Elliptic and hyperbolic trajectories
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Table of contents
Lemma 15
If from the two foci S, H, of any ellipsis or hyberbola, we draw to any third point V the right lines SV, HV, where one HV is equal to the principal axis of the shape (the axis in which the foci are situated, the other SV is bisected in T by the perpendicular TR that falls on it. That perpendicular TR will touch the conic section somwhere. Vice versa, if it does touch it, HV will be equal to the principal axis of the shape.
Let the perpendicular TR cut the right line HV in R and then join SR. TS and TV are equal. Therefore the right lines SR and VR and the angles TRS, TRV, will also be equal.
Point R will be in the conic section, and the perpenpendicular TR will touch the same.
Proposition 18 PROBLEM 10
From a focus and the principal axes given, describe elliptic and hyperbolic trajectories, which shall pass through given points, and touch right lines given by position.
Let:
- S be the common focus of the shapes.
- AB is the length of the principal axis of any trajectory
- P is a point through which the trajectory should pass
- TR is a right line which it should touch.
The center P has an interval AB-SP. If the orbit is an ellipsis, or AB-SP, if the orbit is an hyperbola, describe the circle
Around the centre P, with the interval AB - SP, if the orbit is an ellipsis, or AB + SP, if the orbit is an hyperbola, describe the circle HG. On the tangent TR let fall the perpendicular ST, and produce the same to V, so that TV may be equal to ST; and about V as a centre with the interval AB describe the circle FH. In this manner, whether two points P, p, are given, or two tangents TR, tr, or a point P and a tangent TR, we are to describe two circles. Let H be their common intersection, and from the foci S, H, with the given axis describe the trajectory: I say, the thing is done. For (because PH + SP in the ellipsis, and PH - SP in the hyperbola, is equal to the axis) the described trajectory will pass through the point P, and (by the preceding Lemma) will touch the right line TR. And by the same argument it will either pass through the two points P, p, or touch the two right lines TR, tr. Q.E.F.
Proposition 19 Problem 11
From a given focus, draw a parabolic passing through given points, and touch right lines given by position.
Let:
- S is the focus
- P a point
- TR a tangent of the trajectory to be drawn
With P as a centre, with the interval PS, draw the circle FG. From the focus let fall ST perpendicular on the tangent, and produce the same to V, so as TV may be equal to ST. After the same manner another circle fg is to be described, if another point p is given; or another point v is to be found, if another tangent tr is given; then draw the right line IF, which shall touch the two circles FG, fg, if two points P, p are given; or pass through the two points V, v, if two tangents TR, tr, are given: or touch the circle FG, and pass through the point V, if the point P and the tangent TR are given. On FI let fall the perpendicular SI, and bisect the same in K; and with the axis SK and principal vertex K describe a parabola: I say the thing is done. For this parabola (because SK is equal to IK, and SP to FP) will pass through the point P; and (by Cor. 3, Lem. XIV) because ST is equal to TV, and STR a right angle, it will touch the right line TR. Q.E.F.