Wikipedia"s section on rigid forms does not show up to actually define what a rigid shape is. Quite it defines "same shape" and also "rigid transformations" without giving any kind of definitions that what is necessary and sufficient for a shape to be taken into consideration rigid.

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For instance, I"ve watched the following image:

I recognize intuitively why the triangle is rigid and quadrilateral is non-rigid. It is additionally my expertise that inserting a single diagonal link into the quadrilateral would make the shape rigid? However, provided the adhering to image (labeling the shapes \$S_1\$, \$S_2\$ and also \$S_3\$ in succession left to right):

If \$S_1\$ had actually a single diagonal connection AB, it would certainly be possible to upper and lower reversal ADB end axis q to obtain a shape similar to \$S_2\$ through the enhancement of connection abdominal muscle (my explanation might not it is in the greatest). This form is plainly not the very same as \$S_1\$, so how have the right to we speak \$S_1\$ with a single diagonal link CD is rigid? i feel i am unsure on what the meaning of rigidity in reality is.

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edited Dec 23 "18 in ~ 21:14

Glorfindel
asked Sep 11 "13 in ~ 5:20

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The factor you space finding these principles hard come reconcile is because the diagram is introduce to a physical notion of rigidity, if the wiki web page you"re analysis is centered around a geometrical notion of rigidity. (I don"t discover the short article you connected to it is in particularly plainly written either.)

## Geometry

In geometry, we don"t talk about rigid shapes really, us talk about rigid transformations. Shapes in geometry are just sets that points, no physical objects with resistance come bending and also stretching. They are at the mercy that transformations used to them.

Assuming we are working in a geometry that has notions of exactly how to measure up angles and also distances:

A rigid revolution preserves all distances and also angle procedures (and depending upon your taste, orientation too)

The idea is the no issue what form we start out with, any kind of shape you draw in the airplane at all will look the same after a rigid revolution is applied, other than that the location and also situation can be various from what it used to be. (It might also be its winter image, if you have permitted transformations to upper and lower reversal the orientation that the plane.)

Many of those shapes would be changed if us picked a nonrigid transformation, because that example, the change given by \$x\mapsto x\$ and \$y\mapsto 2y\$ in the Cartesian plane. This would readjust a circle at the origin right into an ellipse.

If you space interested in geometry the is not founded upon distance, then you can adopt some geometry axioms that assume notions of congruence the segments and also angles. Rigid transformations of the plane would then be ones which perform not annoy segment congruence no one angle congruence.

## Physics

Now, over there is a concept of rigidity in truth that has an ext to perform with that is resistance to transforming shape. This is called structural rigidity. This is really not the same pet as rigidity of form in geometry, back it"s obviously related.

In the diagram friend supplied, the allude seems come be the we space assuming the segments perform not readjust length, yet that the joints are on hinges. You space able to use physical pressures to both, and see just how they behave. One would certainly classify the triangle as a strictly shape due to the fact that none the its edgelengths or angles would change because that the intrinsic shape of the object. The square at the same time isn"t geometrically limited to having 90 degree angles once you use pressure to it, so it can adjust into a rhombus quite quickly.

Moreover, you can easily imagine building an object with length an altering segments and rigid hinges, so that a square could be pulled into a rectangle, however not propelled over into a rhombus. Ns don"t think that object would certainly be thought about "rigid" either.

## Conclusion

Hopefully I"ve express a bit around the difference between these 2 studies.

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I don"t median to say that the physics notion is entirely disjoint native betterworld2016.orgematics: for certain physical rigidity can be analyzed through betterworld2016.orgematics. It"s just that the wiki web page on geometry was not whereby you want be if you"re interested in structure rigidity.