Points in a plane rectangular coordinate system can be represented by abscissa and ordinate.

Of course, it can also be expressed in other forms.

Set point a, the distance from a to the origin is ρ (some books are denoted by r).

The angle formed by the connecting line between point A and the origin and the positive semi-axis of X axis is recorded as θ.

Therefore, the points in the plane rectangular coordinate system can be the same as those in the polar coordinate system.

Form a one-to-one relationship

Accord to that geometric relation of triangle,

x =ρcosθ； y=ρsinθ

Parabola: y=a(x-b)∧2+c

The polar coordinates are ρ sin θ = a (ρ cos θ-b) ∧ 2+c.

Simple parabola y=x∧2

Polar coordinates ρ sin θ = (ρ cos θ) ∧ 2 → sin θ = ρ (1-sin θ) ∧ 2.

That is, x in rectangular coordinates is changed to ρcosθ.

Change y to ρsinθ.

The corresponding polar coordinate equation can be obtained.

In addition to polar coordinate replacement, there are

1. Universal polar coordinate substitution

2. Spherical coordinate replacement

3. Cylindrical coordinate replacement

4. Natural coordinates

5. Universal coordinate replacement

All coordinate substitutions can be summed up as follows

Universal coordinate replacement

Establishment of polar coordinate system;

Take a fixed point o on the plane, which is called the pole, and draw a ray Ox, which is called the polar axis, and then choose a length unit and the positive direction of the angle (usually counterclockwise). For any point M on the plane, ρ represents the length of the line segment om, θ represents the angle between Ox and OM, ρ indicates the polar diameter of M, θ indicates the polar angle of M, and the ordered number pairs (ρ, θ) are called the polar coordinates (θ) of point M. If point M is at the pole, its polar coordinate is ρ=0, and θ can take any value. At this time, the polar coordinates of point M can be expressed in two ways: (1) ρ > 0, M(ρ, π+θ) (2) ρ > 0, M(-ρ, θ) in the same way, (ρ, θ). Therefore, the polar coordinates of a point are not unique. However, if ρ > 0, 0 ≤θ < 2π or-π < θ≤π are defined, then all points on the plane can correspond to polar coordinates one by one except poles. 2. Methods and steps to solve the curve polar coordinate equation: Establish an appropriate polar coordinate system at 1, set the coordinates of the moving point M as (ρ, θ)2, and write a set of points M suitable for the conditions. 4 simplify the equation. Prove that the equation is a curve equation. (3) Unified polar coordinate equation of three conic curves. The intersection point f is perpendicular to the directrix L, the vertical foot is k, the focus point f is the pole, and the reverse extension line Fx of Fk is the polar axis, so as to establish a polar coordinate system. Let M(ρ, θ) be on the curve. B. Let the distance from the focus F to the directrix L be |Fk|=p, and let |MF|=ρ, |MA|=|Bk|=p+ρcosθ, which is the unified polar coordinate equation of ellipse, hyperbola and parabola. When 0 < e < 1, the equation represents an ellipse, and the fixed point f is its left focus. This equation only represents the right branch of hyperbola, the fixed point F is its right focus, and the fixed line L is its right directrix. If ρ < 0 is allowed, the equation represents the whole hyperbola. 3. The conversion between polar coordinates and rectangular coordinates takes the origin of rectangular coordinates as the pole, and the positive half axis of X axis as the polar axis, and takes the same length unit in the two coordinate systems. Let m be any point on the plane, with rectangular coordinates (x, y) and polar coordinates (. Defined by trigonometric function, x=ρcosθ, y =ρsin θ. Note: In general, when the angle θ is determined by tgθ, the minimum angle can be taken according to the quadrant where point M is located.