The practice of dihedral angles, how to make the plane angles of dihedral angles

Dihedral angle is a type of problem in the college entrance examination, and almost every year the science paper will involve the method of finding the dihedral angle. Some students often don't know where to start when solving this content, and today we will sort out the common methods, hoping to help you.

1. Definitions.

It refers to a method of making a straight line perpendicular to the edge in the two planes at any point on the edge of the dihedral angle, then the angle formed by the two straight lines is the plane angle of the dihedral angle, and then the plane angle is found in the plane.

Second, three perpendicular line method.

It refers to the method of using the three-perpendicular theorem to construct the plane angle of the dihedral angle according to the idea of "perpendicular to the projection, it is also perpendicular to the oblique line", and then find the plane angle.

3. Vertical surface method.

It refers to the use of a plane perpendicular to the edge to cut the dihedral angle, then the two faces of the section and the dihedral angle must have two intersection lines, and the angle formed by these two intersection lines is the plane angle of the dihedral angle, and then find a method of its plane angle.

Fourth, the area projection method.

The so-called area projection method is based on the relationship between the figure and its projective area on a certain plane, and uses the area of the projection to calculate the cosine value of the original area equal to the dihedral angle. This method is often used for edgeless dihedral corners.

5. Normal vector method.

The normal vector method is a method of finding the dihedral angle by finding the angle formed by two vectors perpendicular to the dihedral angle, and then using the relationship that this angle is equal or complementary to the plane angle of the dihedral angle. (The question of how to determine equality or complementarity will be announced in the near future).

Sixth, the perpendicular method.

It refers to the use of the pending coefficient method to determine the vertical foot, and then the use of the formula to find the size of the dihedral angle.

1. Definition method (respectively to the intersection line as a perpendicular line, to find the angle between the two lines) 2, three perpendicular line method: through a point in a certain half of the plane to the other half of the plane and the intersection line as a perpendicular line, to make a projection by the tan angle to solve, where cos dihedral angle = projective area of the original area. 3. Vertical surface method:

Find the perpendicular surface of the intersection line, and make the intersection line between the vertical plane and the half-plane, and find the angle. 4. Vector method first establish a Cartesian coordinate system and find the coordinates of each point. Let the normal vector of the surface s1 and the normal vector of the surface s2.

Then sum the angles of the cosine. According to the image observation and direction. If one of the two normals points inside the dihedral and the other points outside the dihedral, then the magnitude of the dihedral is .

If both normals point to the inside or outside of the dihedral at the same time, the magnitude of the dihedral is -

How will the three-dimensional geometric space angle test center be tested in the college entrance examination, and is there a more convenient and faster way to solve the problem? The A+ classroom area will give you a different spatial angle solution today!

Dihedral angle is a frequent visitor to science and mathematics in the college entrance examination, with a probability of more than 80%, usually the most direct test is to let you find the dihedral angle or various trigonometric values of the dihedral angle, and the other is to give the trigonometric value of a certain dihedral angle as a condition.

When students encounter the first situation, they usually solve it by establishing a spatial Cartesian coordinate system and finding the normal vector, but when they encounter the second situation, more students are at a loss.

The reason is that since the introduction of the spatial Cartesian coordinate system in the college entrance examination in 2004, many students have begun to pay less attention to the essence of dihedral angles or other spatial angles, and do not know how to draw the plane angles of space angles, let alone use other methods to solve this type of problem. However, a lot of precious time was lost, and the final score did not meet the ideal.

If the plane angle of the dihedral angle is required, then the most common method is the turning method of the three-perpendicular theorem. It is to make a perpendicular line from one point to another, and draw a perpendicular line from the vertical foot to the edge of the dihedral angle, and the plane angle of the dihedral angle appears.

If it is convenient to calculate the area, the original area is used to remove the projected area, which is the cosine value of the dihedral angle, then the area projection method can be used; The angle between the normal vectors of two planes is found by the spatial coordinate system; Find a straight line perpendicular to the edge of the dihedral angle from the two sides (not necessarily intersecting), and then the size of the dihedral angle is transformed into the problem of the angle between the two straight lines of the opposite plane.

Representation method. Corners are usually represented by three letters: the letters of the dots on the two sides are written on both sides, and the letters on the vertices are written in the middle. The corners in the overview diagram are represented by AOBs.

However, in the absence of confusion, it is also represented directly by the letter of the vertex, such as the corner o. In mathematical formulas, the Greek letter ( , which is used to indicate the size of the horn.

To avoid confusion, symbols are generally not used to indicate angles.

The length of the arc cut by the angle on the circle divided by the radius of the circle is generally denoted as rad. The radian is a measure of the angle specified in the International System of Units, but it is not the legal unit of measurement in China, and the angle is the legal unit of measurement of the angle in China. In addition, radians have a wide range of applications in mathematics and trigonometry.

Dihedral angle is a concept used in 3D geometry to measure the angle between two planes. For two planes that do not coincide, the angle between them can be represented by a dihedral angle. The value range of the dihedral angle is [0, that is, from 0 to , including 0 and .

When the two planes coincide, the dihedral angle is 0, indicating that the two planes overlap each other and there is no included angle.

When two planes are perpendicular to each other, the dihedral angle is 2, indicating that the two planes are orthogonal to each other, forming a right angle.

When two planes are not parallel but also not perpendicular, the dihedral value of the angle between them is between 0 and 2.

It should be noted that the range of values for dihedral angles only applies to the plane and line relations in Euclidean beats. In non-Euclidean geometry, such as spherical or hyperbolic geometry, the range of values for dihedral angles can vary dramatically.

There are two methods for solving dihedral angles: geometric method and vector method.

1. Geometric method:

1. Make a plane angle of dihedral angle.

2. Prove that the angle is a plane angle.

3. Summarize to find the angle of the triangle.

2. Vector method:

1. Establish a Cartesian coordinate system first, and find out the standard of each point.

2. Find the two vectors of the plane, and then find the normal vector.

3. Finally, find the cosine of the included angle.

1) Definition method (basic): find the angle between the two lines as perpendicular lines to the intersection line;

2) Perpendicular surface method (less used): find out the perpendicular surface of the intersection line, and make the intersection line of the vertical plane and the half-plane, and find the angle of the potato ling;

3) Three-perpendicular line method (commonly used): through a point in a certain half of the plane to the other half of the plane and the intersection line to make a perpendicular line, make a projection and solve by the tan angle;

4) Vector method (universal): make two half-plane normal vectors respectively, which are obtained by the vector angle formula, note that the angle is not a dihedral angle, but its complementary angle!

5) Projective area method (commonly used): the cosine value of the dihedral angle is equal to the ratio of the area of the projection of a certain half-plane in the other half-plane to the area of the plane itself.

Definition: The figure composed of two half-planes from a straight line is called a dihedral angle, this straight line is called the edge of the dihedral angle, and these two half-planes are called the faces of the dihedral angle.

The magnitude of the planar angle of a dihedral angle is independent of the position of its vertices on the edge. If the two dihedral angles can coincide completely, then they are said to be equal If the plane angles of the two dihedral angles are equal, then the two dihedral angles are equal. Conversely, the plane angles of equal dihedral angles are equal.

The size of the dihedral angle can be measured by its plane angle, and the plane angle of the dihedral angle is several degrees, which is said to be several degrees. The dihedral angle can also be seen as a half-plane from a straight line that rotates around this straight line, and the rock Ashi's initial and final positions are a figure.

1. Half plane: A straight line of the plane divides the plane into two parts, each of which is called a half plane.

2. Plane rough suspicion angle: take any point on the common straight line of the dihedral angle as the endpoint, and make two rays perpendicular to the common straight line in the two planes, and the angle formed by these two rays is called the plane angle of the dihedral angle. The size of the dihedral angle can be expressed in terms of planar angles.

3. Straight dihedral angle: The dihedral angle where the plane angle is a right angle is called a straight dihedral angle. Planes perpendicular to each other: Two planes that intersect at right angles are called planes perpendicular to each other.