Problem 3    Applications of Inversions to the Peaucellier-Lipkin Linkage

 

a.         Show that for the linkage in figure below the points P and Q are the inversions of each other through the circle of inversion with center at C and radius . 

Draw the circle with center R and radius d and note that C, P, Q, are collinear. Use Problem 1b.

 

 

b.  Show that the point Q in the linkage in this figure always traces a straight line.

 

Apply Problem 2c. 

 

 

 

 

If we modify the Peaucellier-Lipkin linkage by changing the distance between the anchor points then: 

c. The point Q in the linkage in the figure always traces the arc of a circle. Why? Show that the radius of the circle is expressed by   r2f / (g2  f 2).

Apply Problem 2d and look at the inversive images of the end points of the diameter of the circle with radius f.

 

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Introduction

Problem 1. Circles in the Plane, which explores some geometry of circles.

Problem 2. Inversions in Circles, which explores properties of inversion in circles.

Problem 3. Applications of Inversions to the Peaucellier-Lipkin Linkage, which explores the applications of Problems 1 and 2 to an understanding of the linkage.