The figure formed by two half planes starting from a straight line is called dihedral angle. This straight line is called the edge of dihedral angle. The two half planes are called the face of dihedral angle.[1]
1. Half planeA straight line divides a plane into two parts, each of which is called a half plane.
2. Plane angle:Take any point on the common line of the dihedral angle as the endpoint, and make two perpendicular to the common line in two planesradialThe angle formed by these two rays is called dihedral anglePlane angle。The size of dihedral angle can be represented by plane angle.
3. Straight dihedral angle: Plane angle isright angleThe dihedral angle of is calledStraight dihedral angle。Mutually perpendicular planes: two planes intersecting at right angles are called mutually perpendicular planes.[2]
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A straight line in a plane divides the plane into two parts, and each part is called a half plane.The figure formed by two half planes starting from a straight line is called dihedral angle.This straight line is called the edge of the dihedral angle, and the two half planes are called the faces of the dihedral angle.The size of a dihedral angle can be measured by its plane angle. When the plane angle of a dihedral angle is several degrees, it is said that the dihedral angle is several degrees.[3]Dihedral angle can also be seen as a figure formed by the initial position and final position of a half plane starting from a straight line rotating around the straight line.
The size of the plane angle of a dihedral angle is independent of the position of its vertex on the edge.If two dihedral angles can coincide completely, they are equal. If the plane angles of two dihedral angles are equal, the two dihedral angles are equal.Conversely, the plane angles of equal dihedral angles are equal.
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The properties of dihedral angle are:
(1) Any two plane angles of the same dihedral angle are equal, and the plane angle of the larger dihedral angle is larger.
(2) The plane angle corresponding to the sum or difference of two dihedral angles is the sum or difference of the plane angles corresponding to the original two dihedral angles.
(3) A dihedral angle can be bisected, and the bisecting face is unique.
(4) The dihedral angles of opposite edges are equal.[4]
Plane angle method
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The common methods for making plane angles of dihedral angles are as follows:
1、Definitional method: Take a point A on the edge, and then make the perpendicular lines of point A on the edge in two planes.Sometimes, perpendicular lines of edges can be made in two planes, and then pass through one of themPerpendicular footMake a parallel line to another vertical line.
2、Vertical plane methodIf the plane is perpendicular to the edge, the angle formed by the intersection of the vertical plane and the dihedral angle is the plane angle of the dihedral angle
3、Area projective theoremThe cosine value of the dihedral angle is equal to the ratio of the projected area of one half plane on another half plane to the area of the plane itself.That is, the formula cos θ=S'/S (S' is the projective area and S is the slope area).The key to using this method is to find out the oblique polygon and its projection on the relevant plane, and their area can be easily obtained.
5、Vector method: Make the normal vectors of the two half planes respectively, and obtain them from the vector angle formula.The dihedral angle is the included angle or its supplementary angle.
6. Transformation method: find a point P on one half plane α of the dihedral angle α - l - β, find the distance h from P to β and the distance d from P to l, then arcsin (h/d) (dihedral angle is acute angle) or π - arcsin (h/d) (dihedral angle is obtuse angle) is the size of dihedral angle.
7、Distance of straight lines in different planesMethod: Assume the dihedral angle is C-AB-D, where AC and BD are mutually non planar straight lines and AC ∨ AB, BD ∨ AB (that is, AB is the common vertical line of non planar straight lines AC and BD).Let AB=d,CD=l，AC=m，BD=n，according to
To find the angle θ formed by the straight line of the different surface.When using this method to calculate θ, it is necessary to judge whether the dihedral angle is acute or obtuse from the image.If it is a sharp angle, take a positive sign;Obtuse angle, then take the minus sign.After finding θ, if the dihedral angle is an acute angle, then the size of the dihedral angle is θ;Obtuse angle, then the dihedral angle is π - θ.
Among them, points (1) and (2) are mainly used to find the plane angle of dihedral angle according to the definition.
The dihedral angle is generally on the intersection line of two planes, and the appropriate point is taken, usually the endpoint and midpoint.Go through this point and do it on two planesIntersecting lineOfvertical, and then put the two perpendicular lines into a triangle.Sometimes two perpendicular lines are often madeParallel lineMake them in a more ideal triangle.
Solution method
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Geometric method
(1) Plane angle for making dihedral angle
A: Use the midpoint of the base of isosceles (including equilateral) triangle as the plane angle;
B: Use the vertical line of the face (three vertical line theorem or itsInverse theorem）Make plane angle;
C: Use a straight line perpendicular to the edge to make a plane angle through the vertical surface of the edge;
D: Use two parallel lines without edge dihedral angle as plane angle.
Direction of.If one of the two normal vectors points to the inside of the dihedral angle and the other points to the outside of the dihedral angle, then the size of the dihedral angle is θ.If two normal vectors point inside or outside the dihedral angle at the same time, the dihedral angle is π - θ.
