The statement above is, indeed, a bold statement to start. A vector product in the form of ab (geometric product) is not familiar to most people. The a⋅b (also called inner product) is the familiar linear algebra dot product, and is a scalar. The a ⋀ b signifies a "wedge" product (also called exterior product) and is a bivector signifying a directed area of |a||b|sin(ϴ) where ϴ is the angle between a and b. The directed nature of the wedge product has to do with the well-known notion of the cross product - the direction/orientation is "orthogonal" to the a/b plane, and follows the familiar right hand rule. At θ = 90°, sin(θ) = 1, and the area is simply |a||b|, a parallelogram. At other angles the trapezoidal area interpretation applies. The location of the directed area is not known in ℝ³ (see comment below).
Two vectors a and b having a common origin define a plane. If this plane is represented by the x, iy plane then a and b can be characterized by a = |a| e^iα and |b| characterized by b = |b| e^iβ. Then the product ab can be represented ab* = |a|b| e^i(α-β) = |a||b| cos(α-β) + i|a||b| sin(α-β), and we have the familiar dot product involving the cos(θ) and the "wedge" product involving the sin(θ). θ = α-β, the angle between a and b.
Hestenes makes a comment to the extent that we are held "prisoner" by the notion of a coordinate system, and a misconception of vectors. A good example is the velocity vector of MH370 in a local tangent plane - a plane tangent to the surface of the earth at the location of the aircraft. When we write V = vxi + vyj + vzk, the vector can be anywhere in ℝ³. The i,j,k and the vx,vy,vz merely specify a magnitude and a direction - not the location of the vector. We instinctively place the vector at the surface of the earth at the location of the aircraft, but there is nothing in the vector representation which mandates this location, and, in fact, it is downright misleading (but it feels good). A strength of geometric algebra is to break the bonds we have relative to a general misconception of what a coordinate system actually represents.
Just like vectors sum, so does the exterior or wedge product.
Analogous to integrated Doppler which only depends on the end points and time, and not on the specific path. The exterior product does not care about the details of the particular path taken to get from point x to point y, only the distance.
(u + v) ∧ w = u ∧ w + v ∧ w
So, you do not need to concern yourself with a detailed path. You can pick specific points in space and time and construct a perfectly valid geometric algebra analysis.
The meaning of the geometric product in the context of MH370 remains elusive to me. The dot product part obviously represents Doppler, and that is quite clear and easily interpreted from conventional analytics. I have no insight into the physical meaning of the wedge product part.
Not being one to let lack of insight be an impediment, I constructed a blade experiment using the following direct paths. A blade being the directed area of the wedge product. In conventional linear algebra the cross product is a vector which only has meaning in ℝ³. In geometric algebra the wedge product is a scalar area in two dimensions and an oriented scalar area in ℝ³.
path A = IGARI to 19:40 (the same for all paths B and C below)
path B = 19:40 location to 18S, 22S, 26S, 30S, 34S, 38S on the 00:11 arc
path C = IGARI to same 00:11 arc positions as path B
A diagram of the paths (except 18S) is shown in the Google Earth screen capture below for an arbitrarily selected 19:40 position of 0N 93.5E.
I then compared IGARI ∧ 19:40 + 19:40 ∧ (arc positions) to IGARI ∧ (arc positions) where:
A_ = IGARI ∧ 19:40 = path A ∧ 19:40 satellite position vector at 19:40
B_ = 19:40 ∧ (arc positions) = path B ∧ 19:40 satellite position vector at 19:40
C_ = IGARI ∧ (arc positions) = path C ∧ 19:40 satellite position vector at 19:40
I did this calculation for 19:40 positions of 0N 93.5E and 8N 93E. The values plotted below are % difference as a function of terminal latitude on the 00:11 arc.
% diff = (A_ + B_ - C_) / C_
Plot (with trend lines) shown using signed values of % difference.
Plot (with trend lines) shown using absolute values of % difference.
My speculation is that there exists a 19:40 location (probably around 5N-6N) where the trend line will "flatten" and exhibit a pure noise characteristic i.e. the blue trend line will "warp" toward the red trend line as the 19:40 position is moved Northward. I will test this later. The calculations, even with fairly good software support, are tedious. (I need people for this.)
It should be noted that the % variations above are very small, and any of the 00:11 positions can probably be supported by standard Doppler path calculations (with appropriate BFO error tolerances).
What is interesting to me is that the region of 25S to 30S is "highlighted" in a "null" fashion above. This could be real or it could be an artifact related to systematic errors in Google Earth, systematic errors related to the geometry of the earth ellipsoid effects on the wedge product, or some other systematic error I cannot hypothesize at this time.
Of course, without a physical interpretation of what the wedge product represents in this calculation, all the above could be completely meaningless. I believe including the 38S data point for the 19:40 location of 8N is not realistic given limitations to aircraft speed, but the wedge product does not know that.
BTW, I did the above calculations for a "common vector" using the nominal satellite position, the vector to 0N 93.5e, and the vector to 8N 93E. The results other than for the actual satellite position vector at 19:40 were "gobble-d-gook". There is "something special" about 25S to 30S, but I am totally unable to explain what it might be.
OK. So as mentioned above, I did the calculations for a 19:40 position of 5N. The results are shown below for 0N, 5N, and 8N at 19:40
As I expected, the "warping" flattened at 5N with noise like errors.
Signed values of percentage difference below (most interesting, IMO).
Absolute values of % difference.
Conclusion
My conclusion is that there is no added value to be derived from the wedge product. I believe the warping is due to some interaction with the geometry that I have not able to find. The noise is simply that - small random errors in my values for the various locations. The interpretation of the wedge product, I believe, is a representation of the area swept out by the aircraft path relative to the satellite position vector at 19:40. The sum of the directed area from IGARI to 19:40 plus the directed area from 19:40 to an arc position should be the same (in the absence of measurement errors or geometry errors) as the area from IGARI to the arc position. Basically, I do not believe that geometric algebra adds anything to the determination of a terminal location.
Sorry about that. A lot of time spent for no added information.
Afterthought
Or it might be that running the calculation again at 6N at 19:40 would bring the orange line down to near zero errors and flatten? Refer to signed value difference. Is that telling us something? I have no idea.
Percentage signed value chart repeated below with select paths labeled with Doppler residual in Hz at 19:40 using average speed and track to 00:11 ring point. The reported Doppler residual was -38.4 Hz. So the average speed from 19:40 to the 00:11 ring latitude is best at ~34S, ~26S, ~22S for 0N, 5N, 8N respectively when this speed is used at 19:40 to calculate BFO.
Example Path
"Quick and dirty" path below. There are many such paths. A single example merely illustrates feasibility, and I put very little time into it.
more 9/28/17
I still have low confidence in what the blades are telling me if anything, so don't place a great deal of weight on these ramblings.
I re-did the blade area work above using vectors at 19:40 @0N, 5N, and 8N, a common vector at IGARI, and vectors at 18S, 22S 26S, 30S, 34S, and 38S on the 00:11 arc. All vectors were from the center of the earth to the surface of the earth. WGS-84 coordinates were used in all cases. Plot of the results shown below.
Another major difference below is the use of only position vectors in the wedge products. In the previous work above the blades were generated by position vectors and "weighted" velocity vectors. For example the velocity vectors from 19:40 to the arc positions were weighted by 4.5, the velocity vector from IGARI to 19:40 by 2.33, and the velocity vector from IGARI to the arc positions by 7.83.
Once again, one can conclude that 26S at 00:11 (27S @ 00:19) is an "indicated" terminus. In this case a 19:40 location of 8N is preferred. The Iannello and Godrey McMurdo path fits this conclusion extremely well, or a path using the Cocos as an LNAV weigh point subsequently overflown with a magnetic heading.
