GEOMETRY OF MOTION. 



[16. 



II. Plane Motion. 



16. The position of a plane figure in its plane is fully deter- 

 mined by the positions of any two of its points since every 

 other point of the figure forms with these two points an invari- 

 able triangle. But the position of the figure can of course be 

 determined in other ways ; for instance, by the position of one 

 point and that of a line of the figure passing through the point ;. 

 or by the position of two lines of the figure. 



17. Let us now consider the motion of a plane figure F in its. 

 plane from any initial position F Q to any other position F^ 

 The displacement F Q F 1 can be brought about in various ways. 



Thus, it would be suffi- 

 cient to bring any two 

 points A, B (Fig. 2) of 

 the figure F from their 

 initial positions A Q , B Q in 

 FQ to their final positions 

 ' B i A v B l in F v This can 

 be done, for instance, by 

 first giving the whole fig- 

 ure a translation through 

 a distance A Q A 1 and then 

 a rotation by an angle 

 ^ ; or by such a rota- 



Fig. 2. 



and 



equal to the angle between A Q 

 tion followed by the translation. 



Instead of A we might have selected any other point of the 

 figure. But it is important to notice that the angle of rotation 

 required for a given displacement F Q F l is always the same, while 

 the translation will differ according to the point selected as 

 centre. 



18. This leads us to inquire whether the centre of rotation 

 cannot be so selected as to reduce the translation to zero. 

 Now any rotation that is to bring A from A Q to A 1 must have 



