A first course in projective geometry . Xat Fig. 56a. In the former (hyperbola), since of the two conjugate lines whichproject into the axes, one cuts the circle in points which areon opposite sides of the vanishing line, and the other does not. \at C =X rt/ CO Fig. 566. cut the circle in real points, one axis in the projected figurecuts both branches of the curve and the other cuts neither. 112 PROJECTIVE GEOMETRY In the latter case (parabola) one conjugate line is thevanishing line itself, and the other passes through its point ofcontact with the circle. The parabola, therefore, has but oner
A first course in projective geometry . Xat Fig. 56a. In the former (hyperbola), since of the two conjugate lines whichproject into the axes, one cuts the circle in points which areon opposite sides of the vanishing line, and the other does not. \at C =X rt/ CO Fig. 566. cut the circle in real points, one axis in the projected figurecuts both branches of the curve and the other cuts neither. 112 PROJECTIVE GEOMETRY In the latter case (parabola) one conjugate line is thevanishing line itself, and the other passes through its point ofcontact with the circle. The parabola, therefore, has but onereal finite axis, the other axis being at infinii;y. If the secondconjugate line cut the circle in A, AY is a tangent to the circleand projects into the tangent to the parabola at the finite endof its axis. § 14. The Symmetry of the Conic. Lengths of § 12 of this chapter, it follows that each axis is anaxis of symmetry, as shown in the figures. Central conies, therefore, possess double symmetry. Moreover, the tangents at the points where the axes cutthe curve are perpendicular to those axes. These points are called the Vertices. The lengths of the axes are the intercepts made by thecurve on them. They are both real only in the
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