Minimizers for Curvature-Based Surface Functionals," CAD Conf. Proc., Waikiki, Hawaii, June 25-29, 2007, pp
P. Joshi and C. H. Séquin, "Visualizing High-Order Surface Geometry," Computer-Aided Design and Applications, Vol 6, No 2, pp , 2009.
P. Joshi and C. H. Séquin, "An intuitive explanation of third-order surface behavior," Computer-Aided Geometric Design, Volume 27, Issue 2, pp , Feb
And here are some references, if you would like to follow up on this.
=>> QUESTIONS ?

76. === SPARES ===