By C. Zwikker

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**Extra info for The Advanced Geometry of Plane Curves and Their Applications (Dover Phoenix Editions)**

**Example text**

Z z -zz z. The radial of the circle z =exp(ju) is again this circle. The radial of the circle evolvente z=(l-ju)exp(ju)tums out to be: z,= ~ exp (iu) J and this is a spiral for which the modulus is proportional to the argument (spiral of Archimedes). CHAPTER III THE STRAIGHT LINE 1. Collinearity of three points, concurrency of three straight lines We have already used the formula Z = 1 + j u for the straight line, passing through the point 1 in the direction of the vector j. If we wish to represent the line passing through the pojpt Zl in the direction of the vector Z2' we can generalize the above formula to Z = Zl + Z2 u.

If 'U stands for the time Izi is the absolute value of the kinematic velocity. Special attention should be paid to the case where we choose the arc-length 8 along the curve as parameter; dz and d8 are, apart from the argument factor, identical, Izl = jdzjd81 = 1. For this choice of the parameter the curve is described with constant velocity 1, only the direction of the velocity being variable. Plotting the velocity vector from the origin for all values of 8 brings the extremities of this vector on the unit circle, the 8 scale, marked on this circle is characteristic for the curve under consideration.

136 and 144). But even if we start with the special parameter 8, we are already compelled to follow the general procedure as soon as we consider the second derivative, this being the velocity of the velocity indicatrix, on which 8 is no longer identical with the arc-length. du .... 28 II. Geometrical interpretation of analytic operations Examples: a. Circle: z = exp (ju); i Total perimeter: b. = j exp (ju); Iii = s = du = 2n r 1. o Circle evolvente: Z= (1 - ju) exp (ju); i = u exp (ju); Iii = u u Arc-length: I udu = tu 2 o c.