linkages
I have always been fascinated by the idea of linkages. I love the way they do really complicated movements to produce something simple. On the left is a picture of one I made called Peaucellier's Linkage, invented by a French army officer in 1864. It was the first linkage that could draw a straight line, and solved the problem of creating straight pistons in engines.
The video 'How to draw a straight line' on the excellent 'How round is your circle' shows it in action. On that website there are also some excellent geogebra files that stoke the interest further. After watching this video I was determined to work out how it worked, and then make one myself! I have also made a pantograph, a different type of linkage here.
The mathematics of Peaucellier's linkage involves a really interesting area of geometry called inversion that seems to pop up all over the place!
The video 'How to draw a straight line' on the excellent 'How round is your circle' shows it in action. On that website there are also some excellent geogebra files that stoke the interest further. After watching this video I was determined to work out how it worked, and then make one myself! I have also made a pantograph, a different type of linkage here.
The mathematics of Peaucellier's linkage involves a really interesting area of geometry called inversion that seems to pop up all over the place!
inversion
Given the large circle centre O, radius r, the two points P and P' are inverse points if they are collinear with O and OP x OP' = r ^ 2.
If P follows a circle through O, then it can be shown that P' will follow a straight line. Can you see how? The proof is really nice and simple if you can see it!
Hint: It involves the fact that they are inverse points (see definition above), similar triangles and the circle theorem 'the angle in a semi-circle is 90 degrees'.
If P follows a circle through O, then it can be shown that P' will follow a straight line. Can you see how? The proof is really nice and simple if you can see it!
Hint: It involves the fact that they are inverse points (see definition above), similar triangles and the circle theorem 'the angle in a semi-circle is 90 degrees'.
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Relating this to this image of the linkage, C is the point P on the diagram above. C follows a circle through O (because it is constructed so that OQ = CQ).
It can be shown that OC x OP = constant using Pythagoras, and because OCP is a line, then C and P are inverse points with respect to some (other) circle centre O. So P follows a straight line. For more on linkages, look at the howround website, or read Mathematical Models by Cundy and Rollett. I found out about inversions when working through the very good UKMT book Crossing the Bridge by Gerry Leversha. And I really would recommend making one - it was lots of fun!! |