Curvature

Folium Curve - First discussed by Descartes in 1638 but, although he found the correct shape of the curve in the positive quadrant, he believed that this leaf shape was repeated in each quadrant like the four petals of a flower.

Folium of Descartes: Cartesian equation: x+y = 3axy or x = 3at/(1+t), y = 3at/(1+t)

The curve is sometimes known as the noeud de ruban.

The curve passes through the origin at t = 0 and approaches the origin a second time as it goes to infinity.

C O N S I D E R:
Curvature [Vision & Invention, An Introduction to Art Fundamentals]
We look to curves of several types for secrets of natural or organic form-development. Since only the perfect circle, or absolute centric symmetry (which does not exist in nature), expresses equal tensions on all sides and in all directions, we recognize from the start that unequal tensions are the true order of nature, and that all of nature's forms are the result of dynamic--static synthesis.

Stress and strain, tension and compression, follow curves. Growth follows a curve, liquids follow lines of flow that are curved, objects hurtling through space follow curves (according to Einstein, space itself is curved), erosion creates curved forms and surfaces. Therefore, whether in the living event, the happening, or in the forms themselves, we recognize curves as the expression of the dynamic principle. We experience this in our own bodies as we gain more and more motor control in work and play. As Arnheim says: "The lever construction of the human body favors curved motion. The arm pivots around the shoulder joint, and subtler rotation is provided by the elbow, the wrist, the fingers." He points out that animals reveal mastery of circular movements in the paths they make through woods and fields. A colony of pigeons will practice the most beautiful banking or three-dimensional slalom movements, aerial choreography. The curve is associated with joy, with consummate skill.

The purest , most universal form of motion is the spiral: the counterclockwise spiral, the levogyre, is the one found most often in nature. We discover it in the growth of trees and in other members of the vegetable kingdom, in the majestic sweep of the great spiral galaxies, in the long bones of animals, and in seashells. There are also many instances of double spirals--the two intersecting curves in the seed heads of ripe sunflowers, in the centers of daisies., the seed cones of fir trees. Like curves, spirals are not all the same. The equiangular or logarithmic spiral of the elegant chambered nautilus is one type, seen also in ram's horns. It is interesting to note that the double spiral of the sunflower corresponds to the ratio fo the Fibonacci Series. If you count the number of seeds in a clockwise spiral and in an intersecting counterclockwise spiral, the two figures will be that of a sequence in Fibonacci's magic chain.

Wright, in his later work, emulates the spiral of a seashell. Buckminster Fuller, the architect-engineer-mathematician-utopian, used thousands of equilateral triangles conjoined as icosahedrons to form a geodesic dome of great economy of means and strength. The geodesic lines take the place of straight lines of plane geometry to form the shortest paths across the dome's surface. Le Corbusier moved deliberately toward the use of large curved forms in his later works; he said that the inspiration for the roof of his chapel Notre-Dame-du-Haut, Ronchamp, France, came from a crab shell that he had found on a beach. His experience as sculptor and painter must have contributed to his sensitivity to curved relationships, to color, light, and decoration in the chapel. These men show a deepening interest in the geometry of energy, the geometry of growth, worthy of comparison with that of Leonardo. The language and imagery of biology were used with greater frequency in talks and writings by Wright and Le Corbusier--their increasing references to "Nature" and the "organic."

Algebraic geometry and the geometry of growth appear to have some things in common. We can identify curves in living forms by those that pertain to the realm of pure mathematics--the parabola, the hyperbola, the ellipse, the spiral, and so forth. But we have to take into account another important factor: the role of nature as sculptor. The curve of the seashell or crab shell--of each seashell and crab shell--is the result of the give and take between biological geometry (replication and the structure of heredity via the DNA helix) and the environment; That is, forces working from within against forces, pressures from without; one kind of physical substance or system against another. It would be difficult to think of living form, except submicroscopic forms (viruses and the like), perhaps, that would serve as a model of mathematical perfection throughout. The surface configuration of most forms would reveal curves of far greater variance, having been sculpted by erosion, by action of wind and water by a process of subtraction in the case of earth formations. Or an erosion pattern would have been built in by nature over perhaps millions of years, as in the forms of most aquatic creatures--fish, shellfish, seals, squids, and the like, their "pre-eroded" surfaces having been designed by nature so as to offer the least resistance to the force and density of water.

These curves--the serpentine lines of flow, the analytic curves or conic sections (hyperbola, parabola, ellipse), the logarithmic spiral, catenary curves, flat curves and banking curves--are the repertory of curves with which the artist-designer works, consciously or not. We discover by analysis that lines and form profiles of particular beauty and strength are those that reveal considerable contrast in the kinds of curves that follow one another, that flow in and out of one another. They consist of deep or slow curves (ellipses or parabolas) and shallow or fast curves (hyperbolas), long and short curves. They flow into one another with an inevitability and naturalness usually absent from the pure curves of mathematics, especially that of the circle.

Diagonals and curves, expressed either two-dimensionally as lines or three-dimensionally as flat and curved planes--that is, in drawings or in pieces of sculpture--are the artist's form-equivalents of kinetic energy. Graham collier writes of overt or kinetic energy in terms of thrust and identifies three common manifestations:

(1) point thrust, seen in the arrow, the column or shaft, the steeple;

(2) centripetal thrust, [helical] seen in the clock spring, the spiral seashell, in all spiral forms, natural or man-made, where energy uncoils from a center; and

(3) pressure or pneumatic thrust, seen in a balloon, in ripe fruit and vegetables, eroded earth forms, sea creatures, the back of the human skull, the head of the femur, the egg, and all forms that are the result of tensions fairly evenly distributed.

(4) I believe he could have included a fourth type of thrust, seen in the wheel, in certain seed heads, in explosions, the Universal Mode System, and the like: radial thrust.

These kinds of thrust are often seen in combination with elements of potential energy, as in a tent, a clothesline, or a suspension bridge, where the point thrust of pole or nylon is complemented by the catenary arc of the canvas, the line or the cables, or contrariwise in the association of continuous and discontinuous patterns in the formations of mountains.

[Harlan, Calvin. Vision & Invention, An Introduction to Art Fundamentals. Englewood Cliffs, NJ: Prentice-Hall, 1986.]

The Law of Curvature - Ruskin
. . . . You must ascertain, by experiment, that all beautiful objects whatsoever are thus terminated by delicately curved lines, except where the straight line is indispensable to their use or stability; and that when a complete system of straight lines, throughout the form, is necessary to that stability, as in crystals, the beauty, if any exists, is in colour and transparency, not in form. Cut out the shape of any crystal you like, in white wax or wood, and put it beside a white lily, and you will feel the force of the curvature in its purity, irrespective of added colour, or other interfering elements of beauty.

Well, as curves are more beautiful than straight lines, it is necessary to a good composition that its continuities of object, mass, or colour should be, if possible, in curves, rather than straight lines or angular ones . . . . Now it is almost always possible, not only to secure such a continuity in the arrangement or boundaries of objects which, like these bridge aches or the corks of the net, are actually connected with each other, but--and this is a still more noble and interesting kind of continuity--among features which appear at first entirely separate . . . . on a larger scale [for example] the reader may easily perceive that there is a subtle cadence and harmony among them [towers]. The reason of this is, that they are all bounded by one grand curve, traced by the dotted line; out of the seven towers, four precisely touch this curve, the others only falling back from it here and there to keep the eye from discovering it too easily.

And it is not only always possible to obtain continuities of this kind: it is, in drawing large forest or mountain forms, essential to truth . . . .

Graceful curvature is distinguished from ungraceful by two characters; first in its moderation, that is to say, its close approach to straightness in some part of its course; and secondly, by its variation, that is to say, its never remaining equal in degree at different parts of its course.

This variation is itself twofold in all good curves.

A. There is, first, a steady change through the whole line, from less to more curvature, or more to less so that no part of the line is a segment of a circle, or can be drawn by compasses in any way whatever . . . . a) is a bad curve because it is part of a circle, and is therefore monotonous throughout; but b) is a good curve, because it continually changes its direction as it proceeds . . . . observance of this fact . . . . the springiness of character dependent on the changefulness of the curve [in a bough of leaves, for example]. [Variations of curvature in tree boughs--foliage arranged at the extremities instead of the flanks, etc . . . . ]

B. Not only does every good curve vary in general tendency, but it is modulated, as it proceeds, by myriads of subordinate curves. Thus the outlines of a tree trunk . . . . So also in waves, clouds, and all other noble formed masses. Thus another essential difference between good and bad drawing, or good and bad sculpture, depends on the quantity and refinement of minor curvatures carried, by good work, into the great lines. Strictly speaking, however, this is not variation in large curves, but composition of large curves out of small ones; it is an increase in the quantity of the beautiful element, but not a change in its nature.

[Ruskin, John. On Composition, pgs. 176-180, The Elements of Drawing, John Ruskin, Dover Publications, Inc., New York, 1971 [Originally Published in London, 1857]

R  E  F  E  R  E  N  C  E  S 

Curvilinear adj [L curus + linea line] [1710] 1: consisting of or bounded by curved lines: represented by a curved line 2: marked by flowing tracery [__ Gothic]

1 Curve adj [ME, fr. L curvus: akin to Gk Kyrtos convex. MIr cruinn round] [15c] archaic: bent or formed into a curve.

2 Curve vb [L curvare, fr. curvus] vi [1594]: to have or take a turn, change, or deviation from a straight line or plane surface without sharp breaks or angularity -vt 1: to cause to curve 2: to throw a curveball to [a batter] 3: to grade [as an examination] on a curve

3 Curve n [1696] 1a: a line esp. when curved: as [1] the path of a moving point [2]: a line defined by an equation so that the coordinates of its points are functions of a single independent variable or parameter b: the graph of a variable 2: something curved; as a: a curving line of the human body b pl: Parenthesis 3a: curveball b: trick, Deception 4: a distribution indicating the relative performance of individuals measured against each other that is used esp. in assigning good, medium, or poor grades to usu. predetermined proportions of students rather than in assigning grades based on predetermined standards of achievement