What are the methods to find dihedral angle?

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. Projective 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.

4. Three-perpendicular theorem and its inverse theorem method: First, find the perpendicular of a plane, and then connect two vertical feet with the perpendicular of the vertical foot as the edge to get the plane angle of dihedral angle.

5. Vector method: Make the normal vectors of two half-planes, respectively, and get them from the vector included angle formula. A dihedral angle is an included angle or its complementary angle.

6. Conversion method

Among them, points (1) and (2) mainly calculate the plane angle of dihedral angle according to the definition, and then use the sine and cosine theorem of triangle to solve the triangle.

The dihedral angle is generally at the intersection of two planes, and appropriate points, usually endpoints and midpoints, are taken. After this point, make the perpendicular lines of the intersection line on two planes respectively, and then put these two perpendicular lines into a triangle to consider. Sometimes, two parallel lines perpendicular to each other are often made into a more ideal triangle.

The plane angle of dihedral angle is directly obtained by the formula s projection =S inclined plane cosθ. The key to using this method is to find out the inclined polygons and their projections on the relevant planes from the graphics, and their areas are easy to obtain.

You can also use analytic geometry to find the coordinates of the normal vector N 1 and N2 of two planes. Then according to n1N2 = | n1| N2 | cos α, θ = α is the included angle between two planes. It should be noted here that if both normal vectors are vertical planes and point to two planes, then the included angle θ between the two planes is = π-α.