Directrix Sentence Examples
The surface formed by revolving the catenary about its directrix is named the alysseide.
The cartesian equation referred to the axis and directrix is y=c cosh (x/c) or y = Zc(e x / c +e x / c); other forms are s = c sinh (x/c) and y 2 =c 2 -1-s 2, being the arc measured from the vertex; the intrinsic equation is s = c tan The radius of curvature and normal are each equal to c sec t '.
In one solution of the former problem is the first recorded use of the property of a conic (a hyperbola) with reference to the focus and directrix.
The line FL perpendicular to the axis, G D and passing through the focus, is the semilatus rectum, the latus rectum being the focal chord parallel to the directrix.
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.
The pedal equation with the focus as origin is p 2 =ar; the first positive pedal for the vertex is the cissoid and for the focus the directrix.
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.
The only surface of revolution having this property is the catenoid formed by the revolution of a catenary about its directrix.Advertisement
This catenoid, however, is in stable equilibrium only when the portion considered is such that the tangents to the catenary at its extremities intersect before they reach the directrix.
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.
The radius of curvature of a catenary is equal and opposite to the portion of the normal intercepted by the directrix of the catenary.Advertisement
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.
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 T, the point of intersection of Pp and Qq, must be the centre of similitude and must be on the common directrix.
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.Advertisement
The condition of stability of a catenoid is therefore that the tangents at the extremities of its generating catenary must intersect before they reach the directrix.
Take any point R on the directrix, and draw the lines RAM, RSN; draw SL so that the angle LSN =angle NSA'.
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.Advertisement
The focus of the parabola was discovered by Pappus, who also introduced the notion of the directrix.
To construct the parabola when the focus and directrix are given, draw the axis CD and bisect CF at G, which gives the vertex.
A fundamental property of the curve is that the line at infinity is a tangent (see Geometry, Projective), and it follows that the centre and the second real focus and directrix are at infinity.
The orthocentre of a triangle circumscribing a parabola is on the directrix; a deduction from this theorem is that the centre of the circumcircle of a self-conjugate triangle is on the directrix ("Steiner's Theorem").