A surface is the trajectory of a line or curve under certain constraints. This moving straight line or curve is called curved surface a bus or bus bar ； The generatrix at any position on the surface is called a isoline. The constraints on the movement of the bus are called kinematic constraints. In the constraint conditions, the straight line or curve that controls the movement of the bus is called the conductor; The plane that controls the movement of the bus is called the guide plane. [2] When the moving line moves according to certain rules, the surface formed is called regular surface; When a moving line moves irregularly, the surface formed is called an irregular surface. The generatrix that forms the surface can be a straight line or a curve. If the surface is formed by linear motion, it is called a linear surface (such as cylindrical surface, conical surface, etc.); The surface formed by curve movement is called curve surface (such as sphere, torus, etc.). Two continuous straight isolines of a straight face are parallel or intersect each other (that is, they are on the same plane). This kind of surface that can be unfolded into a plane without deformation is a developable surface. If two consecutive straight isolines cross each other (that is, they are not on the same plane), they are undevelopable surfaces.
The representation of curved surface is similar to that of plane. The most basic requirement is to determine the projection of each geometric element of the curved surface, such as bus, wire, guide surface, etc. In addition, in order to clearly express a surface, it is generally necessary to draw the outline of the surface to determine the scope of the surface. [3]
According to different classification standards, surfaces have many different classification methods.
1) Classification according to bus movement mode
(1) Revolving surface - the curved surface formed by the generatrix rotating around an axis;
(2) Non rotating surface the surface formed by the movement of the generatrix according to other constraints.
2) Classification according to the shape of bus
(1) Ruled surface - any surface that can be formed by the movement of straight generatrix, such as cylindrical surface circular cone surface , elliptical cylinder, elliptical cone Hyperbolic paraboloid 、 Conical surface And columnar surface; (2) Hyperbolic surface - a surface that can only be formed by the movement of curve generatrix, such as sphere, torus, etc.
The same surface may be formed by several different forms of motion. For example, a cylindrical surface can be seen as either a straight line rotating around an axis parallel to it, or a circle translating along the axis.
3) Classification according to whether the surface can be developed into a plane
(1) Developable surface: a surface that can be expanded into a plane. Such as cylindrical surface and conical surface;
(2) Undevelopable surface: a surface that cannot be expanded into a plane, such as an elliptical surface Elliptic paraboloid Curved revolving surface. Generally, only ruled surfaces can be divided into developable surfaces and undevelopable surfaces. Hyperbolic surfaces are all undevelopable surfaces. [2] The surface formed by the straight generatrix moving along a curved wire and always parallel to another straight wire is called a cylinder. Cylindrical surface is usually named after the shape of the intersection line obtained by cutting the surface with a section plane (normal section) perpendicular to the cylindrical isoline. If the shape of the intersection line is circle, it is called cylindrical surface; If the intersection line is an ellipse, it is called an elliptical cylinder. [1]
The front projection of the oblique elliptic cylinder is parallelogram The upper and lower sides are the projections of the top and bottom surfaces of the oblique elliptical cylinder, and the left and right sides are the projections of the front turning contour of the oblique elliptical cylinder. The top turning contour line is tangent to the horizontal projection of the top circle and the bottom circle. The side projection of an oblique elliptical cylinder is a rectangle. [4] When a straight generatrix moves along a curved wire and always passes through a fixed point (guide point), the resulting surface is called a cone. Similar to a cylinder, a cone is named after the intersection of the normal section perpendicular to the axis and the cone. If the shape of the intersection line is a circle, it is called a conical surface; If it is an ellipse, it is called an elliptical cone. [1]
If the axis of the elliptical cone is inclined to the bottom of the cone, it is called an oblique elliptical cone. The front projection of an oblique elliptical cone is a triangle. The main difference between it and the front projection of a regular cone is that this triangle is not an isosceles triangle. There are two dotted lines in the triangle, one of which coincides with the bisector of the cone top angle, is the axis of the cone, and the other is the line connecting the center of the circle. The horizontal projection of the inclined elliptical cone is a circle reflecting the real shape of the bottom circle (conductor) and two turning contours tangent to the circle. The side projection of an oblique elliptical cone is an isosceles triangle.
There are two ways to draw a cone: ① draw several straight lines with density in the view reflecting the real length of the axis, and draw several even straight lines in the view reflecting the real shape of the arc at the bottom of the cone; ② Several grade lines are drawn in each view of the cone. Note that the direction of the taper slope line should point to the cone top. [4]
The surface of revolution formed by a straight line rotating around an axis at the intersection with it is called Hyperboloid of one sheet 。 The smallest circle formed by the rotation of the point closest to the axis on the bus is called throat circle. The plane containing the axis is used to cut the hyperboloid of one sheet, and the shape of its cross section line is hyperbola. Therefore, the hyperboloid of one sheet can also be regarded as rotating around its imaginary axis with hyperbola as the generatrix. [1] The straight generatrix moves along the cylindrical axis of the curved conductor and the cylindrical axis of the straight conductor, and always intersects with the axis of the cylinder at an angle of 90 °. The surface thus formed is called a positive helicoid. The adjacent two isolines of a positive helicoid cross each other, so it is an undevelopable ruled surface. [1]
