Cone (geometry)

“ Of such surfaces an infinite number exists. Every ray through S1 or S2 which is not a secant determines one of them.
If, however, the rays a1 and a2 are secants meeting at A, then the ruled quadric surface becomes a cone of the second order, having A as centre. Or all lines of the congruence which pass through a point on the twisted cubic k form a cone of the second order. In other words, the projection of a twisted cubic from any point in the curve on to any plane is a conic. ”

“ Thus in representing a cone of revolution with a vertical axis we get in the plan a circular trace of the surface whose centre is the plan of the vertex of the cone, and in the elevation the contour, consisting of a pair of lines intersecting in the elevation of the vertex of the cone. The circle in the plan and the pair of lines in the elevation do not determine the surface, for an infinite number of surfaces might be conceived which pass through the circular trace and touch two planes through the contour lines in the vertical plane. ”

“ Theorem. Every section of a cone made by a plane passing through its vertex is a triangle.
At this time, or in speaking of the preliminary definitions, reference should be made to the conic sections. Of these there are three great types: (1) the ellipse, where the cutting plane intersects all the elements on one side of the vertex; a circle is a special form of the ellipse; (2) the parabola, where the plane is parallel to an element; (3) the hyperbola, where the plane cuts some of the elements on one side of the vertex, and the rest on the other side ”

“ A cone is cut by every plane through the vertex in lines, and if it is a cone of revolution by planes perpendicular to the axis in circles.
A cylinder is cut by every plane parallel to the axis in lines, and if it is a cylinder of revolution by planes perpendicular to the axis in circles.
A sphere is cut by every plane in a circle. ”

“ The nodes of the singular surface are points whose complex cones are coincident planes, and the complex conic in a singular tangent plane consists of two coincident points. This configuration of sixteen points and planes has many interesting properties; thus each plane contains six points which lie on a conic, while through each point there pass six planes which touch a quadric cone. In many respects the Kümmer quartic plays a part in three dimensions analogous to the general quartic curve in two ”

“ And from a point unto a point is this justly demanded to be done, not unto points; For neither doe all points fall in a right line: But many doe fall out to be in a crooked line. And in a Spheare, a Cone & Cylinder, a Ruler may be applyed, but it must be a sphearicall, Conicall, or Cylindraceall. But by the example of a right line doth Vitellio, 2 p j. demaund that betweene two lines a surface may be extended: And so may it seeme in the Elements, of many figures both plaine and solids, by Euclide to be demanded; That a figure may be described, at the 7. ”

“ In a sphericall it is onely a crooked line; But in a conicall or Cylindraceall it may bee a right line, as is the side of a Cone and Cylinder. Therefore in the conversion or turning about of a line making a periphery, there is considered onely the distance; yea two points, one in the center, the other in the toppe, which therefore Aristotle nameth Rotundi principia, the principles or beginnings of a round.
12. An Helix is a crooked line which is unequally distant from the middest of the space, howsoever inclosed. ”

“ For here the base is to be added to the variable surface.
2 If variable solids have their axes proportionall to their bases, they are alike. 24. d xj.
It is a Consectary out of the 19 e, iiij. For here the axes and diameters are, as it were, the shankes of equall angles, to wit, of right angles in the base, and perpendicular axis.
3 A mingled body is a Cone or a Cylinder.
The cause of this division of a varied or mingled body, is to be conceived from the division of surfaces. [291]
4 A Cone is that which is comprehended of a conicall and a base. ”

“ If the conic be a circle the conoid is a sphere (q.v.) ; if an ellipse a spheroid (q.v.) ; if a parabola a paraboloid; if a hyperbola the surface is a hyperboloid of either one or two sheets according as the revolution takes place about the conjugate or transverse axis, and the surface generated by the asymptotes is called the “asymptotic cone.” If two intersecting straight lines be regarded as a conic, then the principal axes are the bisectors of the angles between the lines ”

“ And it then appears that there are two kinds of non-singular cubic cones, viz. the simplex, consisting of a single sheet, and the complex, consisting of a single sheet and a twin-pair sheet; and we thence obtain (as for cubic curves) the crunodal, the acnodal and the cuspidal kinds of cubic cones. It may be mentioned that the single sheet is a sort of wavy form, having upon it three lines of inflection, and which is met by any plane through the vertex in one or in three lines; the twin-pair sheet has no lines of inflection, and resembles in its form a cone on an oval base. ”

“ Conic Projections A conic projection is produced by transferring points from the surface of the earth to a cone or series of cones. This cone is then cut along an element and spread out flat to form the chart. When the axis of the cone coincides with the axis of the earth, then the parallels appear as arcs of circles, and the meridians appear as either straight or curved lines converging toward the nearer pole. Limiting the area covered to that part of the cone near the surface of the earth limits distortion. A parallel along which there is no distortion is called a standard parallel. ”

“ In general a cone consists of one or more single or twin-pair sheets, and if we consider the section of the cone by a plane, the curve consists of one or more complete branches, or say circuits, each of them the section of one sheet of the cone ”

“ This excess of diameter at the base being determined, the reader is to ask himself how 86 most easily and simply to smooth the column from one extremity to the other. To cut it into a true straight-sided cone would be a matter of much trouble and nicety, and would incur the continual risk of chipping into it too deep. Why not leave some room for a chance stroke, work it slightly, very slightly convex, and smooth the curve by the eye between the two extremities? you will save much trouble and time, and the shaft will be all the stronger. ”

“ The profiles of both bases, or ends, will be circles. The smaller circle is generally called the upper base, and the larger circle the lower base of the frustum. The lateral area of a frustum of a cone is found by adding together the circumferences of the upper and lower bases, dividing the sum by 2, and then multiplying by the slant height. This is often expressed as a formula for area of a frustum:
A frustum whose lower base has a circumference of 40" and whose upper base has a circumference of 22" will have a surface area of
if the slant height of the frustum is 16". ”

“ We are accustomed in ordinary geometry to admit that every circle passes through the two imaginary circular points at infinity. This is the specialized form of the general theorem which asserts that every circle has double contact with the fundamental conic. The two theorems indeed coincide if the fundamental conic degrades to the infinity of ordinary measurement. A critical case is presented when the chord through coincides with either of the two tangents which may be drawn from to the fundamental conic. ”

“ A conic determines at every point in its plane an involution of rays, corresponding rays [pg 89] being conjugate with respect to the conic. The double rays, if any exist, are the tangents from the point to the conic. In particular, taking the point as the center of the conic, we find that conjugate diameters form a system of rays in involution, of which the asymptotes, if there are any, are the double rays. Also, conjugate diameters are harmonic conjugates with respect to the asymptotes. ”

“ The diameter of this cone at bottom is about 60 yards, its height 70 feet; and the area on its top large enough to receive a battery of cannon, with which it is hereafter to be fortified. Its solid contents are 2500 French toises; which, in our measure (allowing the French foot to be to the English as 144 to 135) , will amount to 24,250 cubic yards nearly. Several other cones, of [47] equal dimensions, are sunk at convenient distances from each other, forming the line of the pier: their number, when complete, it is said, will be forty. ”

“ These conjugate lines pass through the pole of the line, the pole lies therefore at infinity, and the line is a diameter, hence by the last property an axis.
It follows that all foci lie on one axis, for no line joining a point in one axis to a point in the other can be an axis.
As the conic determines in the pencil which has its centre at a focus a circular involution, no tangents can be drawn from the focus to the conic. Hence each focus lies within a conic; and a directrix does not cut the conic. ”

“ Using cones tangent at various parallels, a secant (intersecting) cone, or a series of cones varies the appearance and features of a conic projection.
312. Simple Conic Projection A conic projection using a single tangent cone is a simple conic projection (Figure 312a) . The height of the cone increases as the latitude of the tangent parallel decreases. At the equator, the height reaches infinity and the cone be-
comes a cylinder. ”

“ The in plano definition, i.e. as the conic having an eccentricity greater than unity, is a convenient starting-point for the Euclidian investigation. In projective geometry it may be defined as the conic which intersects the line at infinity in two real points, or to which it is possible to draw two real tangents from the centre. ”

“ If the plane SS2P has not this property, then we draw a plane SS1P. This cuts each surface in a conic, and these conics have in common the points S, S1, one point on each of the conics α1, β1, and one point on one of the conics through S and S2 which lie on both surfaces, hence five points. They are therefore coincident, and our theorem is proved.
§ 96. The following propositions follow:—
A quadric surface has at every point a tangent plane.
Every plane section of a quadric surface is a conic or a line-pair. ”

“ If we imagine a series of cones, each of which touches one of a selected series of parallels, the apex of each cone will lie on the prolonged axis of the spheroid; the generators of each cone lie in meridian planes, and if each cone is unrolled and the generators in any one plane are superposed to form a straight central meridian, we obtain a projection in which the central meridian is a straight line and the parallels are circular arcs each of which has a different centre which lies on the prolongation of the central meridian, the radius of any parallel being ν cot φ. ”