Euler angles with "standard" rotations

 

We can specify a general rotation \(R\) of the 3D space by mean of 3 angles \(\alpha\), \(\beta\) and \(\gamma\). Fix three axis \(x\), \(y\) and \(z\), and a rotation will be determined by the image of this three axis, \(X\), \(Y\) and \(Z\). If we call \(N\) to the intersection of the \(xy\) plane with the \(XY\) plane, we can carry the first frame to the second one by means of

  • A rotation along \(z\) of angle \(\alpha\), to carry \(x\) to \(N\). Say \(R_z^{\alpha}\).
  • A rotation along \(N\) (the new \(x\)) of angle \(\beta\), to carry \(z\) to \(Z\). Call it \(R_N^{\beta}\).
  • A rotation along \(Z\) (the new \(z\)) of angle \(\gamma\), to carry \(N\) to \(X\). Say \(R_Z^{\gamma}\).



This way,
\[
R=R_{Z}^{\gamma}R_N^{\beta}R_{z}^{\alpha}
\]

This description has never satisfied me, because the axis of rotation are not defined extrinsically, that is, they are not universal, they do not work for any rotation. They depends on the rotation in particular. But with a little of  algebraic manipulation we can recycle this angle to use them with standard rotations along the axis \(x\), \(y\) and \(z\).

First, observe that
\[
R_N^{\beta}=R_{z}^{\alpha}R_x^{\beta}R_{z}^{-\alpha}
\]
since we can express a rotation in a different axis going and coming back.


Second, by the same reason,
\[
R_{Z}^{\gamma}=(R_N^{\beta}R_{z}^{\alpha}) R_{z}^{\gamma} (R_{z}^{-\alpha}R_N^{-\beta})
\]

But then,
\[
R_{Z}^{\gamma}=((R_{z}^{\alpha}R_x^{\beta}R_{z}^{-\alpha})R_{z}^{\alpha}) R_{z}^{\gamma} (R_{z}^{-\alpha}(R_{z}^{\alpha}R_x^{-\beta}R_{z}^{-\alpha}))=
\]
\[
=R_{z}^{\alpha}R_x^{\beta} R_{z}^{\gamma}R_x^{-\beta}R_{z}^{-\alpha}
\]

And therefore our original rotation can be expressed with standard axis
\[
R=(R_{z}^{\alpha}R_x^{\beta} R_{z}^{\gamma}R_x^{-\beta}R_{z}^{-\alpha})(R_{z}^{\alpha}R_x^{\beta}R_{z}^{-\alpha})R_{z}^{\alpha}=R_{z}^{\alpha}R_x^{\beta} R_{z}^{\gamma}
\]

Comments

Post a Comment

Popular posts from this blog

About the definition of angle, from Geometric Algebra

Image
In GA, bivectors represent  2-dimensional directions. Any simple bivector can be written as \[ \theta e_1e_2 \] with \(\theta \in \mathbb{R}\) and \(\{e_i\}\) unitary and orthogonal vectors. The unitary vectors specify the 2- direction itself, and \(\theta\) is the size (analogous to the length of a vector, that is a 1-dimensional direction). Given two vectors \(a\) and \(b\) (that we take to be unitary, wlog) we say that the simple bivector \(\theta e_1e_2\) is the angle formed by them if \[ ab=e^{\theta e_1 e_2} \] where we take as a definition \[ e^A=1+A+A^2 /2+\cdots=\lim_N (1+\frac{A}{N})^N \] Real analysis let us assure that this expression is well defined when \(A\) is a 2-blade. In fact, since \(e_1e_2e_1e_2=-1\), we can write \[ e^{\theta e_1 e_2}=cos(\theta)+e_1 e_2sin(\theta) \] where \(cos\) and \(sin\) are defined by their power series. Oserve that \(b=a e^{\theta e_1 e_2}\) and then \[ b=a(cos(\theta)+e_1 e_2sin(\theta))=a\cdot cos(\theta)+ ae_1e_2 \cdot sin(\theta) \]
Read more

Spherical, hyperbolic and parabolic geometry

Image
Let's consider the space \(\mathbb{M}^4\), that is, the manifold \(\mathbb{R}^4\) with coordinates \((x,y,z,t)\) but with the pseudo-Riemannian metric given in each tangent space by \[ ds^2=dx^2+dy^2+dz^2-dt^2. \] This metric is left invariant by the group \(O(3,1)\) when acting on \(\mathbb{M}^4\). It also leaves invariant the cone \[ t^2-x^2-y^2-z^2=0 \] so is logical that we pay attention to it. Now, following TRTR from Penrose page 423, we can consider the family of hyperplanes inside \(\mathbb{M}^4\) given by \[ z+t+\lambda(t-z)=2 \] intersecting our cone in different 2-dimensional submanifolds \(P_{\lambda}\) of \(\mathbb{M}^4\). Here we have a picture from that book, where we consider only \((x,z,t)\) for purposes of drawability. If we analyse the case \(\lambda=0\), we observe that we obtain the submanifold \(P_0\), \(E\) in the picture with a shape of a parabola, given by the embedding \begin{align*}   \phi \colon \mathbb{R}^2 &\longrightarrow \mathbb{M}^4\\   (u_1,u_2
Read more

4-dimensional spheres, or haunting an ant

Image
In this post, we will try to explain a little bit what it feels like to live in a 4-dimensional world. First, we will see that we can enter the inside of a normal sphere without breaking the wall. And secondly, we will try to describe what we would feel if our world were not the classic 3D universe, known as \(\mathbb{R}^3\), but another 3-dimensional space called \(S^3\). But before we start, let's name things. The circles we all know have a more technical name for mathematicians: \(S^1\). On the other hand, a sphere (let's say, a ball) is called \(S^2\). This similarity in names is not accidental: since we were children, we intuitively knew that circles and spheres have something in common. In fact, they are "the same" except for the detail that circles are 1-dimensional (you can only move forward or backward), and spheres are 2-dimensional (if you lived in one, you could move forward-backward, but also left-right. Oops, actually, we live in one). Now let's
Read more