Spheriosty and Parallel Transport on the Sphere

Today I was writing some unit tests for Spheriosity and I discovered a flaw with the code that currently handles parallel transported lines. For those of you who are unfamiliar with the concept of parallel transport I will do my best at describing this to you.

Basically start with any two intersecting lines:

Two intersecting lines

Two intersecting lines

Now, slide the first line along the second keeping the angle between the two lines the same. You have just parallel transported your first line! By definition parallel transport merely keeps the angle between those two lines the same as they move.

Parallel transported line

Parallel transported line

Why is that so dang special? But wait! Isn’t that just a fancy way to say “keep the lines parallel”? Well, on the Euclidean plane, yes. However on a sphere there is no such thing as parallel. Two straight lines [known as great circles on a sphere] will always intersect. If you don’t believe me you can always try it out for yourself. Any true mathematician probably doesn’t believe me, as I have not provided a proof 🙂 .

Parallel transport of a great circle

Parallel transport of a great circle

Notice in the above picture that we have transported along the horizontal line. It is obvious that the original line (right line) and the transported line (left line) will intersect.

It is also interesting to note that on a sphere parallel transport is really the same as rotation! (Awesome… I know!) So where to rotate… well you can probably figure it out by simply looking at the sphere, but we want to try and put it in writing.  In order to find the axis to rotate on we have to imagine vectors going from the center of our sphere to the two end points that define our line of transport. Lets call those vectors a and b. Then, we take do: a x b (a cross b) and get some vector c. If we place vector c at the center of the sphere then we get our axis of rotation. Let call this primary axis just so we have name for it

So in the context of Spheriosity I want to take a line of transport, a line to transport and a cursor location and transport the ‘line to transport’ to where the cursor is, but making sure it is with respect to the ‘line of transport’. Translating the cursor location to a line parallel transporting is not a one step process, sadly. Since I don’t know where the cursor will be I had to find a way to map out the cursor location to the line being transported. [A big thanks to my professor who figured this one out]

  1. We use the point provided by the user and the axis of rotation for the line of transport to give us a new axis of rotation. Just to have some order here lets call the user point U, the first axis of rotation primary axis, the line of transport line AB, original line we are transporting line CD, and the new axis of rotation secondary axis
  2. Next we measure the angle between the primary axis and the vector defined from the center of the sphere to point C
  3. Now plot the point which is the intersection of the axis of rotation with the sphere. There are two poential points, so plot whichever one you wish. We will call this point E. Rotate point E with the secondary axis by the angle computed from step 2
  4. Now the first point is in place. We will call this point F. We measure the angle between the vectors defined by going from the center of the sphere to point F and point C.
  5. Using the angle from step 4 we rotate point D using the primary axis
  6. Enjoy the glory of seeing the line parallel transported

There are actually some problems with that method. The first of which is figuring out which direction to rotate point D. The .angle() function of the Vector3f class in Java3D only returns an angle from 0 to PI. The way I figure this out is actually by use of tangent vectors and dot products. To get the tangent vector I actually cheat and only approximate the angle vector by rotating the point C a very small amount (.0001 radians) with the primary axis and use the original point C and the rotated point C to make my vector (vector t). Yes, a crude trick, but it’s quick and easy. Next I define a vector from the center of the sphere to the point the cursor is at (vector c). I take the dot product of t and c,  and if the dot product is negative I know to take the angle from .angle() and rotate that in the opposite direction. Perhaps I will write a separate entry and how the dot product is my hero, but for now you will have to trust that it makes sense 🙂 .

The last issue, which I just discovered today, is that if the line we want to transport has one of it’s endpoints as the axis of rotation the aforementioned method does not work. So I simply special case this, and rotate the end point which isn’t the axis of rotation(point D) by the angle formed by the vectors made from the center of the sphere to the cursor point, and point D. Then I use the same tangent vector trick to figure out which way to rotate.

That’s all for now. Hopefully you at least understand parallel transport a little better :). Please let me know if some of that was unclear and I will do my best to make it sound better.

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