Tim Panczak
I pulled into the parking lot of a small print shop in an industrial-looking area in Dublin, Ireland. I looked back at the one lane bridge I had just crossed. It was an ordinary looking bridge spanning a small canal and a rail line. I walked to the bridge, crossed over it, and then crossed back again. This time I stopped in the middle to have a look around. I could see a large cell phone tower, high-tension power lines, a rail station, some warehouses, and a bit of graffiti.
I was wondering if I was in the right place.
I had expected a little bit of pomp. Some celebratory recognition of what had happened here, or at least some kind of informational sign.
Then I noticed a gate to a narrow switch back path that leads down to the water. I followed it. As I turned back to see the bridge, I noticed the intricate stonework and how old it really was. The modern asphalt covering the road deck had hidden its true character. Despite the spray paint here and there, it was still an attractive arched stone bridge.
I came upon was a lady who was enjoying the sunshine, sitting against the wall while knitting. A nice way to spend a lunch break, I thought.
She looked up and gave me a slight but knowing smile, as if she had seen this many times before. Another pilgrim, she must have thought.
And then I saw the plaque. This was indeed the place!
I was at Broom Bridge, where Irish physicist, mathematician, and astronomer Sir William Rowan Hamilton was struck by an inspiration that would have a profound effect on mathematics and engineering. In his excitement, he scratched his famous formula for quaternion multiplication on one of the bridge’s bricks in the fall of 1843.
Eight years earlier, Hamilton became intrigued with the relationship between complex numbers and two-dimensional geometry. He was looking to invent an algebra that could be extended into three-dimensional geometry. He was working with triples, and had deduced rules for addition, but could not reconcile multiplication.
His young son asked him frequently at breakfast, “Well, Papa, can you multiply triplets?”
He was always obliged to reply, with a sad shake of the head, “No, I can only add and subtract them."
He recounts this incident and his discovery of quaternions in a letter written August 5, 1865 to his now-grown son, Rev. Archibald H. Hamilton[1]
“An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery.”
For a long time he had played with the idea of a scalar and two imaginary components, forming a triple. What he had suddenly realized is was that he needed four numbers: a real scalar, and a vector of three imaginary components. The imaginary basis components, described by i, j, and k, are all the square root of minus one. But his insight was how they multiply together, non-commutatively, in a unique way.
Sir William Rowan Hamilton
Source: //www.thefamouspeople.com/profiles/sir-william-rowan-hamilton-552.php
The study of quaternions was popular in his day, and laid the foundation for many mathematical concepts, including vector calculus, which we engineers employ on a regular basis today. The more accessible vector analysis eventually displaced the use of quaternions in applied engineering.
Hamilton spent the rest of his life working on quaternions, and commented on how the scalar part was similar to time, in that it only had one dimension, and the imaginary vector part was similar to space and its three dimensions. In his poem The Tetracyts[2] he wrote: “And how the One of Time, of Space the Three, Might, in the Chain of Symbol, girdled be …” Impressively, he had foreshadowed the modern use of quaternions in special relativity.[3]
The revival of quaternions in modern times has also found applications in control theory, computer graphics, and orbital mechanics, primarily due to its ability to describe rotational transformations.
Why did I make that pilgrimage to Broom Bridge? Because we also use quaternions here at CRTech in the formulation of Thermal Desktop and TD Direct’s Curved Elements to describe how the anisotropic conduction tensor varies over the domain of the element. Tiny flat elements can get away with describing the conduction tensor at a single point in the center of the element. However, that is not an acceptable approximation for large curved elements.
A curved quad element saves the orientation of the principle axes of conduction at nine separate locations on the element. This allows complex anisotropic systems to be modeled, such as carbon fiber layups that follow spiral or curved paths. The heat flow between control volumes is computed by integrating the flux normal to the control volume surface and so the conduction tensor must be generated at each integration point.
We have a set of known conduction tensors at discrete points on the element. The problem is to figure out what the conduction tensor is at any other point on the element. Quaternions can represent a transformation from one coordinate system to another. However, the real beauty of using quaternions is that it is possible to interpolate smoothly and directly between two arbitrary coordinate systems.
Source: http://caig.cs.nctu.edu.tw/course/CA/Lecture/slerp.pdf
Coordinate system transformations are typically done with a 3x3 matrix; multiplying a vector by this matrix transforms it from one coordinate system to another. Unfortunately, it is not possible to interpolate between two transformation matrices. Linearly scaling the components does not lead to valid intermediate transformations.
A coordinate system transformation can also be described by three Euler angles, say yaw, pitch, and roll. If we have two such sets of such yaw, pitch and roll angles: {yaw1, pitch1, roll1}, and {yaw2, pitch2, roll2}, then we can interpolate between them and have valid intermediate transformation matrices. (Yaw is interpolated between yaw1 and yaw2, for example.) However, the intermediate transformations can go through non-uniform and sometimes severe angular accelerations. Singularities can also occur, commonly referred to as “gimbal lock.”
A third option is to use a unit quaternion to describe the orientation of a coordinate system. There are four values in a quaternion, but since it is a unit quaternion, it only has three degrees of freedom. A unit quaternion is constrained such that the sum of the squares of the components equals unity, and this constraint reduces the dimension degrees of freedom by one. These three remaining degrees of freedom are the minimum necessary to describe an orientation of a coordinate system (as does the set of yaw, pitch and roll angles). The typical 3x3 matrix has nine values, but it is subject to many other constraints. Each row and column must be a unit vector, and all columns must be orthogonal to each other, as well as all rows must be orthogonal to each other. A unit quaternion, on the other hand, is a compact representation of a spatial orientation.
The unit quaternion representing a spatial orientation can be imagined as a vector in four dimensions. This orientation can be represented by a point on a four-dimensional unit sphere in quaternion space. All spatial orientations are represented by points on this unit sphere. To interpolate between two orientations, a great arc is constructed on the surface of the sphere between the two points.
Material orienters displayed in Thermal Desktop
Computer animators have used this method to automatically construct smooth and realistic motion between two key frames. Key frames are constructed by the animator at important points in the motion of the object. Intermediate frames are constructed automatically by interpolating between these key frames. Quaternion interpolation produces physically realistic motion, whereas Euler angle interpolation can produce jerky and unnatural motions.
We thermal engineers use this method to construct smooth and valid representations of the anisotropic conduction tensor as we integrate the heat flux over the surface of a control volume. The orientation of the principle axes of conduction are represented by a 3x3 matrix at discrete points. These are converted to quaternions, then interpolation is carried out for the desired location, and the resulting quaternion converted back to a 3x3 conduction tensor.
The orientation of the principle axes of conduction is represented by a 3x3 matrix at discrete points. These matrices are converted to quaternions, interpolation is carried out for the desired location, and the resulting quaternion converted back to a 3x3 conduction tensor.
Maybe you can see now why it was very so satisfying to stand on the very same bridge that Sir Hamilton did, and pay homage to the dogged effort that lead to his mathematical discovery. And even as important as quaternions are, they are just a part of his immense mathematical legacy that benefits us today.
Broom Bridge will probably never be a stop for a tour bus. But if you ever get the chance, it is well worth the visit.
Footnotes:
[1] http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Letters/BroomeBridge.html
[2] http://against-the-day.pynchonwiki.com/wiki/index.php?title=Brougham_Bridge
[3] http://www.ime.unicamp.br/~deleo/Pub/p07.pdf
Further Reading:
Who Gave You the Epsilon? & Other Tales of Mathematical History, Anderson, Katz, and Wilson. 2009.