Representation of a Circle. 421 



presentation of the line in the drawing referred to co-ordinate 

 axes, Oy coinciding with the intersection of the picture and 



the plane xy 

 O z coinciding with the intersection of the picture and 

 the plane x z 

 is ^{ay — bA) = u{az — c A); and the coordinates of the 



vanishing point of this line are and — ; it is easy to show 



that the vanishing points of all lines which are situated in the 

 same plane are in the same straight line, which line is called 

 the vanishing line of the plane. 



It is evident from the pre- 

 ceding theorems : that if V 

 and V' (fig. 1.) are the va- 

 nishing points of any two 

 lines, and if OCK be drawn 

 perpendicular to V V cut- 

 ting it in C, and if CK be 

 taken so that CK-= 00^ + 

 A", the angle VKV is equal 

 to the angle contained by 

 these lines. This theorem is 

 the foundation of the whole 

 practice of perspective. 



Suppose it were required to draw a circle ; having drawn 

 any diameter AB and V V (fig. 2. ) 

 the vanishing line of the plane in 

 which the circle is situated by 

 making the angle VKV con- 

 stantly equal to a right angle, and 

 joining AV, B V, as many points 

 as required may be found by the 

 intersection P of AV and BV ; 

 and this is also perhaps the sim- 

 plest method of describing the 

 curves of the second order. If 

 the chord AB is given of a seg- 

 ment including any given angle, 

 the same construction obtains, 

 making VKV = to this given 

 angle. If the circle which is to 

 be represented be wholly with- 

 out a plane passing through the 

 eye ol the spectator and parallel 

 to the picture, the representation 

 will be an ellipse, if it only meet 

 this plane it will be a parabola ; if it cut tlic plane il will be 



a I type r- 



Fiff. 2 



