Chapter I Space Geometry

1. 1 Structural characteristics of columns, cones, platforms and balls

1.2 Three Views and Straight Views of Space Geometry

1 three views:

Front view: from front to back

Side view: from left to right

Top view: from top to bottom

2 the principle of drawing three views:

Long alignment, high alignment and equal width

3 orthographic drawing: oblique drawing.

4. The steps of oblique two drawing method:

(1). Lines parallel to the coordinate axis are still parallel to the coordinate axis;

(2) The length of the line parallel to the Y axis becomes half, while the length of the line parallel to the X and Z axes remains unchanged;

(3) the painting method should be written well.

5. Step of drawing a cuboid obliquely: (1) Draw an axis (2) Draw a bottom (3) Draw a side (4) for drawing.

1.3 surface area and volume of space geometry

(A) the surface area of space geometry

1 Surface area of prism and pyramid: the sum of the areas of each face.

2 Surface area of cylinder

3 Surface area of cone

4 Surface area of frustum of a cone

5 Surface area of ball

(2) the volume of space geometry

1 volume of cylinder

2 the volume of the cone

3 Volume of platform body

4 the volume of the sphere

Chapter II Position Relationship between Straight Line and Plane

2. 1 The positional relationship among points, lines and surfaces in space

2. 1. 1

1 plane meaning: the plane is infinitely extensible.

2 plane drawing and representation

Drawing method of (1) plane: the plane placed horizontally is usually drawn as a parallelogram with an acute angle of 450, and the horizontal side is drawn as twice as long as the adjacent side (as shown in the figure).

(2) Plane is usually represented by Greek letters α, β, γ, etc. Such as plane α, plane β, etc. , you can also use capital letters to represent four vertices or two opposite vertices of a planar parallelogram, such as planar AC and planar ABCD.

Three axioms:

(1) axiom 1: If two points on a straight line are in a plane, then the straight line is in this plane.

Symbols are expressed as

A∈L

b∈L = & gt； L α

A∈α

B∈α

Axiom 1 is used to judge whether a straight line is in a plane.

(2) Axiom 2: When three points that are not on a straight line intersect, there is only one plane.

The symbols are as follows: A, B and C * * * line => There is only one plane α,

Make a∈α, b∈α, c∈α.

Axiom 2 function: the basis for determining a plane.

(3) Axiom 3: If two non-coincident planes have a common point, then they only have a common straight line passing through the point.

The symbol is: p ∏ α ∪ β = > α∪β= L, and P ∈ L.

Axiom 3 function: the basis for judging whether two planes intersect.

2. The positional relationship between straight lines in1.2 space

Two straight lines in 1 space have the following three relationships:

Intersecting straight lines: there is only one common point on the same plane;

Parallel straight lines: there is no common point in the same plane;

Out-of-plane straight lines: they are different in any plane and have nothing in common.

Axiom 4: Two lines parallel to the same line are parallel to each other.

The symbol is expressed as: Let A, B and C be three straight lines.

a∑b

c∑b

Key point: Axiom 4 essentially says that parallelism is transitive and applies to both plane and space.

Axiom 4 function: the basis for judging the parallelism of two straight lines in space.

3 Equiangular Theorem: If the two sides of two angles in space are parallel, then the two angles are equal or complementary.

4 note:

The angle formed by (1)a' and b' is only determined by the mutual position of A and B, and has nothing to do with the choice of O. For simplicity, the point O is generally taken on one of the two straight lines;

② The angle θ∈(0,) formed by two straight lines in different planes;

(3) When the angle formed by the straight lines of two different planes is a right angle, we say that the straight lines of two different planes are perpendicular to each other and marked as A ⊥ B;

(4) Two straight lines are perpendicular to each other, including vertical plane and vertical plane;

⑤ In calculation, the angle formed by two straight lines in different planes is usually converted into the angle formed by two intersecting straight lines.

2. 1.3—2. 1.4 positional relationship between straight line and plane, and plane and plane.

1, there are three positional relationships between a straight line and a plane:

(1) The straight line is in the plane-there are countless things in common.

(2) A straight line intersects a plane-there is only one common point.

(3) Straight lines are parallel in the plane-there is no common point.

It is pointed out that when a straight line intersects or is parallel to a plane, it is called a straight line out of the plane, which can be expressed by α.

AαA∪α= A A∪α

2.2. Determination of parallelism between straight line and plane and its properties

2.2. 1 Determination of parallelism between straight line and plane

1. Theorem for judging that a straight line is parallel to a plane: If a straight line out of the plane is parallel to a straight line in the plane, it is parallel to the plane.

Abbreviation: if the lines are parallel, the lines are parallel.

Symbolic representation:

An alpha

bβ= & gt； a∧α

a∑b

2.2.2 Determination of parallelism between planes

1. Theorem for judging the parallelism of two planes: If two intersecting lines in one plane are parallel to the other plane, the two planes are parallel.

Symbolic representation:

a β

b β

a∩b = Pβ∩α

a∧α

b∧α

2. There are three ways to judge that two planes are parallel:

(1) definition;

(2) Judgment theorem;

(3) Two planes perpendicular to the same straight line are parallel.

2.2.3—2.2.4 The property that a straight line is parallel to a plane and a plane is parallel to a plane.

Theorem: A straight line is parallel to a plane, so the intersection of any plane passing through this straight line and this plane is parallel to this straight line.

Abbreviations are: lines are parallel to faces, and lines are parallel to lines.

Symbolic representation:

a∧α

aβa∨b

α∪β= b

Function: This theorem can be used to solve the problem of parallelism between straight lines.

2. Theorem: If two planes intersect with the third plane at the same time, their intersection lines are parallel.

Symbolic representation:

α∥β

α∧γ= a a∨b

β∪γ= b

Function: From the plane parallel to the plane, it can be concluded that the straight line is parallel to the straight line.

2.3 Determination and characteristics of vertical lines and planes

2.3. 1 Determination of straight line perpendicular to plane

1, definition

If the straight line L is perpendicular to any straight line in the plane α, we say that the straight line L and the plane α are perpendicular to each other, which is called L⊥α, and the straight line L is called the perpendicular of the plane α. As shown in the figure, when the straight line is perpendicular to the plane, their only common point P is called vertical foot.

L

p

α

2. Decision Theorem: If a straight line is perpendicular to two intersecting straight lines on a plane, the straight line is perpendicular to the plane.

Note: a) The condition of "two intersecting straight lines" in the theorem cannot be ignored;

B) Theorem embodies the mathematical idea of mutual transformation between "straight line perpendicular to plane" and "straight line perpendicular to straight line".

2.3.2 Determination of the plane perpendicular to the plane

1, the concept of dihedral angle: it represents a graph consisting of two half planes starting from a straight line in space.

A

Shuttle l β

B

α

2. The symbol of dihedral angle: dihedral angle α-l-β or α-AB-β.

3. Theorem for judging two planes perpendicular to each other: If one plane intersects the perpendicular of the other plane, the two planes are perpendicular.

2.3.3—2.3.4 The perpendicularity between straight line and plane, and between plane and plane.

Theorem: Two straight lines perpendicular to the same plane are parallel.

Property Theorem: If two planes are perpendicular, the straight line perpendicular to the intersection line in one plane is perpendicular to the other plane.

Block diagram of knowledge structure in this chapter

Chapter III Linear Sum Equation

3. 1 Angle and slope of straight line

3. 1 inclination and slope

1. Concept of straight line inclination: When the straight line L intersects with the X axis, the angle α formed by the positive direction of the X axis and the upward direction of the straight line L is called the inclination of the straight line L, especially when the straight line L is parallel or coincident with the X axis, α = 0 is specified.

2. Angle range α: 0 ≤α < 180.

When the straight line L is perpendicular to the X axis, α = 90.

3. Slope of the straight line:

The tangent of the inclination angle α (α ≠ 90) of a straight line is called the slope of this straight line, and the slope is often expressed by the lowercase letter k, that is,

k = tanα

(1) When the straight line L is parallel or coincident with the X axis, α = 0 and k = tan0 = 0.

⑵ When the straight line L is perpendicular to the X axis, α = 90, and k does not exist.

Therefore, the inclination angle α of the straight line L must exist, but the slope k does not.

4. Straight line slope formula:

Given two points p 1 (x 1, y 1), p2 (x2, y2), x 1 ≠ x2, the slope of the straight line P 1P2 is expressed by the coordinates of two points:

Slope formula:

3. 1.2 Parallelism and verticality of two straight lines

1, both straight lines have slopes and do not overlap. If they are parallel, then their slopes are equal; On the contrary, if their slopes are equal, they are parallel, that is,

Note: The above equivalence is established on the premise that two straight lines do not coincide and the slope exists. Without this premise, the conclusion will not be established. That is to say, if k 1=k2, then there must be l 1∑L2.

2. Both straight lines have slopes. If they are perpendicular to each other, their slopes are negative reciprocal. On the other hand, if their slopes are negative reciprocal, they are perpendicular to each other, that is,

3.2. Oblique equation of1point line

1. Point inclination equation of a straight line: the straight line passes through a point with a slope of.

2. Oblique equation of straight line: it is known that the slope of straight line is, and the intersection point with axis is.

3.2.2 Two-point linear equation

1. Two-point equation of a straight line: two of them are known.

2. Interception equation of straight line: It is known that the intersection of straight line and axis is A, and the intersection of straight line and axis is B, where

3.2.3 General equation of straight line

1. General equation of straight line: On binary linear equation (A and B are not 0 at the same time)

2. The mutual transformation between various linear equations.

3.3 Formula for coordinates and distance of intersection points of straight lines

3.3. 1 Intersection coordinates of two straight lines

1. For example: the coordinates of the intersection of two straight lines.

L 1 :3x+4y-2=0

L 1:2x+y +2=0

Solution: Solve the equation.

X=-2，y=2。

So the coordinate of the intersection of L 1 and L2 is m (-2,2).

3.3.2 Distance between two points

Distance formula between two points

3.3.3 Distance formula from point to straight line

1. Distance formula from point to straight line:

The distance from a point to a straight line is:

2, the distance between two parallel lines formula:

The general equation for the sum of two parallel lines is,

:, the distance is.

The fourth chapter circle sum equation

4. Standard equation of1.1circle

1, standard equation of circle:

Equation of a circle with center A(a, b) and radius r

2, the judgment method of the relationship between point and circle:

(1) > The emphasis is outside the circle.

(2) =, the point is on the circle

(3)& lt； The focus is on the circle.

4. General equation of1.2 circle

1, general equation of circle:

2, the characteristics of the general equation of the circle:

(1) ① The coefficients of x2 and y2 are the same and not equal to 0.

② There are no quadratic terms such as xy.

(2) There are three specific coefficients D, E and F in the general equation of a circle, so as long as these three coefficients are found, the equation of the circle is determined.

(3) Compared with the standard equation of a circle, it is a special binary quadratic equation with obvious algebraic characteristics, while the standard equation of a circle points out the coordinates and radius of the center and has obvious geometric characteristics.

4.2. positional relationship between1circles

1, use the distance from point to straight line to judge the positional relationship between straight line and circle.

Let a straight line:, a circle:, the radius of the circle is, and the distance from the center of the circle to the straight line is, then the basis for judging the positional relationship between the straight line and the circle is as follows:

(1) When the straight line is out of the circle;

(2) When a straight line is tangent to a circle;

(3) When a straight line intersects a circle;

4.2.2 positional relationship between circles

The positional relationship between two circles.

Let the length of the connecting line between two circles be, then the basis for judging the positional relationship between circles is as follows:

(1) When the circle is separated from the circle;

(2) In time, the circle is circumscribed with the circle;

(3) When the circle intersects with the circle;

(4) In time, circles are inscribed;

(5) If, circle and circle contain;

4.2.3 Application of Linear and Circular Equations

1, and use the plane rectangular coordinate system to solve the positional relationship between the straight line and the circle;

2. Process and method

Steps to solve geometric problems by coordinate method;

Step 1: Establish an appropriate plane rectangular coordinate system, express the geometric elements in the problem with coordinates and equations, and transform the plane geometric problem into an algebraic problem;

Step 2: Solve algebraic problems through algebraic operations;

Step 3: Transform the result of algebraic operation into geometric conclusion.

4.3. 1 space rectangular coordinate system

1 and point m correspond to a uniquely determined ordered real array, and, and are the coordinates of p, q, r on the,, and axes respectively.

2. Ordered real array, corresponding to a point in the space rectangular coordinate system.

3. The coordinates of any point M in space can be expressed by an ordered real array, which is called the coordinates of point M in the rectangular coordinate system of this space, and m is called the abscissa, ordinate and ordinate of point M. ..

4.3.2 Distance formula between two points in space

1, the distance formula between any point in space