Degrees of Freedom of a Moving Rigid Body

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This Demonstration explains how six coordinates can describe the position of a moving rigid body. Varying the six coordinates for the position of the body (three for the position of a point of the body and three for the orientation of the body), it is possible to observe the corresponding displacement of the rigid body.

Contributed by: Luca Mannella (January 2014)
Open content licensed under CC BY-NC-SA



Consider three points , , of a moving rigid body of a given shape. Six coordinates are necessary and sufficient to describe the position of the rigid body. The distances between the three points (as well as the distances between any pair of points of the rigid body) are left unchanged by the motion, and the rigidity condition fixes the number of degrees of freedom of the body to six. For example, in a given frame of reference, the position of a rigid body can be described by three coordinates for the position of the point (or of any other point of the body) and three angles (, , ) for its orientation (or, more precisely, for the position of , relative to ).

The distances between the points , , have to satisfy the triangle inequalities; each of the distances has to be less than or equal to the sum of the other two distances; for example, . Also, each of the distances has to be greater than or equal to the positive difference of the other two distances; for example, . If these conditions are not satisfied, the two yellow spheres (one with center at of radius and the other with center of radius ) do not intersect.

is the zenith angle between the segment and the axis.

is the azimuthal angle between the projection of on the - plane and the axis.

The angle for the position of is the rotation around the segment .

Varying (within the specified range for , , ) the six coordinates , , , , , , it is possible to obtain any position of the rigid triangle and consequently any position of a rigid body of fixed shape around that triangle.

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