# How to use *Intersections* in a sentence

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.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.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.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.AdvertisementIf 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.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.The

**intersections**of two curves are obtained by combining their equations; viz.But the resultant equation may have all or any of its roots imaginary, and it is thus not always that there are m real

**intersections**.AdvertisementThe 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.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.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 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).AdvertisementBut 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.AdvertisementThirdly, 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.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.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.AdvertisementIt 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.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.A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a conic became the

**intersections**of the tangents from the circular points at infinity; the centre of a conic the pole of the line at infinity, &c. In analytical geometry the line at infinity plays an important part in trilinear co-ordinates.The

**intersections**of the lines drawn from corresponding points are points on the ellipse.The great dodecahedron is determined by the

**intersections**of the twelve planes which intersect the Platonic icosahedron in five of its edges; or each face has the same boundaries as the basal sides of five covertical faces of the icosahedron.The

**intersections**of the diagonal streets left a number of small, triangular parks, which, as well as the larger ones, are well shaded.The only allowable spinning states are at their

**intersections**.A police escort led the cavalcade through the heart of the city, with no hold ups for lights or

**intersections**.Both organizations were set up to explore

**intersections**between artistic disciplines.This paper takes a cultural studies approach to investigate the

**intersections**of history, technology, food, and national identity.The police had mobilized a much larger force and were coming toward us from multiple

**intersections**.Widely scattered massive sulfide

**intersections**, however, demonstrate considerable exploration potential.To keep the traffic running on the highway in peak time a ramp metering system is operating on 3 highway

**intersections**.Thinking Classics Essays on classical subjects; film reviews course on Greek scansion and metrics; commentary on

**intersections**of antiquity and modernity.At wall

**intersections**, you'll have to cut the paper, match the design and continue to the adjacent wall.Install metal flashing in roof valleys and

**intersections**for extra protection, and secure it in place with galvanized nails.People who live in apartments or near busy

**intersections**may experience constant external noise.At strategic locations throughout the park - near restrooms, along the Main Midway, and at prominent

**intersections**- guests will find "you are here" maps they can use to orient themselves.At each of the

**intersections**is a colored marble, and you can swap adjacent marbles (connected by the spun threads) in order to make sets of three (or more) of the same color.Villages are different, as they are built on 3-hex

**intersections**.This ensures that three

**intersections**near a village are empty throughout the game.Many youth groups set up near major

**intersections**on weekend mornings offering doughnuts for sale to passersby.Other driving skills you'll learn in driving classes include how to handle 4-way

**intersections**, how to drive through rotary circles, and how to get on and off the interstate.The driving test involves a series of driving situations, many of which include some fairly complicated and busy

**intersections**.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.Cramer, that, when a certain number of the

**intersections**of two algebraical curves are given, the rest are thereby determined.