The description of Bernoulli's double knot line first appeared in 1694 when Jacob Bernoulli regarded it asellipseMore than a kind of processing.An ellipse is a point whose distance to two fixed points is the sum of fixed valuestrajectory。andCassini ovalIt is the track of the point whose fixed value is the product of the distance to two fixed points.When this fixed value makes the track pass through the midpoint of two fixed points, the track is Bernoulli double knot line.Bernoulli called this curve lemniscate, which means "hanging ribbon" in Latin.Bernoulli double button line has also been widely used in science and technology and light industry. Bernoulli also applies Bernoulli double button line to gambling.
Bernoulli double newline, also known as double newline, was first described in 1694, and Jacob Bernoulli treated it as a kind of ellipse.Set line segment
The length is 2a, if the moving point M meets
, then the track of M is called Bernoulli double knot line.Double button cable can pass throughEquiaxed hyperbolaIt is obtained through inversion, that is, it is the inverse figure of hyperbola about the circle whose center is at the center of hyperbola.Bernoulli double knot curve is a special case of Cassini oval curve and sine spiral curve, which means Bernoulli double knot curve plays an important role in the study of connecting all curves, so it is particularly important and urgent to discuss Bernoulli hyperbola.The double knot line plays a crucial role in the field of mathematical curves. The study of Bernoulli double knot line will help us to better study other related curves and achieve the effect of understanding by analogy.The Bernoulli double newcastle line has been widely and appropriately applied in light industry and technology. Therefore, the research on Bernoulli double newcastle line is of great practical significance.
The double button line is a functional graph, which not only reflects the symmetry, harmony, abstraction, simplicity, accuracy, unity, singularity and mutation of mathematical beauty, but also has special and valuable artistic beauty. It is the cornerstone of forming other common beautiful patterns, and also the main geometric element of many artists' design works.The figure outline of the double newline function is like the "8" in the Arabic numerals. In China, 8 is a simple number, but modern people give it a richer meaning.In the south, it means to get rich, because it is homophonic with the Chinese character "fa".Through the extension and connotation of the double knot line, on the basis of not deforming it, the function graph of the double knot line can be summarized with a schema, and on this basis, many excellent works of art can be created[1]。
equation
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In mathematics, for a fixed line segment of length 2a
Bernoulli binomial line is also expressed concisely in polar coordinates[2]：
In the bipolar coordinate system, the equation of Bernoulli binomial line is similar:
nature
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(1) OnCartesian coordinate systemIn, the Bernoulli double knot line is symmetrical about the coordinate origin, which is a node and inflection point with tangent y=± x.From any point M on the Bernoulli double knot line to the given two points
The product of the distances of is equal to
The square of the distance between.The shape of the curve is similar to the horizontal Arabic numeral 8 or the symbol of infinity
。
(2) The curvature of Bernoulli binomial line can be expressed as:
(3) The radius of curvature of Bernoulli kink line is:
(4) The area enclosed by each loop of Bernoulli Double New Line is:
application
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(1) Application in textile: Bernoulli double button thread is widely used as a pattern in textile. The cloth woven with double button thread has beautiful appearance, compact structure, repeatability and gradual change.
(2) Application in supercharger: Bernoulli double button line impact free double inlet widened flow supercharger is widely used in industry.
(3) Application in gambling: In Jacob Bernoulli's "The Art of Guessing", Bernoulli double button line was widely used in gambling.
Available schemata
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Repetitive
Figure 2 Repetitive
Repetitive composition refers to the repeated appearance of the same image in the same way, making use of the repetition in nature and human life to produce pleasure and sense of order.There are many natural and man-made repetitions, such as twins;The never-ending surf;Video replay in sports competitions;Repetition of door, window and column decoration in architecture;Terraced fields, etc.
As shown in Figure 1, the use of repeated composition to repeat the composition of the double newline function graph can be used as lace or shading in textiles for decoration, thus increasing interest and beauty.
Involute form
Fig. 3 Progressive type
Gradual is the continuous change of group image.It shows the increase and decrease in the same direction with certain regularity, such as the water ring formed by throwing stones into the water;Progression caused by perspective relationship of telegraph poles on the highway;The decreasing relationship of the same cornices on each floor of the ancient Chinese pagoda.The gradual change of each time cannot be too big. If it is too big, it will not be gradual. If it is too small, it will form repetition. The key is a word of "delivery".In the composition, various image relationships can be drawn from top to bottom, from more to less, from large to small, from strong to weak, and from black to white, from solid to virtual, from cold to warm.
Gradual composition is more and more used in modern textiles, which is easy to give people the sense of beauty of order visually. The use of double button line function graphics for gradual composition can be highlighted in many textiles.As shown in Figure 3.
Symmetrical type
Fig. 4 Symmetrical type
The images on both sides or around the axis are equivalent.Symmetrical composition is divided into overall symmetry and local symmetry.Global symmetry refers to the symmetry of large skeleton, while local symmetry refers to the symmetry of some details.Pay attention to the subtle changes of symmetry, such as the symmetrical replacement of images, the reversal of orientation, and the adjustment of volume.In textiles, symmetrical composition is very common.As shown in Figure 3, the figure of the double knot line function itself has a symmetrical aesthetic feeling, and it can be made into a variety of symmetrical forms, which is suitable for textiles.
Overlapping type
Figure 5 Overlapping type
The overlap between shapes sometimes forms a new overlapping image, and sometimes two or more images that do not affect each other exist in the same composition.Overlapping can be understood as the relationship between up and down, front and back.Overlap cannot be absent, otherwise there is no hierarchy.But not too much, too much overlap, concealment will make the image and structure confused and unclear.When the outline and structure of large overlapping images are not clear, "through overlapping" can also be used. Through overlapping means that images and images overlap transparently to form new decorative graphics.The overlapping effect can break the dullness caused by large areas of dark shadow, increase black and white or color levels, and activate the picture atmosphere.As shown in Figure 4, a fancy figure is formed by rotating and overlapping the double newline function figure, which can increase its visual effect in textiles.
Emission type
Figure 6 Emission type
Emission is a kind of special repetition, which means that the figure of double knot line function spreads outward or concentrates inward around one or more central points.The blooming flowers in nature belong to the shape of emission.In addition, emission can also be said to be a special gradient.It is the same as gradient, the figure of binomial function should be changed orderly.However, the emission has two significant characteristics:
(1) The emission has a strong focus, which is usually located in the center of the picture;
(2) The launch has a deep sense of space, and the dynamic sense of optics enables all graphics to focus on the center or spread from the center to the surrounding.
As shown in Figure 5, the emission type schema is composed of double button line function graphics. This type of schema is widely used in national textiles, and its visual impact is strong[1]。
