How to find dihedral angles (the more detailed, the better)

A straight line in the plane divides the plane into two parts. Each part is called a half-plane. The figure composed of two half-planes starting from a straight line is called a dihedral angle (this straight line The edges are called dihedrals, and each half-plane is called a dihedral face). The size of the dihedral angle can be measured by its plane angle. The plane angle of the dihedral angle is how many degrees the dihedral angle is. A dihedral angle whose plane angle is a right angle is called a right dihedral angle.

Taking any point on the common straight line of the dihedral angle as the endpoint, draw two rays perpendicular to the common straight line in the two planes. The angle formed by these two rays The plane angle is called the dihedral angle. The size of the dihedral angle can be expressed by the plane angle.

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

The definition of two planes being perpendicular: if two planes intersect and the dihedral angle they form is a straight dihedral angle, the two planes are said to be perpendicular to each other.

0≤θ≤π (not less than 0°, not more than 180°)

(Note: Since the dihedral angle is a three-dimensional figure in space, we can convert 180° to 360 The other side of ° is regarded as 0°~180°)

There are six commonly used methods to calculate the plane angle of the dihedral angle:

1. Definition method?: Take it on the edge point A, and then draw perpendiculars through point A on the edge in two planes. Sometimes you can also draw the vertical lines of the edges in two planes, and then draw the parallel line of the other vertical line through one of the vertical feet.

2. Vertical plane method?: Draw a plane perpendicular to the edge, then the angle formed by the intersection of the vertical plane and the dihedral angle is the plane angle of the dihedral angle

< p>3. Area projection theorem: The cosine of a dihedral angle is equal to the ratio of the area of ??a certain half-plane projected onto another half-plane and the area of ??the plane itself. That is, the formula cosθ=S'/S (S' is the projective area, S is the slope area). The key to using this method is to find the slope polygon and its projection on the relevant plane from the figure, and their areas are easy to find.

4. The three perpendicular theorem and its converse method: first find the perpendicular of a plane, then pass the perpendicular foot to make the vertical line of the edge, connect the two perpendicular feet to get the plane angle of the dihedral angle .

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 of the half-planes α of the dihedral angle α-l-β, and find the distance h from P to β and the distance d from P to l, then arcsin( h/d) (the dihedral angle is an acute angle) or π-arcsin(h/d) (the dihedral angle is an obtuse angle) is the size of the dihedral angle.

7. Trihedral angle cosine theorem method: see related entries for details.

8. Three sine theorem method: see related entries for details.

9. Distance method of straight lines with different faces: Let the dihedral angle be C-AB-D, where AC and BD are straight lines with different faces and AC⊥AB, BD⊥AB (that is, AB is a straight line with different faces) the common perpendicular of straight lines AC and BD). Assume AB=d, CD=l, AC=m, BD=n, and according to

, find the angle θ formed by straight lines on different planes. To use this method to find θ, you must first determine whether the dihedral angle is an acute or obtuse angle from the image. If it is an acute angle, take a positive sign; if it is an obtuse angle, take a negative sign. After θ is found, if the dihedral angle is an acute angle, then the size of the dihedral angle is θ; if it is an obtuse angle, then the size of the dihedral angle is π-θ.

Among them, points (1) and (2) are mainly used to find the plane angle of the dihedral angle according to the definition, and then use the sine and cosine laws of the triangle to understand the triangle.

The dihedral angle is generally on the intersection line of two planes, taking appropriate points, often the endpoints and midpoints. Draw the perpendiculars of the intersecting lines on the two planes through this point, and then put the two perpendiculars into a triangle and consider them. Sometimes it is also common to make parallel lines between two perpendiculars so that they form a more ideal triangle.

Geometry method

(1) Determine the plane angle of the dihedral angle

A: Use the midpoint of the base of an isosceles (including equilateral) triangle to calculate Plane angle;

B: Use the perpendicular line of the surface (the three perpendicular theorem or its converse theorem) to make the plane angle;

C: Use the straight line perpendicular to the edge, by making the edge The vertical plane is used as the plane angle;

D: Use two parallel lines with no edge dihedral angle as the plane angle.

(2) Prove that the angle is a plane angle

(3) Summarize the angle calculation into triangles