An Elementary Course in Synthetic Projective Geometry

An Elementary Course in Synthetic Projective Geometry

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Excerpt: 63 110. Classification of conics. Conics are classified according to their relation to the infinitely distant line. If a conic has two points in common with the line at infinity, it is called a hyperbola; if it has no point in common with the infinitely distant line, it is called an ellipse; if it is tangent to the line at infinity, it is called a parabola . 111. In a hyperbola the center is outside the curve ( 101), since the two tangents to the curve at the points where it meets the line at infinity determine by their intersection the center. As previously noted, these two tangents are called the asymptotes of the curve. The ellipse and the parabola have no asymptotes. 112. The center of the parabola is at infinity, and therefore all its diameters are parallel, for the pole of a tangent line is the point of contact. The locus of the middle points of a series of parallel chords in a parabola is a diameter, and the direction of the line of centers is the same for all series of parallel chords. The center of an ellipse is within the curve. Fig. 28 113. Theorems concerning asymptotes. We derived as a consequence of the theorem of Brianchon ( 89) the proposition that if a triangle be circumscribed about a conic, the lines joining the vertices to the points of contact of the opposite sides all meet in a point. Take, now, for two of the tangents the asymptotes of a hyperbola, and let any third tangent cut them in A and B (Fig. 28). If, then, O is the intersection of the asymptotes, -and therefore the center of the curve, - pg 64 then the triangle OAB is circumscribed about the curve. By the theorem just quoted, the line through A parallel to OB, the line through B parallel to OA, and the line OP through the point of contact of the tangent AB all meet in a point C . But OACB is a parallelogram, and PA = PB . Therefore The asymptotes cut off on each tangent a segment which is bisected by the point of contact. 114. If we draw a line OQ more

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  • Paperback | 40 pages
  • 189 x 246 x 2mm | 91g
  • United States
  • English
  • Illustrations, black and white
  • 1236719476
  • 9781236719478