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Geometric meaning of points, lines and surfaces.
In the related branches of geometry, topology and mathematics, a point in space is used to describe a special object in a given space, which has something like volume, area, length or other high-dimensional analogues. Points are zero-dimensional objects. As the simplest geometric concept, point is usually the most basic component in geometry, physics, vector graphics and other fields. Points form lines, lines form surfaces, and points are the most basic components in geometry. Generally speaking, points are regarded as zero-dimensional objects, lines as one-dimensional objects and surfaces as two-dimensional objects. Points become lines, and lines become faces.

The Basic Elements of Plane Space-Point, Line and Plane

"Relying on the spiritual investigation of personal art, this element analysis is a bridge to the internal rhythm of the work." -Wassily Kandinsky's Point, Line and Surface (China Social Sciences Press)

Any art has its own language, and the language of plastic arts consists of the following morphological elements: point, line, surface, body, color, texture and so on. First, let's learn about points, lines and surfaces. 1, Zhifen

Point, Ci Hai's explanation is: tiny traces. In geometry, a point only has its position, while in morphology, it also has modeling elements such as size, shape, color and texture. In nature, the sand and stones on the seashore are dots, the raindrops on the glass windows are dots, the stars at night are dots, and the dust in the air is dots.

2. Point expression

The image of specific points can be expressed by various tools. Different forms of points show different visual effects, and with the increase of their area, the feeling of points will also weaken. If we look down at the pedestrians in the street in the sky, we will feel "point", and when we return to the ground, the feeling of "point" will disappear.

In the picture space, a point has a strong centripetal force, which can form the focus of vision and the center of the picture, showing the front of the point; On the other hand, dots will also make the picture space present a chaotic state, showing the negativity of dots, which is also a noteworthy problem in the specific application of dots.

Points also have the characteristics of dominance and invisibility. Implicit points exist at the intersection of two lines, the top or end of lines, etc.

3, the composition of the point

(1), the composition of ordered points: here, the shape, area, position or direction of the main points are arranged in a regular form, or the same repetition, or orderly gradual change. Points are often arranged sparsely and densely, which forms the performance requirements of graphics in space. At the same time, rich and orderly dot composition will also produce a delicate sense of space and form a three-dimensional sense. In composition, points form a whole relationship, and their arrangement is combined with the whole space, so the visual tendency of points is line and surface, which is the reasonable composition of points.

(2) Composition of free points: The shape, area, position or direction of the main points here are arranged in a free and irregular form, which often presents rich, flat and scattered visual effects. If you use this to represent the part in space, you can give full play to its advantages, such as symbolizing the stars in the sky or as a decoration for the light and dark levels of graphics.

4, the relationship between point and line

The points move into a line. 1, cognitive line

A straight line is the locus of point motion and the starting point of surface motion. In geometry, lines only have position and length, while in morphology, lines also have modeling elements such as width, shape, color and texture. During Bauhaus's teaching, the painter Klee once defined lines as follows: lines are points in motion. More importantly, he divided lines into three basic types: positive line, negative line and neutral line. The positive ground line is free and constantly moving, whether there is a specific destination or not; Once a line copies a consistent figure, it becomes a neutral line; If this figure is colored again, then this line will become a negative line again, because at this time, color has already acted as a positive factor. (See Frank whitford's Bauhaus)

Linearly speaking, lines have neat geometric lines and freehand brushwork lines. The image itself has no lines, but a line is formed at the turning point of the plane, and the shape is defined by lines, which is what we call contour lines. It is the general expression of the artist's material.

Usually we divide the lines into the following two categories:

1, straight line: parallel line, vertical line (vertical line), diagonal line, broken line, dotted line, broken line, etc. In Ci Hai, a straight line means that a point moves in a certain (including reverse) direction on a plane or space, and the trajectory formed is a straight line. Only one straight line can be drawn when two points are illuminated.

2. Curves: arc, parabola, hyperbola, circle, ripple line (wavy line), serpentine line, etc. The meaning of curve in Ci Hai is: the trajectory of a point whose direction changes on a plane or space due to certain conditions.

3. Expression of lines

As the most basic language of plastic arts, line has always been concerned because of its strong generality and expressiveness. Chinese painting has various linear changes of "eighteen strokes", and also has the pursuit of linear charm such as "using the bone to use the pen" and "pen death" Learning painting always starts with lines, such as sketching and sketching, which mostly appear in the form of lines. In modeling, lines play a vital role. It is not only the contour line that determines the shape of an object, but also can describe and express the internal structure of the object. For example, lines can outline the texture of patterns, and even convey the expression of objects through lines.

William hogarth wrote in The Analysis of Beauty that straight lines are only different in length, so at least they are decorative. The combination of straight lines and curves forms a composite line, which is more diverse than simple curves and therefore more decorative. Corrugated lines are composed of two opposite curves, which change a lot, so they are more decorative and pleasing to the eye. Huo Jiasi called them "beautiful lines". The serpentine line is called "elegant line" because it can rise and fall in different ways at the same time and make people's attention move with its continuous change in a pleasant way. Huo Jiasi also said that when drawing a curve on paper with a pen or pencil, the hand movements are elegant.

Variety of lines is a powerful means of expression in plastic arts, the most expressive and energetic element in image and picture space, and also a visual expression element that artists at home and abroad have always loved. When talking about our Neolithic Mid-levels painted pottery, our aesthete Yang Xin wrote: "Its decorative patterns are lines, all kinds of lines, such as thick lines, thin lines, tooth lines, wavy lines, red lines and black lines. , which is generated from one line. Many regular lines are combined in a repetitive and staggered way, which makes people feel harmonious, as if lines were used to form a' silent symphony'. " (Yang Xin and Gan Lin (aesthetic principle))

4, the relationship between line and surface

The linear motion can be a straight line, a surface or a plane.

5, the state of the line

Intersection; Parallel; Different faces. 1, knowing the surface

Enlarged points form a surface, and closed lines form a surface. Dense points and lines can also form a surface. Morphologically, human face also has modeling elements such as size, shape, color and texture, and at the same time, human face is the presentation of "image", so human face is "shape"

2. The types of cognitive surfaces can usually be divided into the following four categories.

(1), geometric shape: also known as inorganic shape, is a curved surface formed by combining straight lines, curves or straight curves in a mathematical way. Such as special rectangle, square, general rectangle, triangle, trapezoid, diamond, circle, pentagon, etc. , with mathematical simplicity, lightness, composure and sense of order, is widely used in the modeling design of buildings and practical utensils.

(2) Organic form: an organism form that cannot be obtained by mathematical methods. It has natural development, sense of order and regularity, rhythm of life and simple visual characteristics. For example, pebbles, maple leaves, biological cells, melon and fruit shapes and human eyes in nature are all organic.

(3) Accidentality: refers to a natural or man-made accidental form, the result of which is uncontrollable, such as spilling or dripping ink or water at will, wormholes on leaves, inadvertently torn pieces of paper, etc. , with unrepeatable chance and vitality.

(4) Irregularity: refers to the man-made free form, which can use all kinds of free and freehand linear forms at will, and has strong modeling characteristics and distinct personality.

3. Facial expression

Facial expressions are presented in different morphological types. In the two-dimensional range, facial expression is always the most abundant, and the picture often changes with the shape, fidelity, size, position, color and texture of the face (image), forming a complex modeling world, which is the concrete embodiment of the modeling style.

The most representative expressions of "straight face" and "curved face" in Face: straight face (all faces composed of straight lines) has stable and resolute masculine characteristics, and its characteristics are strengthened with the strengthening of various factors. Surfaces (all surfaces formed by curves) have dynamic and soft feminization characteristics, and their characteristic degree is strengthened (or weakened) with the changes of their factors.

4. Composition of the surface

The composition of surface is the composition of form, and it is also the key to learning and mastering in plane composition. It involves basic types, bones and other concepts, which we will discuss one by one in the following chapters. Here, let's first discuss the relationship between faces in plane space. When two or more faces appear in the plane space at the same time (our picture), there will be various relationships between them.

The relationship between faces can be summarized as: (1). Separation: Faces are separated, keeping a certain distance, showing their respective forms in plane space, and space and faces form a mutually restrictive relationship.

(2) Encounter: also known as tangency, refers to the tangency between the surface and the contour line of the surface, and thus forms a new shape, which makes the image in the plane space rich and complex.

(3) Superposition: One surface is superimposed on another surface, thus forming a sense of layering between surfaces in space.

(4) Penetration: The surfaces overlap each other, and the overlapping shapes are transparent. Through the shape of the upper layer, we can see the covered part of the lower layer, and new shapes appear in the overlapping part between faces, thus making the image rich and varied and full of sense of order, which is a good image processing method in composition.

(5) Overlapping: faces overlap with each other, emphasizing the new image generated by overlapping, which can present a new image in the plane space or allow three images to coexist.

(6) Fusion: also known as combination, refers to the overlapping of surfaces. On the same plane level, faces combine with each other to form a new image with a large area, which will make the image in space whole and fuzzy.

(7) Shortening: a part of a face is covered by another face, and the two shapes are subtracted to keep the shape covered on it, leaving a residual image after being covered by another image, an unexpected new image.

(8) Overlapping: The same two faces, one covering the other, form a completely overlapping image, resulting in special image performance, making it meaningless in image composition. 1, the most important function of a point is to indicate its position and focus. A point is formed by comparing with a surface. If the same point covers all or most areas of a plane, it is a surface. If it appears in a plane many times, it can be understood as a point.

2. The connection between points forms a line, or the regular extension of points along some aspects can become a line, which emphasizes direction and shape;

3. The connection of more than three points on the plane can form a surface, and the closing or unfolding of lines on the plane can also form a surface, emphasizing the shape and area;

The above three points can summarize the subtle relationship between points, lines and surfaces.