What Does a Geometer Do?

Geometric solid, also called solid, is one of the basic concepts of solid geometry. The concept of geometry arises from people's mathematical abstraction of various objects in the objective world. When people only consider mathematical properties such as the shape, size, and position of an object, they do not consider its physical, chemical, biological, and social attributes. At that time, the concept of geometry is obtained. In geometry, people refer to a finite body surrounded by a number of geometric surfaces (planes or curved surfaces) as the geometry, and the surface surrounding the geometry is called the interface or surface of the geometry. The intersection of different interfaces The line is called the ridgeline of the geometry. The intersection of different ridgelines is called the apex of the geometry. The geometry can also be regarded as a limited space area divided by several geometric faces in space. The first study of solid geometry is the geometric properties of some simpler geometry. , Such as polyhedrons, rotating bodies, and their combinations ^[1] .

Geometry: (1) When we only study the shape and size of an object, but not other properties (such as color, weight, hardness, etc.), we call this object a geometry, or body for short. For example, the carton and wooden block in the picture, although their color, weight, hardness, and materials are not the same, but as long as their shape and size are the same, we consider them to be two identical geometric bodies. In fact, since the carton and the block are the same shape and size, they are two identical cuboids ^[2]

Figures drawn from the same space geometry viewed from three different locations are called three views. It mainly includes three basic views: the front view, the top view, and the left view. This is an abstract expression of the object geometry in the engineering world. The view projected from the front to the back of the object is called the main view -it can reflect the front shape of the object; the view projected from the top to the bottom of the object is called the top view -it can reflect the top shape of the object; it is projected from the left to the right of the object The resulting view is called the left view -it reflects the shape of the left side of the object. FIG. 4 is a three view of FIG. 3, and FIG. 6 is a three view of FIG. 5.

The body is surrounded by faces. Faces have flat surfaces and curved surfaces. For example, a cuboid is surrounded by six planes; a sphere is surrounded by a curved surface; a cylinder is surrounded by a curved surface and two planes. According to the characteristics of the main element of the body-the surface, the body can be divided into two categories:

The first type is curved surface geometry with curved surfaces participating in it, also called curved solid, such as cylinders and spheres.

The second type is a plane geometry surrounded by planes, that is, a polyhedron surrounded by several plane polygons , such as a prism or a cube.

Geometry plane stereo

The basic geometry surrounded by several planes is called a plane solid. There are two types of planar solids: prisms and pyramids. The ridge lines of the prism are parallel to each other, the ridge lines of the pyramid intersect at one point, and the pyramid is truncated to form a pyramid. Planar solids are named after their number of edges, such as quadrangular prism, hexagonal prism, pentagonal pyramid, triangular pyramid, quadrangular pyramid, etc. ^[3] . As shown in FIG. 3 to FIG. 6, in FIG. 3 and FIG. 4, the prism is surrounded by a prism face, a top face, and a bottom face. The intersection of two adjacent prism faces is called a ridge line. In Fig. 5 and Fig. 6, the pyramid is surrounded by a face and a bottom surface, and each face is a triangle having a common vertex.

image 3	Figure 4
Figure 5	Image 6

Geometry Surface Solid

The basic geometry enclosed by a surface or a surface and a plane is called a curved solid. Common three-dimensional surfaces include cylinders, cones, and spheres. Their curved surfaces can be regarded as the busbars formed around the axis. Therefore, this type of curved solid is also called a rotary body , and its curved surface is called a rotary surface.

Formation of the surface of revolution

FIG. 7 shows the process of forming the turning surface. Figure 7 (a) shows that a straight generatrix rotates around an axis parallel to it to form a cylindrical surface ; Figure 7 (b) shows a straight generatrix rotates around an axis that intersects it to form a conical surface; Figure 7 (c) shows when the generatrix is circular When the axis is its diameter, the generatrix rotates around the axis to form a spherical surface.

Figure 7 (a)

Figure 7 (b)

Figure 7 (c)

Primes and contours

(1) Prime line: Each specific position of the generatrix during rotation is called the prime line of the curved surface. A surface is a collection of prime lines.

(2) Contour prime line: After the position of the curved solid in the three-projection plane system is determined, the prime lines constituting the outline of the object during projection are called contour prime lines. Obviously, when the axis of the cylinder is perpendicular to the H plane, the cylinder has four contour prime lines, two of which are the contour prime lines in the front view (the two leftmost and rightmost prime lines on the cylinder surface), and the other two are in the side view direction. Contour primes (the first and last primes on a cylindrical surface). In the same way, there are four contour lines on the conical surface; there are three contour lines on the spherical surface, which are the maximum circle of the normal flat, the maximum circle of the horizontal and the maximum circle of the side.