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Application of Real Substitute Elements in Graphical Construction Involving Imaginary Elements

Imaginary elements such as points, lines, circles, etc. are familiar to talk about in elementary analytic geometry and algebra. A line cuts a circle into imaginary points if its distance from the center is greater than the radius. Two imaginary lines are said to be drawn tangent to a circle from an inner point. Similarly, a plane may cut a sphere into an imaginary circle, etc. Imaginary elements can be replaced by real elements called "real substitutes". Two imaginary conjugate points E, F whose coordinates are ( il ± , α ) will be replaced by the real pair E* and F* ( l ± , α ). The real line a x = is called the 2 2 2 = + x m y , may be substituted by the real pair mx y ± = . The imaginary circle 0 2 2 2 = + + r y x is substituted by the real circle 2 2 2 r y x = + , etc. With the help of these real substitutes, different graphical constructions, without the help of projective geometry, may be carried out. Some problems were solved. (1) In this paper the ideas were extended to cover the treatment of pairs of conjugate imaginary lines and imaginary ellipses. The following, among others, were discussed: 1. Constructing the pair of conjugate imaginary tangents