. The principles of projective geometry applied to the straight line and conic . of lohich is the projection of agioen internal point. Let C be the point which is to be projected into the centre of the circle and c itspohir with resi)ect to the conic. Draw two pairs of conjugate lines through C. Theangles between these will overlap, since the tangents from this point to the conicare imaginary. Project these angles into right angles and the line c into the line atinfinity (Art. ^5). The projection of C is the centre of the conic obtained by thisprojection. The two pairs of conjugate lines throu
. The principles of projective geometry applied to the straight line and conic . of lohich is the projection of agioen internal point. Let C be the point which is to be projected into the centre of the circle and c itspohir with resi)ect to the conic. Draw two pairs of conjugate lines through C. Theangles between these will overlap, since the tangents from this point to the conicare imaginary. Project these angles into right angles and the line c into the line atinfinity (Art. ^5). The projection of C is the centre of the conic obtained by thisprojection. The two pairs of conjugate lines through C are projected into twopairs of conjugate diameters of the conic obtained by the projection. Since theyare at right angles this conic must be a circle. (Art. 96.) From the above it is seen that a conic may be projected into a circle and anyline, which docs not meet the conic in real points, into the line at infinity. To form the perspective of a conic so that the corresponding curve maij he a circleof which the centre is the point corresponding to a given internal Let C be the given point which is to correspond to the centre of the circle in theperspective figure. Take its polar as vanishing line iv. Let the polars of P and Qany two points on w meet w in Pi and Q^. Then CP, CPi and CQ, CQi are pairs ofconjugate lines through C. On PPi and Q(^i describe semicircles intersecting in ^ as centre of perspective and any line parallel to to as the axis s. The Conic 181 In the corresponding figure the point C, which corresponds to C, is the centreof the conic and the lines corresponding to CP, CPx and to CQ, CQi are pairs ofconjugate lines through C. They are therefore pairs of conjugate diameters and,since they are at right angles, the conic is a circle. (B) Definition. A conic is the locus of a point which moves so that the ratio ofits distance from a fixed point, termed the focus, to its distance from a fixed line,termed the directrix, is constant. This ratio is t
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
