Slerp for multiple vectors?

Question

Hi,

I am trying to average a set of vectors using slerp, but I couldn't figure out the algorithm for it.

I tried using 4 vectors obtained from each side of a pyramid where base is a perfect square. So my vectors in this case is:

{[0,-0.946941,0.321407], [0.946941,0,0.321407], [0,0.946941,0.321407], [-0.946941,0,0.321407]}

and the angles between them:

{35.6355, 35.6355, 35.6355, 35.6355}

I tried using this: http://en.wikipedia.org/wiki/Slerp

How can I combine these 4 vectors so I get the value [0,0,1]?

The reason is, I want to implement a weighted averaging method. So if I want, I could pass the area of polygons as weights to determine the final averaged normal, instead of add them all and divide by the number.

Thanks for the help!

Thanks. But this is non-weighted averaging, which is the same as:

Normalize ((a+b+c+d)/4)

What I need is to assign weights to the vectors, so for this instance, all weights are the same, but for other geometries, I could use weights like face area or angle between 2 edges.

Answer 1

my location does not allow access to wikipedia so I can not look up slerp. I dont't know if this is what you are looking for but if you set A = 0.946941 and B = 0.321407 (just to make it easier for me to write these things) then the vectors are just:

(0,-A,B)
(A,0,B)
(0,A,B)
(-A,0,B)

If you add these 4 vectors you get (0,0,4B). Since you want (0,0,1) just divide all the vectors by 4B (4*0.321407) and then when you add them you will get (0,0,1).

Is this what you want?

Answer 2

In terms of flow management, or better known as "flow control", Speed control and vectoring is mainly handled in the en route environment. Spacing and sequencing to major airports usually starts hundreds of miles away. For example, the line to JFK over the LENDY5 arrival, I have seen flow control measures taken as far away as Western Iowa, and have heard of it going back even further. So now we're (at Chicago Center) told we now need 30 miles in trail for each aircraft going to JFK. We normally know of this well in advance and spacing is usually accomplished through the use of restricted departure times, (EDCT's) and even more restrictive in-trail restrictions to preceeding artcc's. But there always tend to be a few aircraft that need to be moved around one way or the other to meet that in-trail restricrion. The easiest way to do this is to assign speeds that will help you gain mileage or tighten it up. But remember each hundredth of a mach number at the same altitude is roughly 7 knots. Simple math will tell you it's going to take awhile to gain additional of spacing. Here's where vectoring comes in. Taking the winds and other factors into account, (like sector limitations and boundaries) turning an aircraft off-course naturally adds additional spacing in a much quicker fashion. Used in combination with speed control, it's a very useful tool. Sometimes the turns get pretty severe, 30 degrees or more, but the greater the vector, the quicker the aircraft gets back on course. A last resort in the en route environment is taking an aircraft, "around the block" to achieve the necesary spacing. Either by issuing a 360 degree turn, or issuing holding instructions. Holding instructions are almost exclusively used when a receiving facility cannot accept any aircraft to a particular destination due to weather, volume, etc. There's ALOT more to flow control than what I've mentioned here. I just thought to cover the basics of your question. Please pardon any typos and the brevity of my answer.