It is far from obvious, but this composition is equivalent to a rotation about some point other than the origin. The sections following this one provide the tools to prove this. Can you guess where that center o
THE IDEA OF COMPOSITION 1. Can you describe an isometry S (other than I) where S ∘ S ∘ S= I? Can you think of more than one? 2. Suppose you know that a rotation R about the origin followed by a translation is a rotation R′ around some point other than the origin. Given...
In 2D, I can see that any combinations of translation plus rotation about the origin can be transformed into a pure rotation but perhaps about a different point other than the origin. I wonder in 3D, is this possible? And, how to find out the equivalent pure rotation then? thanks! ...
A rotation about an arbitrary line $\mathbf u+t\mathbf v$ in $\mathbb R^3$ can be decomposed into a translation to the line, a rotation about an axis through the origin, and a translation back. In homogeneous coordinates, this can be represented by the $4\times4$ matrix $$M=\left[...
a.acircularmotionof aconfigurationaboutagivenpointorline,withoutachangeinshape b.atransformationinwhichthecoordinateaxesarerotatedby afixedangleabouttheorigin c.anothernameforcurl11Abbreviation(forsense4c):rot 5.(Astronomy) a.thespinningmotionof abody,suchas aplanet,aboutaninternalaxis.Comparerevolution5a...
(5). We establish a direction for the vector of the angular velocity by using aright-hand rule, as shown in the figure.Curl your right hand about the rotating record, your fingers pointing in the direction of rotation. Your extended thumb will then point in the direction of the angular ...
The rules of rotation help us find the new position of a point after it has been rotated around the origin by a certain angle.The types of rotation depend on the angle of rotation and the direction of rotation.When learning about rotation in math, we usually use the coordinate plane to ...
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Example: so3([pi/2 pi/3 pi/4],"eul","ZYZ") rotates a point by pi/4 radians about the z-axis, then rotates the point by pi/3 radians about the y-axis, and then rotates the point by pi/2 radians about the z-axis. This is equivalent to so3(pi/2,"rotz") * so3(pi/3,...
One may extend the above scenario, to systems larger than the galaxy. On a much larger cosmic scale, it may be possible to envisage galaxies, as point particles interacting with each other. It is now shown that the MBG theory is able to explain the rotation curves of the Milky Way and ...