Wikipedia:Reference desk/Archives/Mathematics/2017 December 30

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December 30

What is the minimum amount of information to rotate a point in 3D Cartesian space?

Suppose you are given a 3D cartesian point {x,y,z} = {1,2,3}. What is the minimum amount of (additional) information you need to rotate that point in 3D space to another point in 3D space? For many years I thought all I need is

1. X coordinate of rotation origin
2. Y coordinate of rotation origin
3. Z coordinate of rotation origin
4. X coordinate of the normal vector
5. Y coordinate of the normal vector
6. Z coordinate of the normal vector
7. Angle of rotation in either degrees or radians
8. Rotation using the normal vector using which rule? Right Hand Rule or Left Hand Rule?

But I was mistaken, because I got the wrong answers to my problem. It took me a few days until I realized what my problem is.

I am missing the Cartesian Axis Type.

For example:

Cartesian Axis Type 1: X-axis is towards the observer, Y-axis is to the right (from observer's point of view), Z-axis is upwards (from observer's point of view)

Cartesian Axis Type 2: X-axis is to the right (from observer's point of view), Y-axis is upwards (from observer's point of view), Z-axis is away from the observer

Depending on which Cartesian Axis Type, you will get completely different answers.

So what is the minimum amount of information to rotate a point in 3D Cartesian space? Am I missing anything else? Ohanian (talk) 03:37, 30 December 2017 (UTC)

In application, you have a point to rotate (3 values), a point to rotate around (3 values), and an angle. That is all. The direction of rotation is implied by the angle. If I say rotate 30 degrees and you want to go the wrong way, then I will tell you to rotate 330 degrees. With those 7 values, you translate the two points so that the point to rotate around is the origin. Then, you rotate (a simple rotation matrix). Then, you reverse the original translation. It is three matrix multiplications, and is very standard in any graphics programming, though you normally plan to do the same thing more than once. So, you multiply the translation, rotation, and un-translation matrices together and then use that on the original point. 71.85.51.150 (talk) 03:48, 30 December 2017 (UTC)

The article on the Euclidean group might be useful here. --RDBury (talk) 07:05, 30 December 2017 (UTC)

As Gandalf explains below, in 3D space you rotate about a line, not about a point. And you need 4 values, rather than 3, to specify the line.

In 4D, by the way, you rotate about a plane. And you can also have a rotation about two planes simultaneously; since those intersect at a point, you could call it a rotation about the point, though you'll need several parameters to specify the rotation even once the point is given. -- Meni Rosenfeld (talk) 20:26, 30 December 2017 (UTC)

The minimum amount of information required to uniquely specify a rotation about a general axis in 3 dimensional Euclidean space is 5 real numbers - we say that there are five "degrees of freedom". One way to see this is as follows :

• The direction of the axis can be specified with two real numbers. If we use three numbers to define a vector then we have redundant information because vectors with the same direction but different lengths define the same axis direction - so the mapping from 3 dimensional vectors to axis directions is many-to-one. If we restrict ourselves to, say, vectors with length 1 then we only need two independent values.
• The location of the axis can be specified with two real numbers. Two not three because we only need to specify a point on any plane that is perpendicular to the axis direction, and these planes have two dimensions. Again, If we use three numbers to locate a specific point on the axis then we have redundant information because all the points in any line parallel to the axis direction will define the same axis location.
• The size of the rotation is specified by one more real number. If you are only concerned with the end state and not the route taken to get there then there is redundancy here as well, because you only really need to know the rotation size module 360 if measuring in degrees (or module 2π if using radians). But for some applications the total size of the rotation (whether it involves one full turn, two full turns etc.) may be significant.

The left/right handedness of the axes and the direction of a positive rotation are usually taken as conventions that are defined in advance. They are not an intrinsic part of the Euclidean space.

It may be convenient to use more than five parameters to specify a general rotation - for example, you could use 6 numbers to define a rotation about the origin together with a translation. But then these parameters will involve some redundancy - the mapping from tuples of parameter values to rotations will be many-to-one. Gandalf61 (talk) 11:38, 30 December 2017 (UTC)

I think people are confused, Assuming I am talking to aliens from another galaxy, how can I know what the convention is if all that the aliens gave me is {x,y,z}={1,2,3} what are the minimum amount of information those aliens need to provide me to rotate that point to another location in 3D space? You bet I need to know the handedness at the bare minimum. And I also need to know the Cartesian Axis Type. Ohanian (talk) 00:32, 31 December 2017 (UTC)

Your question is pretty vague. It doesn't really make much sense to "rotate one point to another point". To specify a rotation in 3 dimensions, you need to specify a point in SO(3) (the special orthogonal group over $\mathbb {R} ^{3}$). This is itself a three-dimensional manifold, and so 3 numbers suffice. Then again, as sets, euclidean 3-space has the same cardinality as the line, so you can encode the entire rotation in a single real number. That probably isn't what you mean, but "pieces of information" is pretty vague. Most likely, it's the dimension of SO(3) that's really what you're after (and like I said, it's three). –Deacon Vorbis (carbon • videos) 01:31, 31 December 2017 (UTC)

After rereading this, I guess you want to specify the center of the rotation too. In that case, yes, it would be 6-dimensional instead. –Deacon Vorbis (carbon • videos) 02:55, 31 December 2017 (UTC)

Rotations in 3 dimensions don't have a unique centre - they have an axis, a line of points unchanged by the rotation. By choosing a point in SO(3) you have defined the direction of the axis and the angle of the rotation. To translate that axis through the origin to a parallel axis elsewhere, you only need add two more dimensions, not three. So the space of general 3D rotations is five dimensional. You can use six parameters if you like, but then there is redundancy. Gandalf61 (talk) 07:11, 31 December 2017 (UTC)

Maybe the following observation will be helpful to the OP: a tuple of 3 numbers is not the same thing as a point in three-dimensional space; the correspondence is a consequence of some arbitrary choices (where to draw the axes, etc.). The algebraic language does not depend and cannot distinguish between some of these choices: the result of performing a certain translation (given in coordinates) to a certain point (given in coordinates) has the same coordinates regardless of whether the coordinate system is right- or left-handed, regardless of how your axes are rotated in space, regardless of where the origin is, etc. --JBL (talk) 19:33, 1 January 2018 (UTC)

Why is the interval when a < b and a,b real numbers then [b,a]=[a,a)=(a,a]=(a,a)=empty set?

I don't understand what's probem? — Preceding unsigned comment added by 151.236.179.187 (talk) 09:39, 30 December 2017 (UTC)

Suppose a is 3 and b is 5. How many numbers are 5 or more and 3 or less, i.e. x>=5 and x<=3? How many numbers are 5 or more and up to but not including 5, i.e. x>=5 and x<5? Dmcq (talk) 09:55, 30 December 2017 (UTC)

I was relieved, I was fearing, so every question can be asked here, thank you mathematicians for supporting us you are great! Alireza Badali (talk) 19:59, 30 December 2017 (UTC)

Our relevant articles are:

• Interval (mathematics)#Including or excluding endpoints -- This states the conditions under which an interval represents the empty set, but does not explain the "why". I think it's fine as is without further explanation. That section does link set builder notation.
• Set-builder notation#Sets defined by a predicate -- This did not explicitly state that {x∈E|Φ(x)}=∅ when Φ(x) is false for all x∈E. I first though this should be equally obvious, but formal set theory can be confusing (this very sections mentions Russell's paradox), so I've added, "If the predicate is false for every element of the domain, then the set defined is the empty set."

-- ToE 00:47, 31 December 2017 (UTC)