Summary of calculation methods of dihedral angle

1. Definition: Passing through any point on the edge of dihedral angle and making a straight line perpendicular to the edge in two planes, the angle formed by these two straight lines is the plane angle of dihedral angle.

Second, use the projected area to find the dihedral angle. The dihedral angle formed by plane ABC and plane A is θ, and its projection on plane A is DBC, so the cosine of the dihedral angle formed by plane ABC and plane A is the ratio of the projected area to the original area.

3. Three perpendicular lines method. The triple perpendicular theorem refers to a straight line on a plane. If it is perpendicular to the projection of a diagonal line passing through this plane on this plane, then it is also perpendicular to this diagonal line. According to the idea of three perpendicular lines theorem, the plane angle of dihedral angle is constructed, and then the calculation method of plane angle of dihedral angle is obtained.

Fourth, the normal vector method. It is suitable for the problem of establishing rectangular coordinate system easily. Firstly, the angle formed by two vectors of two planes perpendicular to the dihedral angle is obtained, and the dihedral angle is obtained by using the equal or complementary relationship between this angle and the plane angle of the dihedral angle.

The common methods for making the plane angle of dihedral angle are as follows:

1. Definition method: Take a point A on the edge, and then perpendicular to the point A on the edge in two planes. Sometimes it is also possible to make the vertical lines of the sides on two planes, and then one of them is perpendicular to the other vertical line.

2. Vertical plane method: If 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 projection theorem: the cosine of 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 cosθ=S'/S(S' is the projected area and s is the inclined surface area). The key to using this method is to find out the oblique polygons and their projections on the relevant planes from the diagram, and their areas are easy to obtain.