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Cor. Join Aa; then, by Euc, 1. 47, and the Prop., Aal = AC + Ca* = CF; hence the distance of either vertex of the conjugate diameter from either vertex of the transverse is equal to the focal distance of the centre.
PROPOSITION IV. If in the hyperbola a fourth proportional be taken to the semi-trans
verse diameter, the eccentricity, and the distance of an ordinate to any point in the curve (all estimated from the centre along the transverse axis), then the parts of the transverse axis intercepted between the extremity of the fourth proportional and the extremities of the transverse are respectively equal to the focal distances of that point.
(See the Fig. of last Prop.) Let E be any point in the hyperbola, F and f the foci, and CP a fourth proportional to CB, Cf, and CD; then the segments AP, BP, of the transverse axis, intercepted between P and the extremities A and B of the transverse diameter, will be respectively equal to EF, Ef, the focal distances of the point E.
For by construction, CB : Cf? :: CD2 : CPTM; hence since (Prop. II.) Cf: - CBP = Ca', CBP : Ca :: CDP : CP - CD. But (Prop. 11.),
CB : Ca : : AD. DB : DE2
or CBP : Ca' :: CD – BC : DE. Wherefore, CD : CDTM - BCR :: CPP - CD : DE'.
Proceeding with this proportion as in the analogous case of the ellipse (Prop. Iv.) we get
EF = AP, and Ef = BP. Cor. The difference of the lines drawn from the foci to any point in the hyperbola is equal to the transverse diameter.
PROPOSITION V. The distance of the directrix from any point in the hyperbola has to
the distance of the focus from the same point a constant ratio.
Let NL be the directrix with respect to the focus F, and N'L' that with respect to the other focus f; then E being any point in the curve, the ratio of FE to EL, or of fE to EL' is equal to the constant ratio of CF to CA (C being the centre).
For take in the transverse axis AB, CF : CA :: CA :CN, and draw NL perpendicular to the axis ; then NL is the directrix with respect to the focus F (Def. 15). Also, as in the last Prop., take CP a fourth proportional to CA, CF, and CD. Then we have
CF:CA ::CA: CN, and CF :CA :: CP: CD. llence
CA: CN :: CP: CD, or CA:CP :: CN: CD. Wherefore, by division,
AP: CA :: DN:CN; or since AP = EF (Prop. iv.),
and DN = EL, EF : EL :: CA :CN : : ČF: CA. And if N'L' be the directrix with respect to the other focus f, it may be shown in a similar way, that
Ef: EL' :: CF: CA.
PROPOSITION VI. The straight line which bisects the angle adjacent to that which is contained by two straight lines drawn from any point in the hyperbola to the foci, is a tangent to the curve at that point.
The demonstration of this, which is exactly similar to the analogous case for the ellipse (Prop. vi.), is left for the studeut's exercise.
THE ASYMPTOTES. Def. 16. If through A, one of the vertices of the transverse diameter of the hyperbola, a straight line KAN be drawn, equal and parallel to a b the conjugate diameter, and bisected in A by the transverse diameter ; the straight lines CK, CN drawn through the centre, and the extremities of that parallel, are called asymptotes to the hyperbola. The asymptotes also of two opposite hyperbolas are common to both.
There are many remarkable properties connected with the asymptotes to the hyperbola ; but the limits of this course do not admit of our entering upon them. It may be stated, however, that the hyperbola and its asymptotes, when produced, continually approach to each other, but meet only at an infinite distance. (See Vol. I., p. 306).
CONJUGATE HYPERBOLAS. Dec. 17. If ab be taken for the transverse diameter, and AB for its conjugate, of other two hyperbolas in the spaces KCL, MCN (a and b being the vertices of those diameters); then those two other hyperbolas are said to be conjugate to the former. When all four are mentioned, they are called conjugate hyperbolas.
Additional Problems and Theorems for Exercises. 1. Prove that if any two diameters of a parabola be produced to meet a tangent to the curve; the segments of the diameters between their vertices and the tangent are to one another as the squares of the segments of the taugent intercepted between each diameter and the point of contact.
2. A parabola is given in position, to find its directrix and focus.
3. If any chord be drawn parallel to the tangent of a parabola, to meet the curve in two points; and if from its extremities and from the point of contact ordinates be drawn to the axis, then double the ordinate of the point of contact will be equal to the sum or difference of the other two, according as they are situated on the same or different sides of the axis.
4. If from the point of contact C of a tangent to a parabola, any chord CL be drawn, and another line parallel to the axis to meet the chord, curve, and tangent, in K, E, I ; then IE: EK :: CK: KL.
5. If from the point of contact of a tangent to a parabola, two lines be drawn to the vertices of any two diameters, each to intersect the other diameter; then the line joining the two points of intersection will be parallel to the tangent.
6. If the chord of contact of two tangents to a parabola pass through the focus, the tangents will intersect at right angles to one another in the directrix.
7. Let TP, tp, be tangents to a parabola at the points P and p, and PM, pm, ordinates on the axis from the same points ; then
(1) tan TPM : tan tpm :: PM : pm;
(2) tan TPM – tan tpm = 2 tan p PM. 8. Show how to draw a tangent and normal to a given parabola from a given point without the curve.
9. Prove that any chord of an ellipse drawn through the focus is so divided in that point that four times the rectangle contained by its segments is equal to the rectangle contained by the chord and the parameter of the transverse diameter.
10. Let a tangent to an ellipse intercept the four perpendiculars on the axis, from the centre, the two extremities of the axis, and the point of contact; then will those intercepted perpendiculars be proportionals.
11. Prove that if a right cylinder be cut by a plane obliquely the section will be an ellipse.
12. Show how to draw a tangent to a given ellipse from a given point without the curve.
13. Prove that chords drawn through the foci of an ellipse or hyperbola are proportional to the squares of their parallel diameters.
14. Prove that the segment of the normal, intercepted by the principal diameter of the ellipse or hyperbola, is equal to the distance of the focus from the centre.
DEFINITIONS AND FIRST PRINCIPLES. 1. (a) If lines be drawn from every point of an object on the opposite side of a pane of glass to the eye (representing the rays of light passing from the points of the object), and if the points in which these lines cut the pane be marked upon it, we have the piclure or projection of the object. This projection is the perspective of the object.
(6) If any opaque figure be interposed between a light and a screen, the shadow of the figure upon the screen is the projection of the figure. This is the radial projection of the object.
(c) Although there is a physical difference in the circumstances of these two projections, yet geometrically, the difference is so slight and subordinate as to not justify a separate general discussion or explanation of the two cases. They are therefore usually considered as one.
(d) When the projeeting point (the eye in the first case and the luminous body in the second) is brought nearer to the body, the projection is enlarged; and when taken more distant, the projection is diminished. For as the point approaches the body, the bounding rays become angularly more and more expanded; and as the point is removed from the body, they approach nearer and nearer to parallelism. Each case, however, is subject to limitations which will be pointed out hereafter. The general fact is sufficient for our present purpose.
(e) The general idea of projection as a branch of geometry has been obtained from the considerations above, by ideally generalizing the objects—by abstracting all considerations of the materiality of light, the figure which by interrupting the light causes the shadow, and of the surface upon which the shadow is cast; and thus retaining only the purely geometrical idea of direction in the rays, of the defining conditions of the intercepting or projected figure, and likewise those of the surface upon which it is projected.
2. The only two cases in which the idea thus generalised is applicable directly and immediately, as far as regards the positions of the rays, are:
(a) That they shall be parallel to a given line; .
3. The figure to be projected may either be:-a point, a straight line, a curve line, a definitely bounded plane figure, or a curve surface. This is called simply the figure.
4. The surface upon which the shadow is hypothetically cast, is called the figure (plane, sphere, cone, etc.) of projection. Most frequently and for the most useful applications of the method of projection, this is a plane, and called the plane of projection. Sometimes, however, projections are made upon spherical, conical, or cylindrical surfaces.
5. The hypothetical shadow itself is called the projection of the figure; and the lines which are drawn from each point of the figure to the corresponding one of its projection, are called the projectors.
6. When the projectors emanate from, or converge towards the same point, the method is called polar projection, and comprehends both the cases supposed in (la and 16); when they are parallel, it is called parallel projection ; and when they are perpendicular to the plane of projection, it constitutes the orthographic projection.*
7. The point in which any given line cuts the plane of projection is called the trace of that line; and the line in which any given plane cuts the plane of projection is called the trace of that plane. The same term is also applied to the intersections of any given curves or surfaces with the plane of projection.
It is manifest that the trace of every line in a given plane upon the plane of projection is situated in the trace of that plane upon the plane of projection.
Moreover, when the traces of two lines which are known to be situated in a plane are given, the trace of that plane is given; and it is the indefinite line drawn in the plane of projection through these two traces.
8. Any geometical figure is said to be given when it is fixed by adequate conditions, by conditions which do not allow it to undergo any change without a previous change in one or more of those conditions.
9. By the term inclination, applied to lines or planes, is meant the ordinary geometrical idea attached to them by writers on planes and solids : but it is to be kept in view, that we shall here always consider the inclination to be from one specified side of the point about which a line may revolve or one specified side of the line about which the plane revolves, to form its inclination with the plane of projection. Thus:
If, in Fig. 1, D'AD be the line from which the inclination of AB to the plane MN is to be estimated, it is to be predetermined whether it
This, and the case of the polar projection, which relates to perspective, are the only ones that were studied in this country till a very recent period.