In intro physics, the concept of vectors are always taught using position since the x,y,z coordinate space results in arrows that show you the relative values in a format that is intuitive. Thus it would follow, for position, velocity, acceleration, vector math is more readily grasped, since it's easier to conceptualize.
But then they move into rotations and lose most of the class, because they forgot to mention that vectors aren't *always* representing something you can easily visualize in a one-to-one fashion. In CG, we have another example of vectors we use all the time but where the values aren't easily associated with a spatial position: RGB colors. Three values contained in a single vector (4 for RGBA), but unless you've got a nice color volume that shows you what the values mean, the association between an arrow floating in space and a resulting color is rather disconnected. Meanwhile, you could represent that color using other vectors: hue/brightness/saturation, CMYK, or hue/saturation/value, etc. Notice how CMYK has 4-dimensions, while the others have 3? Yet they all ultimately generate a color output. Also notice how changing the values of one of the dimensions in RGB color interpolates the resulting color differently from changing one of the values in HSV, yet both can result in the same range of colors? And so it goes with rotations. Different heuristics to represent the same result. Euler uses 3-dimensions, whereas angle/axis and quaternion use 4. Each one results in a different interpolation from changing one of the values, and each one is useful in different circumstances. Yet none of them (save maybe angle/axis) is easily visualized with an arrow in space. -Bradley On Tue, Mar 5, 2013 at 11:57 AM, Andy Moorer <andymoo...@gmail.com> wrote: > Hi all, > > As I was working today I realized that despite doing operations on ice > "orientations" regularly I don't have a firm grasp on what they really * > are*. Orientation relative to what? And what form is this orientation in? > > Trying to phrase it differently... > > How is a particle's "orientation" different from a 3x3 matrix describing > the difference in rotation between it's local coordinate space and the > global frame of reference? Both are clearly descriptions of the same thing. > > I know this is kind of an abstract subject, and (being tired at the > moment) my question may not even be clear, but being self-taught and > lacking adequate formal math education I'd be very interested in your > answers and any discussion in general on how you all visualize rotations > and orientation, > > For some people I talk to about rotations (I'm the life of any party) it > seems to be all about manipulating vectors, others seem more comfortable > thinking about rotations as transformations between reference frames... and > I see a similar wide range of how people go about some of this stuff when > looking into various compounds. >
