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Encyclopedia Britannica - Main :: RON-SAC |
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S1S respectively. lines which pass through a u and alit respectively. As A is any given point on the curve, and ul any line through it, we have solved the problems: Problem.To find the second Problem.To find the second point in which any line through a tangent which can be drawn
the curve. to the curve. If we determine in SI (fig. 16) the ray corresponding to the ray S2SI in S2, we get the tangent at SI. Similarly, we can determine the point of contact of the tangents ul or u2 in fig. 17. 51. If five points are given, of which not three are in a line, then we can, as has just been shown, always draw a curve of the second order through them; we select two of the points as centres of projective pencils, and then one such curve is determined. It will be presently shown that we get always the same curve if two other points are taken as centres of pencils, that therefore five points determine one curve of the second order, and reciprocally, that five tangents determine one curve of the second class. Six points taken at random will therefore not lie on a curve of the second order. In order that this may be the case a certain condition has to be satisfied, and this condition is easily ob-tained from the construction in 49, fig. 16. If we consider the conic determined by the five points A, SI, S2, K, L, then the point D will be on the curve if, s, and only if, the points on DI, S, D2 be in a line. This may be stated differently if we take AKSIDS2L (figs. 16 and 18) as a hexagon inscribed in the conic, then AK and DS2 will be opposite sides, so will be KS' and S2L, as well as SID and LA. The first two meet in D2, the others in S and DI respectively. We may therefore state the required condition, together with the reciprocal one, as follows: Theorem.The envelope of second class which is generated by two projective rows contains the bases of these rows as enveloping lines or tangents. Proof.If s and s' are the two rows, then to the point ss' or P' as a point in s' corresponds in s a point P, which is not coincident with P', for the rows are not perspective. But P and P' are joined by s, so that s is one of the enveloping lines, and similarly s'. us PROJECTIVE] Pascal's Theorem.If a hexagon Brianchon's Theorem.If a be inscribed in a curve of the hexagon be circumscribed about second order, then the intersec- a curve of the second class, then tions of opposite sides are three the lines joining opposite vertices points La a line. are three lines meeting in a point. These celebrated theorems, which are known by the names of their discoverers, are perhaps the most fruitful in the whole theory of conics. Before we go over to their applications we have to show that we obtain the same curve if we take, instead of S1, S2, any two other points on the curve as centres of projective pencils. 52. We know that the curve depends only upon the correspondence between the pencils S1 and S2, and not upon the special
auxiliary
drawn
Only one curve of the second Only one curve of the second order can be drawn which passes class can be drawn which touches through five given points. five given lines. We have seen that if on a curve of the second order two points coincide at A, the line joining them becomes the tangent at A. If, therefore, a point on the curve and its tangent are given, this will be equivalent to having given two points on the curve. Similarly, if on the curve of second class a tangent and its point of contact are given, this will be equivalent to two given tangents. We may therefore extend the last theorem: Only one curve of the second Only one curve of the second order can be drawn, of which class can be drawn, of which four four points and the tangent at one tangents and the point of contact of them, or three points and the at one of them, or three tangents tangents at two of them, are and the points of contact at two given. of them, are given. 53 At the same time it has been proved : If all points on a curve of the All tangents to a curve of second second order be joined to any class are cut by any two of two of them, then the two pencils them in projective rows, those thus formed are projective, those being corresponding points which rays being corresponding which lie on the same tangent. Hencemeet on the curve. Hence The cross-ratio of four rays The cross-ratio of the four joining a point S on a curve of points in which any tangent u is second order to four fixed points cut by four fixed tangents a, b, c, d A, B, C, D in the curve is in- is independent of the position of dependent of the position of S, u, and is called the cross-ratio of and is called the cross-ratio of the the four tangents a, b, c, d. four points A, B, C, D. If this cross-ratio equals1 If this cross-ratio equals1 the four points are said to be the four tangents are said to be four harmonic points. four harmonic tangents. We have seen that a curve of second order, as generated by projective pencils, has at the centre of each pencil one tangent; and further, that any point on the curve may be taken as centre of such pencil. Hence A curve of second order has at A curve of second class has on every point one tangent. every tangent a point of contact. 54. We return to Pascal's and Brianchon's theorems and their applications, and shall, as before, state the results both for curves of the second order and curves of the second class, but prove them only for the former. Pascal's theorem may be used when five points are given to find more points on the curve, viz. it enables us to find the point where any line through one of the given points cuts the curve again. It is convenient, in making use of Pascal's theorem, to number the points, to indicate the order in which they are to be taken in forming a hexagon, which, by the way, may be done in 6o different ways. It will be seen that 12 (leaving out 3) 4 5 are opposite sides, so are 2 3 and (leaving out 4) 5 6, and also 3 4 and (leaving out 5) 6 1. If the points 12 3 4 5 are given, and we want a 6th point on a line drawn through 1, we know all the sides of the hexagon with the exception of 5 6, and this is found by Pascal's theorem. If this line should happen to pass through r, then 6 and 1 coincide, or the line 6 is the tangent at i. And always if two consecutive vertices of the hexagon approach nearer and nearer, then the side joining them will ultimately become a tangent. We may therefore consider a pentagon inscribed in a curve of second order and the tangent at one of its vertices as a hexagon, and thus get the theorem: 697 Every pentagon circumscribed about a curve of the second class has the property that the lines which join two pairs of non-consecutive vertices meet on that line which joins the fifth vertex to the point of contact of the opposite side. This enables us also to solve the following problems. Given five points on a curve of Given five tangents to a curve second order to construct the of second class to construct the tangent at any one of them. point of contact of any one of them. If two pairs of adjacent vertices coincide, the hexagon becomes a quadrilateral, with tangents at two vertices. These we take to be opposite, and get the following theorems: If a quadrilateral be circumscribed about a curve of second class, the lines joining opposite vertices, and also the lines joining points of contact of opposite sides, meet in a point. If we consider the hexagon made up of a triangle and the tangents at its vertices, we get If a triangle is inscribed in a If a triangle be circumscribed curve of the second order, the about a curve of second class, points in which the sides are cut the lines which join the vertices by the tangents at the opposite to the points of contact of the vertices meet in a point. opposite sides meet in a point (fig. 20). 55. Of these theorems, those about the quadrilateral give rise to a number of others. Four points A, B, C, D may in three different ways be formed into a quadrilateral, for we may take them in the order ABCD, or ACBD, or ACDB, so that either of the points B, C, D may be taken as the vertex opposite to A. Accordingly we may apply the theorem in three different ways. Let A, B, C, D be four points on a curve of second order (fig. 21), and let us take them as forming a quadrilateral by taking the points in the order ABCD, so that A, C and also B, D are pairs of opposite vertices. Then P, Q will be the points where opposite sides meet, Every pentagon inscribed in a curve of second order has the property that the intersections of two pairs of non-consecutive sides lie in a line with the point where the fifth side cuts the tan-gent at the opposite vertex. If a quadrilateral be inscribed in a curve of second order, the intersections of opposite sides, and also the intersections of the tangents at opposite vertices, lie in a line (fig. 19). and E, F the intersections of tangents at opposite vertices. The four points P, Q, E, F lie therefore in a line. The quadrilateral ACBD gives us in the same way the four points Q, R, G, H in a line, and the quadrilateral ABDC a line containing the four points R, P, I, K. These three lines form a triangle PQR. The relation between the points and lines in this figure may he expressed more clearly if we consider ABCD as a four-point inscribed in a conic, and the tangents at these points as a four-side circumscribed about it,viz. it will be seen that P, Q, R are the diagonal points of the four-point ABCD, whilst the sides of the triangle PQR are the diagonals of the circumscribing four-side. Hence the theoremAny four-point on a curve of the second order and the four-side formed by the tangents at these points stand in this relation that the diagonal points of the four-point lie in the diagonals of the four-side. And conversely,If a four-point and a circumscribed four-side stand in the above relation, then a curve of the second order may be described which passes through the four points and touches there the four sides of these figures. That the last part of the theorem is true follows from the fact that the four points A, B, C, D and the line a, as tangent at A, deter- mine a curve of the second order, and the tangents to this curve at the other points B, C, D are given by the construction which leads to fig. 21. The theorem reciprocal to the last is Any four-side circumscribed about a curve of second class and the four-point formed by the points of contact stand in this relation that the diagonals of the four-side pass through the diagonal points of the four-point. And conversely, If a four-side and an inscribed four-point stand in the above relation, then a curve of the second class may be described which touches the sides of the four-side at the points of the four-point. 56. The four-point and the four-side in the two reciprocal theorems are alike. Hence if we have a four-point ABCD and a four-side abed related in the manner described, then not only mdy a curve of the second order be drawn, but also a curve of the second class, which both touch the lines a, b,, c, d at the points A, B, C, D.The curve of second order is already more than determined by the points A, B, C and the tangents a, b, c at A, B and C. The point D may therefore be any point on this curve; and d any tangent to the curve. On the other hand the curve of the second class is more than determined by the three tangents a, b, c and their points of contact A, B, C, so that d is any tangent to this curve. It follows .that every tangent to the curve of second order is a tangent of acurve of the second class having the same point of contact. In other words, the curve of second order is a curve of second class,. and vice versa. Hence the important theorems Every curve of second order is Every curve of second class is a a curve of second class, curve of second order. The curves of second order and of second class, having thus been proved to be identical, shall henceforth be called by the common name of Conics. For these curves hold, therefore, all properties which have been proved for curves of second order or of second class. We may therefore now state Pascal's and Brianchon's theorem thus Pascal's Theorem.--If a hexagon be inscribed in a conic, then the intersections of opposite sides lie in a line. Brianchon's Theorem.If a hexagon be circumscribed about a conic, then the diagonals forming opposite centres meet in a point. 57. If we suppose in fig. 21 that the point D together with the tangent d moves along the curve, whilst A, B, C and their tangents a, b, c remain fixed, then the ray DA will describe a pencil about A, the point Q a projective row on the fixed line BC, the point F the row b, and the ray EF a pencil about E. But EF passes always through Q. Hence the pencil described by AD is projective to the pencil described by EF, and therefore to the row described by F on b. At the same time the line BD describes a pencil about 13 projective to that described by AD ( 53). Therefore the pencil BD and the row F on b are projective. Hence If on a conic a point A be taken and the tangent a at this point, then the cross-ratio of the four rays which join A to any four points on the curve is equal to the cross-ratio of the points in which the tangents at these points cut the tangent at A. 58. There are theorems about cones of second order and second class in a pencil which are reciprocal to the above, according to 43. We mention only a few of the more important ones. The locus of intersections of corresponding planes in two projective axial pencils whose axes meet is a cone
The envelope of planes which join corresponding lines in two projective flat pencils, not in the same plane, is a cone
Cones of second order and cones of second class are identical. Every plane cuts a cone of the second order in a conic. A cone of second order is uniquely determined by five of its edges or by five of its tangent planes, or by four edges and the tangent plane at one of them, &c. &c. Pascal's Theorem.If a solid angle of six faces be inscribed in a cone of the second order, then the intersections of opposite faces are three lines in a plane. Brianchon's Theorem.If a solid angle of six edges be circumscribed about a cone of the second order, then the planes through opposite edges meet in a line. Each of the other theorems about conics may be stated for cones of the second order. 59. Projective Definitions of the Conics.We now consider the shape of the conics. We know that any line in the plane of the conic, and hence that the line at infinity, either has no point in common with the curve, or one (counting for two coincident points) or two distinct points. If the line at infinity has no point on the curve the latter is altogether finite, and is called an Ellipse (fig. 21). If the line at infinity has only one point in common with the conic, the latter extends to infinity, and has the line at infinity a tangent. It is called a Parabola
separate
6o. That the circle belongs to the curves of the second order is seen ''N P a slightly different form the theorem that in a circle all angles at the circumference standing
If we join a point in space to all points on a circle, we get a (circular) cone of the second order ( 43). Every plane section of this cone is a conic. This conic will be an ellipse, a parabola
6r. Any two tangents to a parabola are cut by all others in projective rows; but the line at infinity being one of the tangents, the points at infinity on the rows are corresponding points, and the rows therefore similar. Hence the theorem The tangents to a parabola cut each other proportionally. End of Article: S1S If you wish, you can link directly to this article.
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