Rectification of the Circular Arc by Cochleoidal Curve
Construction from Paul Sipos by Maple 

dr.B.Laczik, laczik@goliat.eik.bme.hu 

The problem: construct the the exact length  OT of circular arc OC, see Fig. 2.  

This construction is one of the traditional geometric problems. The solution is impossible by euclidian construction methods (apply only compas and straigth slider). 

In this worksheet is demonstrated the exacte rectification procedure of the problem width applying the cochleoidal curve and the geometric package of Maple. 

The principe of contruction is explained by hungarian mathematician Paul Sipos (1759-1816) in "Beschreibung und Anwendung eines mathematischen Instruments f?r die Mechaniker, und unmittelbaren Vergleichung der Circulbogen", in Bull. of Sci. Acad. Berlin, 1795. 

The cochleoid is one of the planar curves. The points , , , ..., of cochleoidal curve are points of circle , , , ..., , where the arc lengths = = = ...= = k. 

The polar radius of curve is OP = , the polar equation of cochleoid is  , see Fig. 1. 

The steps of the construction are shown in Fig 2. 

0. The cochleoid (red curve) is given, the first point O of circular arc (magenta curve) is the origin, and the line OH is the tangent of the arc. (H = is the first point in cochleoid to circle by radii infinity value and polar angle = 0.) 

1. Draw the chordal line OC of the circular arc. (The equation of chordal line is . The direction of OC is the positive).  

2. The  point Q is the intersection of cochleoid and the line OC. 

3. Draw the line QH. 

4. The paralel line to QH over C point intersect the line OH in point T. 

5. The length of circular arc OC is the distance OH. 

Figure 1. 

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The given parameters of procedure are k, r, a : 

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Figure 2.