Articles of rigid transformation

Let $A,B,C$ be the vertices of a triangle. Find the center of the rotations. $R_{C,\pi} \circ R_{A,\frac{\pi}{2}}$

Let $A,B,C$ be the vertices of a triangle. Find the center of the rotations. $R_{C,\pi} \circ R_{A,\frac{\pi}{2}}$ Hey everyone, I am learning about euclidean rigid motions and my book only explains theory but does not focus on computations therefore, half the problems are computations and the other half are proofs. I am having a hard […]

Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane

A cube with vertices $(\pm 1, \pm 1, \pm 1)$ gets projected into the plane perpendicular to vector $\mathbf{n}\in S^2$. The projection is a hexagon, how do I find the area? http://biochemistry.utoronto.ca/steipe/abc/images/2/2b/CubeBasic.jpg I think I can just compute $\mathbf{n}\cdot e_{\mathbf{x}},\mathbf{n}\cdot e_{\mathbf{y}}$ and $\mathbf{n}\cdot e_{\mathbf{z}}$ and the area would just be the some of the 3 […]

Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to optimize $argmin_{\mathbf T} \sum_i \mathbf{e}_i(\mathbf{T}), i = 0..N $ with $\mathbf{T} \in SE(3)$ $\mathbf{e}_i(\mathbf{T}) = exp(a_i log(\mathbf{T}))\cdot P_i – P^*_i $ $a_i \in [0,1]$ […]

A question of H.G. Wells' mathematics

H.G Wells’ short story The Plattner Story is about a man who somehow ends up being “inverted” from left to right. So his heart has moved from left to right, his brain, and any other asymmetries belonging to him. Then H.G Wells’ goes on a slight metaphysical exposition: There is no way of taking a […]

Minimization on the Lie Group SO(3)

Refering to a question previously asked Jacobian matrix of the Rodrigues' formula (exponential map). It was suggested in one of the answer to only calculate simpler Jacobian $\left.\frac{\partial}{\partial \mathbf p}\exp(\hat{\mathbf p})\right|_{\mathbf p=\mathbf 0}$ with $\exp(\hat{\mathbf p})$ being identity. Similar idea is suggested in a technical report on Minimization on the Lie Group SO(3) and Related […]

Finding a Rotation Transformation from two Coordinate Frames in 3-Space

The question I’m trying to figure out states that I have 3 points P1, P2 and P3 in space. In one frame (Frame A I called it) those points are: Pa1, Pa2 and Pa3, same story for Frame B (namely: Pb1, Pb2 and Pb3). Whats the rotation matrix from one to the other? That’s literally […]

Jacobian matrix of the Rodrigues' formula (exponential map)

I am working an algorithm which is supposed to align a pair of images. The motion model, which describes the pose $p$ of an image (with respect to the second) in 3D space, is purely rotational. Theory to my question: According to the Rodrigues’ formula for rotation matrices, the matrix exponential $R(p)=e^{\hat{p}}$ for a vector […]