Curve (geometry)

“ They need to be defined for young people, but most of them are not basal, that is, they are not used in the proofs of propositions. Some of these terms, such as magnitudes, curve line, broken line, curvilinear figure, bisector, adjacent angles, reflex angles, oblique angles and lines, and vertical angles, need merely a word of explanation so that they may be used intelligently. ”

“ The straightedge and the compasses are the only two drawing instruments recognized in elementary geometry. [51] While this involves more than Euclid's postulate, it is a better working assumption for beginners.
3. A straight line is the shortest path between two points. This is easily proved by the method of Euclid [52] for the case where the paths are broken lines, but it is needed as a postulate for the case of curve paths. It is a better statement than the common one that a straight line is the shortest distance between two points ”

“ By joining V ́ A ́ and drawing T ́ S ́ parallel to V ́ F ́, then T ́ S ́ is obtained which will be the length of half the side of the square for drawing the inner spiral. The method for obtaining the inner spiral is the same as for the outer.
Fig. 37 There is no geometric means of drawing a perfect catenary curve; at best we can only obtain{196} it by an approximation in geometry. The curve is formed by suspending a chain from two points and pricking points along the curve of the chain. These
Fig. ”

“ A curve is the envelope of lines which touch it, as well as the locus of points which lie on it. A homogeneous equation of degree above the first in l, m, n is a relation connecting the coordinates of every line which touches some curve, and represents that curve, regarded as an envelope. For instance, the condition that the line of coordinates (l, m, n) , i.e. the line of which the allied point-coordinate equation is lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z) 2 = 0, is readily found to be of the form (A, B, C, F, G, H) (l, m, n) 2 = 0, i.e. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ A curve is a line, or continuous singly infinite system of points. We consider in the first instance, and chiefly, a plane curve described according to a law. Such a curve may be regarded geometrically as actually described, or kinematically as in the course of description by the motion of a point; in the former point of view, it is the locus of all the points which satisfy a given condition ”

“ 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. ”

“ In particular if D = 0, that is, if the given curve be unicursal, the transformed curve is a line, φ is a mere linear function of θ, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of θ ”

“ 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. ”

“ Sometimes the upright lines are omitted and a rule of signs is given:—Let the arc s of the curve be measured from some point along the curve in a chosen sense, and let the normal be drawn towards that side to which the curve is concave; if the normal is directed towards the left of an observer looking along the tangent in the chosen sense of description the curvature is reckoned positive, in the contrary case negative. ”

“ Those curves I call geometrically rational, all the points whereof may be determined by lengths that are defineable by equations, that is, by the complicated ratio's of lengths. Other curves (such as spirals, quadratrixes, and cycloids) I call geometrically irrational. For the lengths which are or are not as number to number (according to the tenth book of elements) are arithmetically rational or irrational. And therefore I cut off an area of an ellipsis proportional to the time in which it is described by a curve geometrically irrational, in the following manner. ”

“ Wherefore if one rectilinear figure is to be transformed into another, we need only transfer the intersections of the right lines of which the first figure consists, and through the transferred intersections to draw right lines in the new figure. But if a curvilinear figure is to be transformed, we must transfer the points, the tangents, and other right lines, by means of which the curve line is defined. ”

“ 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. ”

“ Within the field of these curves we can construct the transversals of the family; that is to say, there is a finite area of the plane, through any point of which there passes one stationary curve of the field and one curve which cuts all the stationary curves of the field transversely. These curves provide a system of curvilinear co-ordinates, in terms of which the value of ∫Fdx, taken along any curve within the area, can be expressed. The value of the integral is the same for all arcs of stationary curves of the field which are intercepted between any two assigned transversals. ”

“ Forms of Real Curves.—We have in much that precedes disregarded, or at least been indifferent to, reality; it is only thus that the conception of a curve of the m-th order, as one which is met by every right line in m points, is arrived at; and the curve itself, and the line which cuts it, although both are tacitly assumed to be real, may perfectly well be imaginary. For real figures we have the general theorem that imaginary intersections, &c., present themselves in conjugate pairs ”

“ Granting the quadratures of curvilinear figures, it is required to found the forces with which bodies moving in given curve lines may always perform their oscillations in equal times.
Plate 20, Figure 2 Let the body T (Pl. 20. Fig. 2.) oscillate in any curve line STRQ, whose axis is AR passing through the centre of force C. Draw TX touching that curve in any place of the body T, and in that tangent TX take TY equal to the arc TR. The length of that arc is known from the common methods used for the quadratures of figures. From the point Y draw the right line YZ perpendicular to the tangent. ”

“ Similarly for the curve h, set the compasses on i and mark the arc g, and from the point where it crosses line 6, draw the curve h. In inking in it is best to draw in all curves or arcs of circles first, and the straight lines that join them afterward, because, if the straight lines are drawn first, it is a difficult matter to alter the centres of the curves to make them [144] fall true, whereas, after the curves are drawn it is an easy matter, if it should be necessary, to vary the line a trifle, so as to make it join the curves correctly and fair. ”

“ Hence if the curves (A) and (B) are tangent, the slopes (C) and (D) will be equal (for the same value of ) , giving
or
By hypothesis (A) and (B) are tangent for every value of ; hence for all values of the point given by (A) must lie on a curve of the family (B) . ”

“ A curve itself is often treated, even in mathematical science, as consisting of a succession of very short straight lines, or as one broken line, constantly changing its direction; and many beautiful theorems were evolved by the early mathematicians on this hypothesis. ”

“ A right line is a direction which is always constant; the curve a direction which is always varied. A direction always varied is a collection of right directions infinitely small. Therefore, the circumference of a circle is considered as a polygon of an infinite number of sides. The curve is therefore formed by the variety of directions reduced to infinitesimal values. This theory which explains the difference of the right line and the curve, is evidently applicable to surfaces and solids. ”

“ In the problem of variable limits, when a terminal point moves on a given guiding curve, the integral cannot be an extremum unless the stationary curve along which it is taken is cut transversely by the guiding curve at the terminal point. A simple example is afforded by the shortest line, drawn on a surface, from a point to a given curve, lying on the surface. The required curve must be a geodesic, and it must cut the given curve at right angles. ”

“ The order of a plane curve is the number of points (real or imaginary) in which the curve is intersected by a straight line; it is equal to the degree (or coefficient of the highest power) of the Cartesian equation expressing the curve. The order of a non-plane curve is the number of points (real or imaginary) in which the curve intersects a plane (see Curve) . The order of a surface is the number of points in which the surface intersects a straight line. ”

“ If we think of the curve which will represent as a function of , it will be such that its slope at any point will be proportional to the ordinate at that point, and will be a negative slope if is positive. So obviously the curve will be a die-away curve (p. 156) , and the solution will contain as a factor. But, without presuming on this bit of sagacity, let us go to work. As both and occur in the equation and on opposite sides, we can do nothing until we get both and to one side, and to the other. To do this, we must split our usually inseparable companions and from one another. ”

“ 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. ”

“ 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. ”