Question

a rigid body exists in n dimensional space, how many co ordinates are needed to specify the position and orientation in this space

Answer 1

Answer on Question #53268, Physics / Mechanics | Kinematics | Dynamics

A rigid body exists in n dimensional space, how many coordinates are needed to specify the position and orientation in this space?

Solution:

The position and orientation of a rigid body in three-dimensional space is defined by three components of translation and three components of rotation, which means that it has six degrees of freedom.

If a body is rigid, then its position can be uniquely specified by a number of generalized coordinates equal to the number of degrees of freedom.

The position of an rigid body in n-dimensional space is defined by the rigid transformation,

$$[ T ] = [ A, d ],$$

where $d$ is an n-dimensional translation and $A$ is an $n \times n$ rotation matrix, which has $n$ translational degrees of freedom and $n(n - 1)/2$ rotational degrees of freedom. The number of rotational degrees of freedom comes from the dimension of the rotation group $\mathrm{SO}(n)$.

Answer: $n + n(n - 1)/2$

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