This may seem like a funny question but, I frequently use mathematics at my job and I have recently started working with 3D simulations. My question is, what exactly is meant by an angle in 3D? Say we have two vectors in a 3D coordinate system and we use the dot product method to figure out the angle between the two. Is this angle with respect to the xy, the xz, or is it some sort of average between the two? This simple concept is evasive to me for some reason. Thanks
If you use the dot product to find this angle then this angle can be interpreted as the angle between them in the plane that they both lay on.
The easiest way of imagining this is like this. If you have any two vectors with x, y, and z components then there will exist some plane, say w, that they both share. The angle that you compute is thus just the angle between the two on this plane.
Does that make more sense?
Yes, I think it makes more sense. My interpretation of what you are saying is that it has nothing directly to do with the xy or xz, but more just a direct angle relationship between the two vectors in the plane that they both lay on. My knowledge of planes is limited. I am having difficulty comprehending planes in 3D and their relationship to vectors and math functions applied to them. To where as a simple projection from a point is quite simple using a bearing and magnitude in 2D, but when considering the 3rd dimension in 3D the whole game seems to change.
So with that in mind, how do you consider a projection in 3D? Say you have an xyz, and you want to project off of that point a magnitude, bearing, and pitch. What role does pitch play in the 3D projection? Deep waters... Thanks for shedding some light already!
What I said before applies specifically to what you were talking about earlier. In other words, when dealing with the dot product of two vectors in x,y,z coordinates the angle between the two is only dealing with the angle between them. This angle is only related to their separation along their mutual plane.
Three dimensional angles in general are a tougher concept altogether. You are now talking about a single vector in x,y,z and not a comparison between two. To be perfectly honest, I myself am not completely comfortable with this notion. From what I can remember though, this type of interpretation does rely on breaking down the xyz vector into two vectors in xy and xz. After this is done then the angle of separation between these two vectors is calculated.
Not too long ago we had an ongoing thread trying to detail the intracacies of steradians which are another interpretation of three dimensional angles. (They are strictly speaking square radians.) They are used by those who deal with three dimensional projections regularly. Perhaps this would be an informative study for you. I know it left me scratching my head for a while. I will mention that I did not find a direct relationship between steradians and normal two dimentional angles other than the strict definition of a steradian itself. Briefly, I know how they calculate the steradian, but I do not see the connection to two dimensional angles. Type steradian into your browser and do some reading on the subject and you will see what I mean.
Perhaps one of the more knowledgable members here can help you out more. Good luck.
This sounds exactly what I was looking for. 2D angles only represent two coordinate systems. I did a little searching and reading and it looks like the steradian is a measure for a 3D angle by using an area of the surface of a sphere. (still not clear on it, but something like that) Thank you a bunch for the help.