MOORE. — MINIMUM GEOMETRY. 217 



Conversely all pseudo circles having the same pair of tangents from F 

 are parallel curves. The form of the equation shows that the parallel 

 is the same whether a is positive or negative. Hence the lines at a 

 distance =±= a from the points of the given curve are parallel tangents 

 to the same parallel curve. 



All the lines which cut a given tangent to the curve /(a-, y) = O, at a 

 constant angle will pass through the same point. The locus of this 

 point will be the dual of the parallel to a given curve. If the curve 

 is given in angle coordinates, 



f(u, v)=0 



then the angle equation of this locus will have the same degree as the 

 reciprocal of the given curve. The curve which corresponds to a 

 pseudo circle is again a pseudo circle. We again have the same curve 

 whether we take the fixed angle as ± a. 



It was shown in A. P. G. that if the line / is the locus of points at a 

 distance k from P then the lines through P all make with / angles equal 



to t. An easy calculation will show that the parallel to a pseudo 



circle corresponding to the distance k and the dual corresponding to 



the angle 7 are one and the same curve. If we wish then to determine 



the point of contact of a given generator / of a parallel curve with its 

 envelope all we need to do is to draw a line through the corresponding 



point of the original curve making an angle — t with the tangent line. 



Where this line cuts I will be the point of contact. 



Evolutes. In ordinary geometry the envelope of the normals of a 

 curve is of considerable interest. Aside from the tangent the normal 

 is the only unique line connected with a curve. Here however each 

 line passing through a point of a curve is unique in the same sense, 

 that is it makes a definite angle with the tangent line and there is no 

 other line passing through this point making the same angle. Then 

 connected with every curve there is an infinite number of involutes, 

 that is, the envelope of lines making a fixed angle with the tangent 

 lines. If the original curve is, 



/(*, y) = 0, 

 the line making the angle k with the curve at the point (x h yi) is 



