Focus by two tangents drawn from a point), and (having given the focus and a double ordinate) he uses the focus and directrix to obtain any number of points on a parabola - the first instance on record of the practical use of the directrix.
The surface formed by revolving the catenary about its directrix is named the alysseide.
1, where P is a point on the curve equidistant from the fixed line AB, known as the directrix, and the fixed point F known as the focus.
The line CD passing through the focus and perpendicular to the directrix is the axis or principal diameter, and meets the curve in the vertex G.
Any number of points on the parabola are obtained by taking any point E on the directrix, joining EG and EF and drawing FP so that the angles PFE and DFE are equal.
Then if a pencil be placed along B C so as to keep the string taut, and the limb AB be slid along the directrix, the A pencil will trace out the parabola.
Again, if a chain pass over a perfectly smooth peg, the catenaries in which it hangs on the two sides, though usually of different parameters, wifi have the same directrix, since by (10) y is the same for both at the peg.
The tangents at the ends meet on the directrix, and their inclination to the horizontal is 56 30.
Since the tension is measured by the height above the directrix these two catenaries have the same directrix.
Every catenary lying between them has its directrix higher, and every catenary lying beyond them has its directrix lower than that of the two catenaries.
Now let us consider the surfaces of revolution formed by this system of catenaries revolving about the directrix of the two catenaries of equal tension.
Hence a catenoid whose directrix coincides with the axis of revolution has at every point its principal radii of curvature equal and opposite, so that the mean curvature of the surface is zero.
14) be two catenaries having the same directrix and intersecting in A and B.
Draw Pp and Qq touching both catenaries, Pp and Qq will intersect at T, a point in the directrix; for since any catenary with its directrix is a similar figure to any other catenary with its directrix, if the directrix of the one coincides with that of the other the centre of similitude must lie on the common directrix.
Also, since the curves at P and p are equally inclined to the directrix, P and p are corresponding, points and the line P p must pass through the centre of similitude.
Hence the tangents at A and B to the upper catenary must intersect above the directrix, and the tangents at A and B to the lower catenary must intersect below the directrix.
A conic section (or as we now say a " conic ") is the locus of a point such that its distance from a given point, the focus, is in a given ratio to its (perpendicular) distance from a given line, the directrix; or it is the locus of a point which moves so as always to satisfy the foregoing condition.
Eccentricity less than unity: this involves the notion of one directrix and one focus; (2) the ellipse is the locus of a point the sum of whose distances from two fixed points is constant: this involves the notion of two foci.
To investigate the form of the curve use may be made of the definition: the ellipse is the locus of a point which moves so that the ratio of its distance from a fixed point (the focus) to its distance from a straight line (the directrix) is constant and is less than unity.
I) be the directrix, S the focus, and X the foot of the perpendicular from S to KX.
Line K'X' parallel to KX such that AX = A'X', then the same curve will be described if we regard K'X' and S' as the given directrix and focus, the eccentricity remaining the same.
The square on the semi-major axis equals the rectangle contained by the distances of the focus and directrix from the centre; and 2a = SP+S'P, where P is any point on the curve, i.e.
A focus or directrix is equal to two conditions; hence such problems as: given a focus and three points; a focus, two points and one tangent; and a focus, one point and two tangents are soluble (very conveniently by employing the principle of reciprocation).
Of practical importance are the following constructions: - (I) Given the axes; (2) given the major axis and the foci; (3) given the focus, eccentricity and directrix; (4) to construct an ellipse (approximately) by means of circular arcs.
2) be the focus, KX the directrix, X being the foot of the perpendicular from S to the directrix.
Take any point R on the directrix, and draw the lines RAM, RSN; draw SL so that the angle LSN =angle NSA'.
For, draw through P a line parallel to AA', intersecting the directrix in Q and the line RSN in T.
One definition, which is of especial value in the geometrical treatment of the conic sections (ellipse, parabola and hyperbola) in piano, is that a conic is the locus of a point whose distances from a fixed point (termed the focus) and a fixed line (the directrix) are in constant ratio.
In the case of the circle, the centre is the focus, and the line at infinity the directrix; we therefore see that a circle is a conic of zero eccentricity.
His proofs are generally long and clumsy; this is accounted for in some measure by the absence of symbols and technical terms. Apollonius was ignorant of the directrix of a conic, and although he incidentally discovered the focus of an ellipse and hyperbola, he does not mention the focus of a parabola.