Curve

“ Plücker first gave a scientific dual definition of a curve, viz.; “A curve is a locus generated by a point, and enveloped by a line—the point moving continuously along the line, while the line rotates continuously about the point”; the point is a point (ineunt.) of the curve, the line is a tangent of the curve. And, assuming the above theory of geometrical imaginaries, a curve such that m of its points are situate in an arbitrary line is said to be of the order m; a curve such that n of its tangents pass through an arbitrary point is said to be of the class n ”

“ Imagine curve lines to be described on any plane, and to revolve about any given axes passing through the centre of force, and by that revolution to describe curve superficies; and that the bodies move in such sort that their centres may be always found in those superficies. If those bodies reciprocate to and fro with an oblique ascent and descent, their motions will be performed in planes passing through the axis, and therefore in the curve lines, by whose revolution those curve superficies were generated. ”

“ In a machine of any kind, each point describes a curve; a simple but important instance is the “three-bar curve,” or locus of a point in or rigidly connected with a bar pivoted on to two other bars which rotate about fixed centres respectively. ”

“ Why does anybody want to differentiate twice over? We know that when the variable quantities are space and time, by differentiating twice over we get the acceleration of a moving body, and that in the geometrical interpretation,
as applied to curves, means the slope of the curve. But what can mean in this case? Clearly it means the rate (per unit of length ) at which the slope is changing—in brief, it is a measure of the curvature of the slope. ”

“ If we draw a curve (fig. 20) to pass through the ends of the span, so that its ordinate represents the value of M′/EI, the corresponding fictitious loads are statically equivalent to a single load, of amount represented by the area of the curve, placed at the point of the span vertically above the centre of gravity of this area. ”

“ So then a 'Line' must be bent, though a 'curve' need not be so? Your client has clearly one merit—great originality of style!
Nie. Here is another definition of 'curve,' which may be more to your taste, 'A curve is a one-way spread, with points as elements.' (p. 10.)
Min. Too much like a dinner à la Russe. I don't like 'spread' at all.
Nie. He illustrates his use of 'spread' by applying it to other subjects. For instance, 'a musical tone allows of variations which form a two-way spread, with different degrees of intensity and of pitch as elements.' (p. 12.) ”

“ If is the vector drawn from a fixed origin to any point on a curve, and the distance of that point measured on the curve from any fixed point, is a unit vector, tangent to the curve and having the direction in which increases; is a vector directed from a point on the curve to the center of curvature, and equal to the curvature; is the normal to the osculating plane, directed to the side on which the curve appears described counter-clockwise about the center of curvature, and equal to the curvature. ”

“ A curve of a given order can in general be rationally transformed into a curve of a lower order; thus a curve of any order for which D = 0, that is, a unicursal curve, can be transformed into a line; a curve of any order having the deficiency 1 or 2 can be rationally transformed into a curve of the order D + 2, deficiency D ”

“ Think of any curve that has a minimum point in it (like Fig. 15) , or like Fig. 34, where the point of minimum is marked , and the curve is concave upwards. To the left of the slope is downward, that is, negative, and is
getting less negative. To the right of the slope has become upward, and is getting more and more upward. Clearly the change of slope as the curve passes through is such that is positive, for its operation, as increases toward the right, is to convert a downward slope into an upward one. ”

“ If we suppose an intelligence capable of conceiving the essence of the curve, by an immediate intuition of the law governing the inflection of points, without the necessity of referring it to any line, whether one axis instead of two suffices, or in any other manner not even imaginable by us; this intelligence will not need to follow all the evolutions which we have made in demonstrating the properties of the curve; for it will perceive them to be clearly contained in the very conception of the curve. ”

“ As the curve is the true line of beauty, and as the circle in Egyptian hieroglyphics is ever used as the symbol of renewed life—the type or sign of the generative principle—so the motion produced by the centripetal and centrifugal forces, seems to be that of nature. ”

“ The cylindrical curve round the stem of a plant binds its fibres together; while the ascent of the stem is in lines of various curvature: so the curve of the horizon and of the apparent heaven, of the rainbow, etc.: and though the reader might imagine that the circular orbit of any moving body, or the curve described by a sling, was a curve of motion, he should observe that the circular character is given to the curve not by the motion, but by the confinement: the circle is the consequence not of the energy of the body, but of its being forbidden to leave the centre ”

“ I suppose that there is no question concerning the form of the curve defined by any equation of the form Let us call such a curve As increases without limit, the curve approaches a limiting form, which may be thus described. ”

“ Many well-known derivative curves present themselves in this manner; thus the variable curve may be the normal (or line at right angles to the tangent) at any point of the given curve; the intersection of the consecutive normals is the centre of curvature; and we have the evolute as at once the locus of the centre of curvature and the envelope of the normal. It may be added that the given curve is one of a series of curves, each cutting the several normals at right angles. ”

“ All curvature, however, is not equally agreeable; but 264 the examination of the laws which render one curve more beautiful than another, would, if carried out to any completeness, alone require a volume. ”

“ The expression for the specific curvature, which is in general somewhat elaborate, assumes a very simple form when a system of geodetics and the system of their orthogonal trajectories are chosen for the parameter curves, the parameter u being made the length of the arc of the geodetic, measured from the curve, u=o selected as the standard. ”

“ We may apply the same reasoning to CURVE and RIGHT lines. Nothing is more apparent to the senses, than the distinction betwixt a curve and a right line; nor are there any ideas we more easily form than the ideas of these objects. But however easily we may form these ideas, it is impossible to produce any definition of them, which will fix the precise boundaries betwixt them. When we draw lines upon paper, or any continued surface, there is a certain order, by which the lines run along from one point to another, that they may produce the entire impression of a curve or right line ”

“ The very curves demonstrate the theorem that a curve is made up of little straight lines, the arches are stiff and unbending, and wherever a public building demands an ornament, a fir-shaped cone of straight lines rises in stoic severity. ”

“ We will suppose in the first instance that the curve is plane. It is often convenient to resolve the forces on an element PQ (= δs) in the directions of the tangent and normal respectively. If T, T + δT be the tensions at P, Q, and δψ be the angle between the directions of the curve at these points, the components of the tensions along the tangent at P give (T + δT) cos ψ − T, or δT, ultimately; whilst for the component along the normal at P we have (T + δT) sin δψ, or Tδψ, or Tδs/ρ, where ρ is the radius of curvature. ”

“ Nor can they be extended immediately to the case where the inquiry is not for potential surfaces but, say, for surfaces of least area, or surfaces of constant positive gaussian curvature, which are to pass through a prescribed twisted curve or to stretch over a given ring surface. ”

“ And so if D = 2, then the transformed curve is a nodal quartic; φ can be expressed as the square root of a sextic function of θ and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a sextic function of θ. ”

“ There are three phases in this natural development: Angular, circular and spiral. The human form poised squarely on both feet is the spiral, the head a convexed curve, the body a concave curve, and the legs a convexed curve, like a wave line. To preserve this spiral line of changing curves, the weight is always thrown against the strong side so as to develop the weak side and maintain an equilibrium. Standing should be principally upon the balls of the feet, and the exercise should be to incline the body to and from the opposite curves. ”

“ In plane geometry the elementary figures are the point and the line; and we then have the curve, which may be regarded as a singly infinite system of points, and also as a singly infinite system of lines. ”

“ Leading lines of Villa-composition.
244. Take a flat line of beauty, a d, fig. 14, for the length of the edifice. Strike a b horizontally from a, c d from d; let fall the verticals, make c f equal m n, the maximum; and draw h f. The curve should be so far continued as that h f shall be to c d as c d to a b. Then we are sure of a beautifully proportioned form. Much variety may be introduced by using different curves; joining parabolas with cycloids, etc.; but the use of curves is always the best mode of obtaining good forms. ”

“ Just in the same way straightness mingled with curvature, that is to say, the close approximation of part of any curve to a straight line, gives to such curve all its spring, power, and nobleness: and even perfect straightness, limiting curves, or opposing them, is often pleasurable: yet, in the abstract, straightness is always ugly, and curvature always beautiful. ”

“ The curve is known to designers as the Curve of Force and is most valuable in all forms of enrichment. Designers even in early ages used it in some form as will be noted from the fragment of Greek sculpture in Figure 72. Its beauty rests in its variety. A circle has little interest due to its rather monotonous curvature. The eye desires variety and the curve of force administers to this need and gives a sense of satisfaction. As designers on wood, how are we to utilize this curve for purposes of outline enrichment? ”

“ There are three kinds of curved lines along which bodies thus doubly driven may move: the circular curve, which goes round a central point at an unvarying equal distance, and returns into itself; the elliptical curve, which returns into itself by a route that is drawn out considerably in one direction; and the hyperbolic curve, that never returns into itself at all, but has, on the other hand, a course which sets outwards each way for ever. ”

“ The superior importance of the vertical and horizontal directions, the specially restful character of the horizontal, and the aspiring aspect of the vertical, the voluptuous nature of the curve as opposed to the severity of the straight line, point to the deeper and fuller experiences of movement, muscular exertion, and repose, which we obtain apart from the eye. Even the value of bilateral symmetry for the eye may owe something to that well-marked rhythmic contrast of right and left, which the movements of the tactual organ yield to us. ”

“ In drawing these curves great exactitude is required to properly find their centres; nothing looks worse in a drawing than an unfair or uneven junction between curves and straight lines. To find the location for these centres, set the compasses to the required radius for the curve, and from the point or corner A draw the arcs b and c, from c mark the arc e, and from b the arc d, and where d and e cross is the centre for the curve f. ”

“ If the body moves in a larger circle, for example on a merry-go-round or on a street car passing{55} around a curve, the mind perceives an inclination of the body towards the convex side of the curve. ”

“ The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. ”

“ LOGOCYCLIC CURVE, STROPHOID or FOLIATE, a cubic curve generated by increasing or diminishing the radius vector of a variable point Q on a straight line AB by the distance QC of the point from the foot of the perpendicular drawn from the origin to the fixed line. The polar equation is r cos θ = a (1 ± sinθ) , the upper sign referring to the case when the vector is increased, the lower when it is diminished. ”

“ Fig. 3 shows the different geometrical and trigonometrical elements of the curve, which can be read upon the various scales, or to which the instrument may be set. An observer standing at C sights the point B directly and the point A by reflection. A staff being set up at each point, he will see them simultaneously, and in coincidence if the instrument be properly set for the curve. If any intermediate position be taken up on the curve, both A and B will be seen in coincidence. ”

“ A spiral curve is a [45] harmonious line, says the designer: repeat it, reversed, and you prolong the harmony; repeat it again, with variations, and you complete the harmony. Or, harmonious effect is produced by recurring form and line. Here is a circular form; here is a meandering line: combine and repeat them, and you get a logical and harmonious border motive. ”