Define a straight line in the plane, divide the plane into two parts, each part is called a half plane, and the figure composed of two half planes from a straight line is called a dihedral angle. (This straight line is called the edge of dihedral angle, and each half plane is called the plane of dihedral angle. The plane angle of dihedral angle takes any point on the edge of dihedral angle as the endpoint, and two rays perpendicular to the edge are made in two planes, and the angle formed by these two rays is called the plane angle of dihedral angle.

A dihedral angle whose plane angle is a right angle is called a straight dihedral angle.

Definition of two planes perpendicular: Two planes intersect, and if the dihedral angle they form is a straight dihedral angle, they are said to be perpendicular to each other. [Edit this paragraph] The range of dihedral angle is 0≤θ≤π.

There are six common methods to find the dihedral angle when the intersection angle is 0 < θ < π, the plane is * *, and θ = π or 0:

1. Definition method

2. Vertical plane method

3. Projective theorem

4. Three vertical theorems

5. Vector method

6. Conversion method

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.

The usual solution of dihedral angle;

(1) defines the plane angle of dihedral angle;

(2) As a vertical plane of dihedral angle, the angle formed by the intersection of vertical plane and dihedral angle is the plane angle of dihedral angle;

(3) Making the plane angle of dihedral angle by using the three perpendicular theorem (inverse theorem);

(4) Find the dihedral angle in the space coordinates.

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.

Basic steps of calculating dihedral angle

(1) Plane angle for making dihedral angle:

A: use the midpoint of the bottom of the isosceles triangle as the plane angle;

B: Use the vertical line of the plane (three vertical lines theorem or its inverse theorem) as the plane angle;

C, a straight line perpendicular to the edge passes through the vertical plane of the edge to form a plane angle;

D: Use two parallel lines with dihedral angles without edges as plane angles.

(2) Prove that the angle is a plane angle;

(3) Inducing into a triangle to find the angle.

In addition, it can also be obtained by using space vectors. The relationship between dihedral angle and plane angle is measured by its "plane angle", and the plane angle of dihedral angle is equal to dihedral angle.