Arc (geometry)

“ It is a very important proposition, theoretically and practically, for it enables us to find the center of a circle if we know any part of its arc. A civil engineer, for example, who wishes to find the center of the circle of which some curve (like that on a running track, on a railroad, or in a park) is an arc, takes two chords, say of one hundred feet each, and erects perpendicular bisectors. It is well to ask a class why, in practice, it is better to take these chords some distance apart. ”

“ What is the chord called?
What is the arc called?
83A segment bounded by a diameter and a semi-circumference is a “semicircle.”
A semicircle is half a circle.
Read four segments each larger than a semicircle.
The part of the circle between the two radii o f, o i, and the arc f i, is called a “sector.”
Read four sectors each less than one-fourth of a circle. [2]
2. Thus, a sector bounded by the two radii o g, o h, and the arc g h.
A sector is a part of a circle bounded by two radii and an arc. ”

“ Teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle, and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface, so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface. ”

“ The problem is a common one in railroad work, only here AP is usually oblique to BQ if they are produced to meet at C, as in the second figure on page 218. It is required to construct an arc so that the tracks shall begin to curve at P and Q, where CP = CQ.
The problem becomes a little more complicated, and correspondingly more interesting, when we have to find the center of curvature for a street railway track that must turn a corner in such a way as to allow, say, exactly 5 feet from the point P, on account of a sidewalk. ”

“ Those who have studied only what is known as "practical geometry" think at once of the ease and accuracy with which a right angle, for example, may be divided into three equal parts. Thus taking the right angle ACB, Fig. 4, which may be set off more easily and accurately than any other angle except, perhaps, that of 60°, and knowing that it contains 90°, describe an arc ADEB, with C for the center and any convenient radius. Now every school-boy who has played with a pair of compasses knows that the radius of a circle will "step" round the circumference exactly six times ”

“ The sine of an arc is the line drawn from one extremity of the arc perpendicular to the diameter of the circle which passes through the other extremity.
The supplement of an arc is the difference, in degrees, between the arc, and a semicircle, or 180 degrees.
The complement of an arc is the difference, in degrees, between the arc, and a quadrant, or 90 degrees.
The tangent of an arc is a line touching the circle in one extremity of that arc, continued from thence to meet a line drawn from the centre through the other extremity; which last line is called the secant of the same arc. ”

“ If the axes be given, the curve can be approximately constructed by circular arcs in the following manner:—Let AA′, BB′ be the axes; determine D the intersection of lines through B and A parallel to the major and minor axes respectively. Bisect AD at E and join EB. Then the intersection of EB and DB′ determines a point P on the (true) curve. Bisect the chord PB at G, and draw through G a line perpendicular to PB, intersecting BB′ in O. An arc with centre O and radius OB forms part of a curve. ”

“ Lay down the base A B (Fig. 7) , making it equal to 6 divisions of the scale; from A with radius equal to 4 divisions, and from B with radius of 5 divisions of the scale [17] describe arcs cutting each other in C; join A C and C B, which will complete the required triangle.
Fig. 8.
To construct an Angle equal to a given angle. —It is required to draw a line making with the line D E (Fig. 8) an angle equal to that contained by the lines B A C. From A, with any radius, draw an arc F G, and from D, with the same radius, draw the arc H I, and make H I equal F G ”

“ With point 1 as a center and a radius equal to the hypotenuse of triangle 1–A, an arc is drawn below line 1–2. This is intersected by an arc drawn from point 2 with a radius equal to the hypotenuse of triangle 2–A. The intersection locates the point A on the pattern. With point 2 as a center and a radius equal to the hypotenuse of triangle 2–B, an arc is drawn bearing away from point A. This arc is intersected by another drawn from point A, whose radius is equal to line AB of Fig. ”

“ In equal circles, angles, whether at the centres or at the circumferences, have the same ratio which the arcs on which they stand have to one another; so also have the sectors.
Let 'ABC and BEF be equal circles, and let BGC and EHF be angles at their centres, and BAC and EDF angles at their circumferences: as the arc BC is to the arc EF so shall the angle BGC be to the angle EHF, and the angle BAC to the angle EDF; and so also shall the sector BGC be to the sector EHF.
Take any number of arcs CK, KL, each equal to BC, and also any number of arcs FM, MN each equal to EF ”

“ It is so: and the measure of an angle is an arc or part of the circumference of a circle, whose angular point is the center: and so many 360th parts as any arc contains, so many degrees the measure of the angle is said to be; thus, 89Z C P (Plate III. fig. 1.) makes an angle of 23-1/2 degrees, because the arc Z P contains 23-1/2 360th parts of the whole circle. Then if A B represent the plane of the ecliptic, and N C S the axis of the earth, as D N contains the same number of degrees as Z P, will not its inclination from a perpendicular be 23-1/2 degrees?
Pupil. Nothing can be plainer. ”

“ We may in fact take, as a varied curve, the arc AQ of a neighbouring stationary curve, the straight line QP drawn from Q to a point of the arc AB, and the arc PB of the stationary curve AB (fig. 6) . The sign of the variation is then the same as that of the function E (x, y, tan φ, tan ψ) , where (x, y) is the point P, ψ is the angle which the straight line QP makes with the axis of x, and φ is the angle which the tangent at P to the curve APB makes with the same axis. We thus arrive at a new necessary (but not sufficient) condition for the existence of an extremum of the integral ∫Fds, viz. ”

“ To realize what is meant when an angle is measured with a “probable error” of a few hundredths of a second of arc, it should be remembered that one second of arc on a circle 10 inches in diameter is less than 1/40000 of an inch. The accuracy which has been attained in the measurement of base lines is not easily realized by a layman. An engineer realizes the practical impossibility of measuring a line twice and obtaining356 precisely the same result to the finest unit of measurement. ”

“ An oblique prism is equivalent to a right prism whose base is equal to a right section of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism.
This is a fundamental theorem leading up to the mensuration of the prism. Attention should be called to the analogous proposition in plane geometry relating to the area of the parallelogram and rectangle, and to the fact that if we cut through the solid figure by a plane parallel to one of the lateral edges, the resulting figure will be that of the proposition mentioned. ”

“ Second. A rectangle, Fig. 121. On the base E F draw the perpendiculars E H, F G, equal to the height of the rectangle, and join G H.
[93]
Fig. 122.
Ex. 17.—To construct a parallelogram of which the sides and one of the angles are given, Fig. 122. Draw the side D E equal to the given length A, and set off the other side D F equal to the other length B, forming the given angle C. From E with D F as radius, describe an arc, and from F, with the radius D E cut the arc at G. Draw F G, E G. Or, the remaining sides may be drawn as parallels to D E, D F.
Fig. ”

“ The arc pattern is first worked: arrange it as directed for pattern 3, and mark the outline with a pencil: in this, the arc began at 280 of the fly-wheel, and ended at 168, making in all 116 numbers; to divide 58 for each half of the arc, divide them thus—4, 5, 6, 7, 8, 9, 10, 3, 4; then count back 4, 3, 10, 9, 8, 7, 6, 5, 4. You will see that the numbers increase one every time or for every circle, which also is enlarged one notch for each. ”

“ The mere inspection of the two angles enclosed between the index and the two beams suffices to show their equality; while if the graduated arc be consulted, the arc from 5 to m is found accurately equal to the arc from 5 to n. The complete expression of the law of reflection is, not only that the angles of incidence and reflection are equal, but that the incident and reflected rays always lie in a plane perpendicular to the reflecting surface. ”

“ The outer arc was divided by continual bisection alone, and a table was constructed by which the readings of the one arc could be converted into those of the other. After the dots indicating the required divisions were obtained, either straight strokes all directed towards the centre were drawn through them by the dividing knife, or sometimes small arcs were drawn through them by the beam-compass having its fixed point somewhere on the line which was a tangent to the quadrantal arc at the point where a division was to be marked. ”

“ We therefore take first, V1 as centre and V1T as radius, and describe arc of circle till it cuts the horizon in M1, which is the measuring point for all lines drawn to V1. Then with radius V2T describe arc from centre V2 till it cuts the horizon in M2, which is the measuring point for all vanishing lines drawn to V2. We have now set out our points. Let us proceed to draw the square Abcd. From A, the nearest angle (in this instance touching the base line) , measure on each side of it the equal lengths AB and AE, which represent the width or side of the square. ”

“ We may, however, ignore the accented letters in translating theorems, for a configuration of lines and its polar with respect to a linear complex have the same projective properties. In Lie’s transformation a linear complex corresponds to the totality of spheres cutting a given sphere at a given angle. A most remarkable result is that lines of curvature in the sphere geometry become asymptotic lines in the line geometry. ”

“ Axes.—Conjugate diameters perpendicular to each other are called axes, and the points where they cut the curve vertices of the conic.
In a circle every diameter is an axis, every point on it is a vertex; and any two lines at right angles to each other may be taken as a pair of axes of any circle which has its centre at their intersection. ”