In high school geometry it is usually formulated as a criterion for proving that two triangles are congruent: This is known as the Side-Side-Side property.
Is it possible to reconstruct everything else about the triangle? Suppose we know the three side-lengths of a given triangle $\Delta ABC$. We can take this as a mathematical characterization of the physical notion of "structural rigidity": If a diagram is completely determined (up to a rigid motion) by its incidence relations and its linear measurements, we can call it a rigid diagram. The original diagram (the square) and the deformed diagram (the rhombus) have exactly the same incidence relationships and linear measurements, but the angles in them are different. To see this, just look at the example in the OP of a "nonrigid" shape: A square that deforms into a rhombus.
Rigid motion geometry definition how to#
That is, even if could reconstruct a diagram from its incidence relations and linear measurements, we would not know where to position it in the plane or how to orient it, because a rigid motion of the plane preserves all incidence and metric properties while possibly changing position and orientation.īut a more significant observation is that knowing the incidence relations and linear measurements of a diagram does not, in general, completely specify a diagram. Is a diagram completely specified by its incidence relations and its linear measurements?Īnother (slightly less formal) way to put this is: If you know how long all the segments in a diagram are, can you draw the diagram?įirst, let's notice that at best we might be able to say that a diagram is completely specified up to a rigid motion. knowing which points are the endpoints of which which line segments) and its linear measurements (i.e.
Suppose we have two kinds of data about a given diagram: its incidence relations (i.e. Let us define a diagram to refer to any finite collection of line segments with labeled endpoints in the plane.
Although the accepted answer by rschwieb is an excellent one, I think there is a connection between the mathematical notion of "rigid motion" and the physical notion of "structural rigidity" that has not been mentioned yet.
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