Damian located the enemy ahead of them, shooting intersections clear as they reached them.
He started the car again and drove through a series of tunnels and intersections, a virtual underground street grid, before arriving at a large garage filled with gleaming cars.
The robed man led her into the fortress and wound his way through bright intersections, down stairs, and into a more opulent part of the building.
They crossed more intersections, descended to the level below, and stopped outside of double doors.
Several more warriors stood at intersections like gargoyles, moving only to point in the direction she needed to go.
Accompanied by two guards, she mounted her favorite bay horse and pounded through familiar roads and intersections to the southern wall., The chill of the ocean crept into its walls.
The image of the star is set updn the intersections of the lines of the central cross, and the positions of the reseau-lines are read off by estimation to - of a division on the glass scale.
These intersections determine the centres of the semicircles CC which form the ends of the respective knuckles.
Cubic equations were solved geometrically by determining the intersections of conic sections.
In these instruments the lines are ruled upon a spherical surface of speculum metal, and mark the intersections of the surface by a system of parallel and equidistant planes, o; of which the middle member passes through the centre of the sphere.
The locus of these intersections is the quadratrix.
Archytas of Tarentum (c. 430 B.C.) solved the problems by means of sections of a half cylinder; according to Eutocius, Menaechmus solved them by means of the intersections of conic sections; and Eudoxus also gave a solution.
This result is modified if the action of the load near the section is distributed to the bracing intersections by rail and cross girders.
But if the load is distributed to the bracing intersections by rail and cross girders, then the shear at C' will be greatest when the load extends to N, and will have the values wXADN and -wXNEB.
The first pair of intersections may be either real or imaginary; we proceed to discuss the second pair.
A further deduction from the principle of continuity follows by considering the intersections of concentric circles.
- The " centres of similitude " of two circles may be defined as the intersections of the common tangents to the two circles, the direct common tangents giving rise to the " external centre," the transverse tangents to the " internal centre."
Then circles having the intersections of tangents to this circle and the line of centres for centres, and the lengths of the tangents as radii, are members of the coaxal system.
The most brilliant are situated at the intersections of the inner halo and the parhelic circle; these are known as parhelia (denoted by the letter p in the figures) (from the Gr.
In this way he established the famous theorem that the intersections of the three pairs of opposite sides of a hexagon inscribed in a conic are collinear.
Corners of signs and intersections of lines were first fixed by minute tube-drill holes, into which the hand tool butted, so that it should not slip over the outer surface.
Descartes used the curve to solve sextic equations by determining its intersections with a circle; mechanical constructions were given by Descartes (Geometry, lib.
John Wallis utilized the intersections of this curve with a right line to solve cubic equations, and Edmund Halley solved sextic equations with the aid of a circle.
36 these various lines and planes are represented by their intersections with a uiiit sphere having 0 as centre.
Now suppose that a body receives first a positive rotation a about OA, and secondly a positive rotation e3 about OB; and let A, B be the intersections of these axes with a sphere described about 0 as centre.
80 by means of the intersections with a concentric spherical surface.
If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their " perpendicular height of incidence," i.e.
Ancient, but now extinct, volcanic upheavals are pretty common at the intersections of the main range with the transverse ranges; of these the most noteworthy are Elbruz and Kasbek.
Cramer, that, when a certain number of the intersections of two algebraical curves are given, the rest are thereby determined.
Gergonne had shown that when a number of the intersections of two curves of the (p+q)th degree lie on a curve of the pth degree the rest lie on a curve of the qth degree.
High, consisting of a six-sided lantern and royal crown, both sculptured, and resting on the intersections of two arched ornamental slips rising from the four corners of the top of the tower.
The intersections of two curves are obtained by combining their equations; viz.
The elimination from the two equations of y (or x) gives for x (or y) an equation of a certain order, say the resultant equation; and then to each value of x (or y) satisfying this equation there corresponds in general a single value of y (or x), and consequently a single point of intersection; the number of intersections is thus equal to the order of the resultant equation in x (or y).
The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry - and it in fact plays an all-important part in modern geometry.
The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m= 2, if the hyperbola xy - 1= o be cut by the line y=0, the resultant equation in x is Ox- 1 = o, and there is apparently only the intersection (x 110, y =0); but the theorem is, in fact, true for every line whatever: a curve of the order in meets every line whatever in precisely m points.
We have, in the case just referred to, to take account of a point at infinity on the line y=0; the two intersections are the point (x=110, y=0), and the point at infinity on the line y= 0.
Intersections of Circles.
The construction in fact is, join the two points in which the third circle meets the first arc, and join also the two points in which the third circle meets the second arc, and from the point of intersection of the two joining lines, let fall a perpendicular on the line joining the centre of the two circles; this perpendicular (considered as an indefinite line) is what Gaultier terms the " radical axis of the two circles "; it is a line determined by a real construction and itself always real; and by what precedes it is the line joining two (real or imaginary, as the case may be) intersections of the given circles.
The intersections which lie on the radical axis are two out of the four intersections of the two circles.
The question as to the remaining two intersections did not present itself to Gaultier, but it is answered in Jean Victor Poncelet's Traite des propeietes projectives (1822), where we find (p. 49) the statement, "deux circles places arbitrairement sur un plan ...
We find in it explicitly the two correlative definitions: " a plane curve is said to be of the p ith degree (order) when it has with a line m real or ideal intersections," and " a plane curve is said to be of the mth class when from any point of its plane there can be drawn to it m real or ideal tangents."
The points of contact are found as the intersections of the curve u= o by a curve depending on the position of the arbitrary point, and called the " first polar " of this point; the order of the first polar is = m - r, and the number of intersections is thus =m(m - I).
But it can be shown, analytically or geometrically, that if the given curve has a node, the first polar passes through this node, which therefore counts as two intersections, and that if the curve has a cusp, the first polar passes through the cusp, touching the curve there, and hence the cusp counts as three intersections.
But, as is evident, the node or cusp is not a point of contact of a proper tangent from the arbitrary point; we have, therefore, for a node a diminution and for a cusp a diminution 3, in the number of the intersections; and thus, for a curve with 6 nodes and K cusps, there is a diminution 26+3K, and the value of n is n= m (m - I)-26-3K.
Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).
But if the given curve has a node, then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve; and the node thus counts as six intersections; so if the curve has a cusp, then the Hessian not only passes through the cusp, but it has there a cusp through which it again passes, that is, there is a cuspidal branch touching the cuspidal branch of the curve, and besides a simple branch passing through the cusp, and hence the cusp counts as eight intersections.
The node or cusp is not an inflection, and we have thus for a node a diminution 6, and for a cusp a diminution 8, in the number of the intersections; hence for a curve with 6 nodes and cusps, the diminution is = 66+8K, and the number of inflections is c= 3m(m - 2) - 66 - 8K.
Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve II = o, which has not as yet been geometrically defined, but which is found analytically to be of the order (m-2) (m 2 -9); the number of intersections is thus = m(rn - 2) (m 2 - 9); but if the given curve has a node then there is a diminution =4(m2 - m-6), and if it has a cusp then there is a diminution =6(m2 - m-6), where, however, it is to be noticed that the factor (m2 - m-6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity.
Xxviii., 1844); in the latter of these the points of inflection are obtained as the intersections of the curve u = o with the Hessian, or curve A = o, where A is the determinant formed with the second derived functions of u.
An investigation by means of the curve II = o, which by its intersections with the given curve determines the points of contact of the double tangents, is indicated by Cayley, " Recherches sur l'elimination et la theorie des courbes " (Crelle, t.
A quartic curve has 28 double tangents, their points of contact determined as the intersections of the curve by a curve II = o of the order 14, the equation of which in a very elegant form was first obtained by Hesse (1849).
To develop the theory, consider the curve corresponding to any particular value of the parameter; this has with the consecutive curve (or curve belonging to the consecutive value of the parameter) a certain number of intersections and of common tangents, which may be considered as the tangents at the intersections; and the so-called envelope is the curve which is at the same time generated by the points of intersection and enveloped by the common tangents; we have thus a dual generation.
Considering the variable curve corresponding to a given value of the parameter, or say simply the variable curve, the consecutive curve has then also 6 and nodes and cusps, consecutive to those of the variable curve; and it is easy to see that among the intersections of the two curves we have the nodes each counting twice, and the cusps each counting three times; the number of the remaining intersections is = m 2 - 263 K.
For real figures we have the general theorem that imaginary intersections, &c., present themselves in conjugate pairs; hence, in particular, that a curve of an even order is met by a line in an even number (which may be = o) of points; a curve of an odd order in an odd number of points, hence in one point at least; it will be seen further on that the theorem may be generalized in a remarkable manner.
Again, when there is in question only one pair of points or lines, these, if coincident, must be real; thus, b line meets a cubic curve in three points, one of them real, and other two real or imaginary; but if two of the intersections coincide they must be real, and we have a line cutting a cubic in one real point and touching it in another real point.
It may be remarked that this is a limit separating the two cases where the intersections are all real, and where they are one real, two imaginary.
First, if the three intersections by the line infinity are all distinct, we have the hyperbolas; if the points are real, the redundant hyperbolas, with three hyperbolic branches; but if only one of them is real, the defective hyperbolas, with one hyperbolic branch.
Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the parabolic hyperbolas; or the twofold point may be a singular point, - viz., a crunode giving the hyperbolisms of the hyperbola; an acnode, giving the hyperbolisms of the ellipse; or a cusp, giving the hyperbolisms of the parabola.
Thirdly, the three intersections by the line infinity may be coincident and real; or say we have a threefold point: this may be an inflection, a crunode or a cusp, that is, the line infinity may be a tangent at an inflection, and we have the divergent parabolas; a tangent at a crunode to one branch, and we have the trident curve; or lastly, a tangent at a cusp, and we have the cubical parabola.
There are in some cases points termed centres, or singular or multiple foci (the nomenclature is unsettled), which are the intersections of improper tangents from the two circular points respectively; thus, in the circular cubic, the tangents to the curve at the two circular points respectively (or two imaginary asymptotes of the curve) meet in a centre.
To a given point (x', y', z') not on the curve u' =0 there corresponds, not a single point, but the system of points (x, y, z) given by the equations x': y': z' = X: Y: Z, viz., regarding x', y', z' as constants (and to fix the ideas, assuming that the curves X = o, Y =0, Z =o, have no common intersections), these are the points of intersection of the curves X: Y: Z, =x': y' : z', but no one of these points is situate on the curve u = o.
To explain this, observe that starting from the equations of x': y' : z'=X: Y: Z, to a given point (x, y, z) there corresponds one point (x', y', z), but that if n be the order of the functions X, Y, Z, then to a given point x', y', z' there would, if the curves X = o, Y =o, Z = o had no common intersections, correspond n' points (x, y, z).
If, however, the functions are such that the curves X =o, Y = o, Z =o have k common intersections, then among the n 2 points are included these le points, which are fixed points independent of the point (x', y', z'); so that, disregarding these fixed points, the number of points (x, y, z) corresponding to the given point (x', y', z') is =' 2 -k; and in particular if k = n 2 -I, then we have one corresponding point; and hence the original system of equations x': y' : z' =X: Y: Z must lead to the equivalent system x: y : z = X': Y': Z'; and in this system by the like reasoning the functions must be such that the curves X' =o, Y' =o, Z' =o have n' 2 -i common intersections.