![]() ![]() The point that we rotate around must by equidistant to the chosen point on each body, therefore it must lie on a line perpendicular to the line joining the two points. So, provided that we can find a suitable point to rotate around (in the above example shown as green point) we can do the translation and rotation in one operation. ![]() However, there is a second option, we can do the translation all in one rotation: Do a rotation about the point so that the solid body is correctly transformed.Do a linear translation such that the point on 'A' is transformed to the point on 'B'.Then we can define a point on A (such as the centre-of-mass) and the corresponding point on B. Imagine that we want to transform the solid body at 'A' into the solid body at 'B' We now want to test the converse, that is, any combination of translation and rotation can be represented by a single rotation provided that we choose the correct point to rotate it around. So the rotational components are the same but the rotation moves the position (to see all the steps see this page) r 00 When these matrices are multiplied this will give the following result for = rotation about origin ( if this is not clear see 1 = inverse transform = translation of point to origin 1 So matrix representing rotation about a given point is: Note for matrix algebra, the order of operations is important, so these translations So if we are using the global frame-of-reference (as explained here) (add P which is translate by +Px,+Py,+Pz) rotate about the origin (can use 3×3 matrix R0).translate the arbitrary point to the origin (subtract P which is translate.With the following 3 simpler transforms which, when done in order, are equivalent: translate about arbitrary point P (Px,Py,Pz). ![]() In order to prove this and to calculate the amount of linear translation we Origin, but it has been translated to a different point on space by the rotation. We now want to apply this same rotation but about an arbitrary point P:Īs we can see its orientation is the same as if it had been rotated about the 'proper' isometry transformation' which means that it has a linear and a rotational component.Īssume we have a matrix which defines a rotation about the origin: In other words rotation about a point is an In order to calculate the rotation about any arbitrary point we need to calculate That is any combination of translation and rotation can be represented by a single rotation provided that we choose the correct point to rotate it around. We will then go on to show that, in two dimensions, that the converse is also true. This represents the same as rotating round the origin but offset by:įor three dimensional rotations about x,y we can represent it with the following 4×4 matrix: r 00 For two dimensional rotations about x,y we can represent it with the following 3×3 matrix: r 00 ![]()
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