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| Torus Knots (Planar) |
As long as the world is turning and spinning, we're gonna be dizzy and we're gonna make mistakes. --Mel Brooks
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| Tommy's Interwoven Paths In His Past and Present Life In His Home Raiding His Father's Refrigerator I See Him Standing by the Fridge His Father Said the Pot Flew Out of a Pantry Kitty-Corner to the Fridge During a Crisis Some Time Later His Kitchen Is Just About Square And Tommy's Path Seems to Be On a Diagonal , His Orientation Turned Around |
Based on the above Scenario, our brave young soldier Tommy goes into his kitchen (at his father's house) just days before he was supposed to report to boot camp for deployment to some vacation spot.
Like any other young man, he goes through the kitchen door and heads due East to raid the refrigerator.
Surely there must be something to eat!
Some time after, Tommy dies of dehydration because his drill Sergeant couldn't allow a break for water during a strenuous exercise before ever getting deployed. Which causes Tommy's father a great deal of distress, and makes his good friend wonder 'what is it about men that makes them act like accomplished women?'
And during a pivotal moment in his father's coping mechanism failure, a sauce pot comes flying out of the NW pantry of the kitchen (no earthquakes reported in that part of town on that particular occasion) and lands right in the middle of the square kitchen (6 ' x 6').
Determine the rotational matrix based on the above scenario.
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| (A bad drawing to kind o' confuse the issue (the planes are offset along a central axis, much like a helix/screw threads/or augers), think 3D and not certain the time). |
{"Sit on it...and Rotate!"--The Fonz (Henry Winkler), popular 1970'a sitcom Happy Days
The Rotational Matrix
In Euclidean geometry, a rotation is an example of an isometry, a transformation that moves points without changing the distances between them.
Rotations are distinguished from other isometries by two additional properties:
- they leave (at least) one point fixed, and
- they leave "handedness" unchanged.
By contrast, a translation moves every point, a reflection exchanges left- and right-handed ordering, and a glide reflection does both.
A rotation that does not leave "handedness" unchanged is an improper rotation or a rotoinversion.
A geometric rotation transforms lines to lines, and preserves ratios of distances between points. A rotation is a linear transformation of the vectors, and can be written in matrix form, Qp.
The fact that a rotation preserves, not just ratios, but distances themselves, it follows that...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, all these properties may be summarize by saying that the n×n rotation matrices form a group, which for n > 2 is non-abelian.
Called a special orthogonal group, and denoted by SO(n), SO(n,R), SOn, or SOn(R), the group of n×n rotation matrices is isomorphic to the group of rotations in an n-dimensional space.
This means that multiplication of rotation matrices corresponds to composition of rotations, applied in left-to-right order of their corresponding matrices.
To rotate points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system use a rotation matrix R; each point is represented by a column vector v, of each point's coordinates (position). The rotated vector is then obtained by matrix multiplication Rv.
Rotation matrices are square matrices (nxn), with real entries; characterized as orthogonal matrices with determinant 1.
The special orthogonal group SO(n) is the set of all such matrices of size n, generally not commutative.
Improper rotations, characterized by orthogonal matrices with determinant -1 (instead of +1) combine proper rotations with reflections (which invert orientation).
The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. -90°).
Basic rotations:
Three basic (gimbal-like) rotation matrices rotate vectors about the x, y, or z axis...
All other rotation matrices can be obtained from these three using matrix multiplication.
...to represent a rotation whose yaw, pitch, and roll are α, β, and γ, respectively.
...whose Euler angles are α, β, and γ (using the y-x-z convention for Euler angles).
Every rotation in three dimensions is defined by its axis — a direction that is left fixed by the rotation — and its angle — the amount of rotation about that axis (Euler rotation theorem).
There are several methods to compute an axis and an angle from a rotation matrix one is based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix.
...which shows that is the null space of...
Viewed another way, [] is an eigenvector of R corresponding to the eigenvalue; every rotation matrix must have this eigenvalue.
To find the angle of a rotation, once the axis of the rotation is known, select a vector perpendicular to the axis. Then the angle of the rotation is the angle between__and__.
It is much easier to calculate the trace (i.e. the sum of the diagonal elements of the rotation matrix)...
In three dimensions, for any rotation matrix , where a is a rotation axis and θ a rotation angle,
(i.e., is an orthogonal matrix)
(i.e, the determinant of is 1)
(where is the identity matrix)
The eigenvalues...
The trace of is equivalent to the sum of its eigenvalues.
Some of these properties can be generalized to any number of dimensions. In other words, they hold for any rotation matrix...
Ambiguities: Alias and alibi rotations
The interpretation of a rotation matrix can be subject to many ambiguities.
The change in a vector's coordinates can be due to a turn of the coordinate system (alias) or a turn of the vector (alibi). Any rotation can be legitimately described both ways, as vectors and coordinate systems actually rotate with respect to each other. Here, the alibi approach is used to describe rotations.
The vector can be pre-multiplied by a rotation matrix (Rv, where v is a column vector), or post-multiplied by it (vR, where v is a row vector). Here rotations are produced by means of a pre-multiplication.
The vector space has a dual space of linear forms, and the matrix can act on either vectors or forms.
In most cases the effect of the ambiguity is equivalent to the effect of a transposition of the rotation matrix.
Two features are noteworthy:
- First, one of the roots (or eigenvalues) is 1, i.e., some direction is unaffected by the matrix. For rotations in three dimensions, this is the axis of the rotation (a concept that has no meaning in any other dimension).
- Second, the other two roots are a pair of complex conjugates, whose product is 1 (the constant term of the quadratic), and whose sum is 2 cos θ (the negated linear term). This factorization is of interest for 3×3 rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthermore, a similar factorization holds for any n×n rotation matrix. If the dimension, n, is odd, there will be a "dangling" eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the two special cases noted). Its a given that the characteristic polynomial will have degree n and therefore n eigenvalues. And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized.
Every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most n⁄2 of them...
The sum of the entries on the main diagonal of a matrix is called the trace...
The large number of options is possible because rotations in three dimensions (and higher) do not commute.
Reversing a given sequence of rotations results in a different outcome.
This implies that two rotations cannot be composed by simply adding their corresponding angles; i.e., Euler angles are not vectors, despite a similar representation as a triple of numbers.
}
(The rest of us just spin on our heels and walk a distance about 6Ö2 along the diagonal heading WNW if we think we are at the fridge or ESE if we think we are at the pantry and want something out of the fridge.
Or become really well acquainted with quaternions to get a sense for what Tommy was seen doing in his 'spare time'. )
Just like those 'I love you' tokens I been hearing of late.
If You Love Me, you would stop this (You Know Exactly what I mean).
Surah 2 Al.Baqarah (The Calf)
وَلاَ تَقُولُواْ لِمَنْ يُقْتَلُ فِي سَبيلِ اللّهِ أَمْوَاتٌ
بَلْ أَحْيَاء وَلَكِن لاَّ تَشْعُرُونَ (2:154
Nay, they Live and yet you perceive it not.
I know, I know, it must be End of the World because i am the smartest girl in the room; don't anybody panic because God can make another dysfunctional one just like it!