CURTIN
, Curtain, or Courtine, in Fortification, that part of a wall or rampart that joins two bastions, or lying between the flank of one and that of another.—The curtain is usually bordered with a parapet 5 feet high; behind which the soldiers stand to fire upon the covert-way, and into the moat.
CURVATURE of a Line, is its bending, or flexure; by which it becomes a curve, of any peculiar form and properties. Thus, the nature of the curvature of a circle is such, as that every point in the periphery is equally distant from a point within, called the centre; and so the curvature of the same circle is every where the same; but the curvature in all other curves is continually varying.—The curvature of a circle is so much the more, as its radius is less, being always reciprocally as the radius; and the curvature of other curves is measured by the reciprocal of the radius of a circle having the same degree of curvature as any curve has, at some certain point.
Every curve is bent from its tangent by its curvature, the measure of which is the same as that of the angle of contact formed by the curve and tangent. Now the same tangent AB is common to an infinite number of circles, or other curves, all touching it and each other in the same point of contact C. So that any curve DCE may be touched by an infinite number of different circles at the same point C; and some of these circles fall wholly within it, being more curved, or having a greater curvature than that curve; while others fall without it near the point of contact, or between the curve and tangent at that point, and so, being less deflected from the tangent than the curve is, they have a less degree of curvature there. Consequently there is one, of this infinite number of circles, which neither falls below it nor above it, but, being equally deflected from the tangent, coincides with it most intimately of all the circles; and the radius of this circle is called the radius of curvature of the curve; also the circle itself is called the circle of curvature, or the osculatory circle of that curve, because it touches it so closely that no other circle can be drawn between it and the curve.
As a curve is separated from its tangent by its flexure or curvature, so it is separated from the osculatory circle by the increase or decrease of its curvature; and as its curvature is greater or less, according as it is more or less deflected from the tangent, so the variation of curvature is greater or less, according as it is more or less separated from the circle of curvature.
It appears, however, from the demonstration of geometricians, that circles may touch curve lines in such a manner, that there may be indefinite degrees of more or less intimate contact between the curve and its osculatory circle; and that a conic section may be described that shall have the same curvature with a given line at a given point, and the same variation of curvature, or a contact of the same kind with the circle of curvature.
If we conceive the tangent of any proposed curve to be a base, and that a new line or curve be described, whose ordinate, upon the same base or absciss, is a 3d proportional to the ordinate and base of the first; this new curve will determine the chord of the circle of curvature, by its intersection with the ordinate at the point of contact; and it will also measure the variation of curvature, by means of the tangent of the angle in which it cuts that circle: the less this angle is, the closer is the contact of the curve and circle of curvature; and of this contact there may be indesinite degrees.
For example, let EMH be any curve, to which ET is a tangent at the point E; then let there be always taken MT : ET :: ET : TK, and through all the points K draw the curve BKF; then from the point of contact E draw EB parallel to the ordinate TK, meeting the last curve in B; and finally, describe a circle ERQB through the point B and touching ET in E; and it shall be the osculatory circle to the given curve EMH. And the contact of EM and ER is always the closer, the less the angle KBQ is. See Maclaurin's Fluxions, art. 366.
Hence it follows, that the contact of the curve EMH and its osculatory circle is closest, when the curve BK touches the arch BQ in B, the angle BET being given; and it is farthest from this, or most open, when BK touches the right line BE in B.
Hence also there may be indefinite degrees of more and more intimate contact between a circle and a curve. The first degree is when the same right line | touches them both in the same point; and a contact of this sort may take place between any circle and any arch of a curve. The 2d is when the curve EMH and circle ERB have the same curvature, and the tangents of the curve BKF and circle BQE intersect each other at B in any assignable angle. The contact of the curve EM and circle of curvature ER at E, is of the 3d degree, or order, and their osculation is of the 2d, when the curve BKF touches the circle BQE at B, but so as not to have the same curvature with it. The contact is of the 4th degree, or order, and their osculation of the 3d, when the curve BKF has the same curvature with the circle BQE at B, but so as that their contact is only of the 2d degree. And this gradation of more and more intimate contact, or of approximation towards coincidence, may be continued indefinitely, the contact of EM and ER at E being always of an order two degrees closer than that of BK and BQ at B. There is also an indefinite variety comprehended under each order: thus, when EM and ER have the same curvature, the angle formed by the tangents of BK and BQ admits of indefinite variety, and the contact of EM and ER is the closer the less that angle is. And when that angle is of the same magnitude, the contact of EM and ER is the closer, the greater the circle of curvature is. When BK and BQ touch at B, they may touch on the same or on different sides of their common tangent; and the angle of contact KBQ may admit of the same variety with the angle of contact MER; but as there is seldom occasion for considering those higher degrees of more intimate contact of the curve EMH and circle of curvature ERB, Mr. Maclaurin calls the contact or osculation of the same kind, when, the chord EB and angle BET being given, the angle contained by the tangents of BK and BQ is of the same magnitude.
When the curvature of EMH increases from E towards H, and consequently corresponds to that of a circle gradually less and less, the arch EM falls within ER, the arch of the osculatory circle, and BK is within BQ. The contrary happens when the curvature of EM decreases from E towards H, and consequently corresponds to that of a circle which is gradually greater and greater, the arch EM falls without ER, and BK is without BQ. And according as the curvature of EM varies more or less, it is more or less unlike to the uniform curvature of a circle; the arch of the curve EMH separates more or less from the arch of the osculatory circle ERB, and the angle contained by the tangents of BKF and BQE at B, is greater or less. Thus the quality of curvature, as it is called by Newton, depends on the angle contained by the tangents of BK and BQ at B; and the measure of the inequability or variation of curvature, is as the tangent of this angle, the radius being given, and the angle BET being a right one.
The radii of curvature of similar arcs in similar figures, are in the same ratio as any homologous lines of these sigures; and the variation of curvature is the same. See Maclaurin, art. 370.
When the proposed curve EMH is a conic section, the new line BKF is also a conic section; and it is a right line when EMH is a parabola, to the axis of which the ordinates TK are parallel. BKF is also a right line when EMH is an hyperbola, to one asymptote of which the ordinate TK is parallel.
When the ordinate EB, at the point of contact E, instead of meeting the new curve BK, is an asymptote to it, the curvature of EM will be less than in any circle; and this is the case in which it is said to be infinitely little, or that the radius of curvature is infinitely great. And of this kind is the curvature at the points of contrary flexure in lines of the 3d order.
When the curve BK passes through the point of contact E, the curvature is greater than in any circle, or the radius of curvature vanishes; and in this case the curvature is said to be infinitely great. Of this kind is the curvature at the cusps of the lines of the 3d order.
As to the degree of curvature in lines of the 3d and higher orders, see Maclaurin, art. 379; also art. 380, when the proposed curve is mechanical.
As curves which pass through the same point have the same tangent when the first fluxions of the ordinates are equal, so they have the same curvature when the 2d fluxions of the ordinate are likewise equal; and half the chord of the osculatory circle that is intercepted between the points where it intersects the ordinate, is a 3d proportional to the right lines that measure the 2d fluxion of the ordinate and first fluxion of the curve, the base being supposed to flow uniformly. When a ray revolving about a given point, and terminated by the curve, becomes perpendicular to it, the first fluxion of the radius vanishes; and if its 2d fluxion vanish at the same time, that point must be the centre of curvature. The same may be said, when the angular motion of the ray about that point is equal to the angular motion of the tangent of the curve; as the angular motion of the radius of a circle about its centre is always equal to the angular motion of the tangent of the circle. Hence the various properties of the circle may suggest several theorems for determining the centre of curvature.
See art. 396 of the said book, also the following, concerning the curvature of lines that are described by means of right lines revolving about given poles, or of angles that either revolve about such poles, or are carried along fixed lines.
It is to be observed that, as when a right line intersects an arc of a curve in two points, if by varying the position of that line the two intersections unite in one point, it then becomes the tangent of the arc; so when a circle touches a curve in one point, and intersects it in another, if, by varying the centre, this intersection joins the point of contact, the circle has then the closest contact with the arc, and becomes the circle of curvature; but it still continues to intersect the curve at the same point where it touches it, that is, where the same right line is their common tangent, unless another intersection join that point at the same time. In general, the circle of curvature intersects the curve at the point of osculation, only when the number of the successive orders of fluxions of the radius of curvature, that vanishes when this radius comes to the point of osculation, is an even number.
It has been supposed by some, that two points of contact, or four intersections of the curve and circle of curvature, must join to form an osculation. But Mr. | James Bernoulli justly insisted, that the coalition of one point of contact and one intersection, or of three interfections, was sufficient. In which case, and in general, when an odd number of intersections only join each other, the point where they coincide continues to be an intersection of the curve and circle of curvature, as well as a point of their mutual contact and osculation. See Maclaurin's Flux. art. 493.
From these principles may be determined the circle of curvature at any point of a conic section. Thus, <*>uppose AEMHG be any conic section, to the point E of which the circle or radius of curvature is to be found. Let ET be a tangent at E, and draw EGB and the tangent HI parallel to the chords of the circle of curvature; then take EB to EG as EI^{2} to HI^{2}; or, when the section has a centre O, as in the ellipse and hyperbola, as the square of the semi-diameter Oa parallel to ET, is to the square of the semi-diameter OA parallel to EB; and a circle EB described upon the chord EB that touches ET, will be the circle of curvature sought.
When BET is a right angle, or EB is the diameter of the circle of curvature, EG will be the axis of the conic section, and EB will be the parameter of this axis; also when the point G, where the conic section cuts EB, and the point B, are on the same side of E, then EMG will be an ellipsis, and EG the greater or less axis, according as EG is greater or less than EB.
The propositions relating to the curvature of the conic sections, commonly given by authors, follow with out much difficulty from this construction.
1. When the chord of curvature, thus found, passes through the centre of the conic section, it will then be equal to the parameter of the diameter that passes through the point of contact.
2. The square of the semi-diameter Oa, is to the rectangle of half the transverse and half the conjugate axis, as the radius of curvature CE is to Oa. And therefore the cube of the semi-diameter Oa, parallel to the tangent ET, is equal to the solid contained by the radius of curvature CE, and the rectangle of the two axes.
3. The perpendicular to either axis bisects the angle made by the chord of curvature, and the common tangent of the conic section and circle of curvature.
4. The chord of the osculatory circle that passes through the focus, the diameter conjugate to that which passes through the point of contact, and the transverse axis of the figure, are in continued proportion.
5. When the section is an ellipse, if the circle of curvature at E meet Oa in d, the square of Ed will be equal to twice the square of Oa. Hence Ed : Oa :: √2 : 1. Which gives an easy method of determining the circle of curvature to any point E, when the semidiameter Oa is given in magnitude and position.
Several other properties of the circle of curvature, and methods of determining it when the section is given; or vice versa, of determining the section when the circle of curvature is given, may be seen in Maclaurin's Flux. art. 375. See also the Appendix to Maclaurin's Algebra, sect. 1.
To determine the Radius and Circle of Curvature by the Method of Fluxions. Let AEe be any curve, coneave towards its axis AD; draw an ordinate DE to the point E where the curve is required to be found; and suppose EC perpendicular to the curve, and equal to the radius of the circle BEe of curvature sought; lastly, draw Ed parallel to AD, and de parallel and indefinitely near to DE; thereby making Ed the fluxion or increment of the absciss AD, also de the fluxion of the ordinate DE, and Ee that of the curve AE. Now put x=AD, y=DE, z=AE, and r=CE the radius of curvature; then is ED=x^{.}, de=y^{.}, and Ee=z^{.}.
Now, by sim. tri. the 3 lines | Ed , de, Ee, |
or | x^{.} , y^{.} , z^{.}, |
are respectively as the three | GE, GC, CE; |
therefore | GC.x^{.}=GE y^{.}; |
becomes | , |
or | . |
Farther, as in any case either x or y may be supposed to flow equably, that is, either x^{.} or y^{.} constant quantities, or x^{..} or y^{..}=to nothing, by this supposition either of the terms in the denominator of the value of r may be made to vanish. So that when x^{.} is constant, the value of r is , but r is when y^{.} is constant.
For example, suppose it were required to find the radius or circle of curvature to any point of a parabola, its vertex being A, and axis AD.—Now the equa- | tion of the curve is ax = y^{2}; hence ax^{..} = 2yy^{.}, and ax^{..} = 2y^{.}^{2}, supposing y^{.} constant, also a^{2}x^{2} = 4y^{2}y^{2}; hence r or , the general value of the radius of curvature for any point E, the ordinate to which cuts offthe absciss AD = x.
Hence, when x or the absciss is nothing, the last expression becomes barely for the radius of curvature at the vertex of the parabola; that is, the diameter of the circle of curvature at the vertex of a parabola, is equal to a the parameter of axis.
Variation of Curvature. See Variation.
Double Curvature, is used for the curvature of a line, which twists so that all the parts of it do not lie in the same plane.