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Pole and Polar

conic, lines, conjugate, respect and triangle

POLE AND POLAR. The secant drawn through the points of contact of two intersecting tangents to a conic is called the polar of the point of intersection with respect to the conic, and the point is called the pole of the secant. Any secant of a conic through a given point 0 is c•ut harmonic-ally by the curve and the polar of 0. Two points are said to be conjugate with re spect to a conic when each lies on the polar of the other. Two straight lines are said to be conjugate with respect to a conic when each passes through the pole of the other. Thus conjugate diameters are conjugate lines through the centre. When the pole is inside the conic, the tangents are imaginary, and the Polar line fails to cut the conic in real points. In this ease the locus of the harmonic conjugate of the pole serves as a more suggestive definition of the polar. If the pole is on the conic. the polar becomes a tangent at the pole. The polar of the focus is the direetrix in the ellipse, hyperbola, and parabola (qq.v.). A few of the relations which give remarkable power to the theory • f polars in the domain of geometry are: (1) The polars of collinear points with respect to a conic are a pencil of lines passing through the pole of the line, and conversely. (•l) if the vertices of a triangle arc the poles of the sides of another triangle, the vertices of the latter are the poles of the sides of the former. Such triangles are said to he conjugate to each other with respect to the conic. (3) If the sides of a triangle are the polars of its own vertices, the triangle is called a self-conjugate triangle. (4) The poles and polar.: of the lines and points of rectilinear plane figures with respect to a co-planar conic fern] a recti linear figure (-ailed the polar reciprocal of the given figure with respect to the auxiliary comic.

The method of reciprocal polar•s obtains from any given theorem concerning the positions of points and lines another theorem in which straight lines take the place of points, and points of straight lines. (See Dirm..err.) Thus a line joining two points in one figure corresponds to a point determined by two intersecting lines in the reciprocal figure. Sine(' pole of any line through the centre of the auxiliary conic is ;It infinity, the points at infinity on the reciprocal curve correspond to the tangents to the original curve from the centre of the auxiliary conic.

Hence the reciprocal of a conic is an hyperbola, parabola. or ellipse, according as the tangents to it through the centre of the auxiliary conic arc real. coincident, or imaginary. Pascal's and Brianchon's theorems are reciprocals. (See CON CURRENCE and COLLINEARITY. ) Conjugate lines or conjugate points project into conjugate lines or points (see PROJECTION ) . hence the relations of pole and polar are unaltered by projection.

The relations between pole and polar were known to the ancients, but Desargues (1639) was the first to develop the theory. To Servois (1810) is due the name pole (in this sense) and to (ler gonne (I812) the mune polar. Steiner (1848) also treated the subject exhaustively. Hesse (1837, 1842) introduced the notion of polar tri angles. polar tetrahedra. and systems of conjugate points as the geometric expressions of analytic relations.

The terms 'pole' and `polar' have other meanings in mathematics than those already mentiomsl. The centre from which radii veetores are drawn in a system of polar et:lilt-din:10.s (see