5. 6. Cartesian coordinate transformation of the plane

§ 6 plane Cartesian coordinate transformation

ITranslation Coordinate Transformation

Definition: If the Cartesian coordinate system {O; I, j} and {O '; I', j '} meet the conditions of I = I', j = J ′, then, the coordinate system {O; I ', j'} can be viewed as translated by {O; I, j}, which is called by the coordinate system {O; I, the transformation from j} to the coordinate system {O '; I', j '} isTranslation Coordinate Transformation.

Translation transformation formula

Set the coordinates of M on the plane under the new series {O '; I', J'} and old series {O; I, j}

(X'', y''), (x, y), and the coordinate of O 'under the old system is (a, B), then

Xi + yj = + = ai + bj + x' I '+ y'j ′

= Ai + bj + x' I + y' j = (a + x') I + (B + y') j

Translate -- the formula for translating coordinates

IIRotating Coordinate Transformation:

Definition: If the two Coordinate Systems {O; I, j} and {O '; I', j '} meet O ≡ O', and the other coordinate (I, j ') = θ

Then, the coordinate system {O '; I', j '} can be viewed as obtained by rotating the θ angle around O in the coordinate system {O; I, j}, which is called {O; I, the transformation from j} to {O '; I', j '} isRotating Coordinate Transformation.

Rotation and Transformation Formula

Since round (I, I ') = 0, round (I, J') = + θ

Then I '= cos θ I + sin θ j, J' = cos (+ θ) I + sin (+ θ) j =-sin θ I + cos θ j

∴ Xi + yj = x 'I' + y 'J' = x' (cos θ I + sin θ j) + y' (-sin θ I + cos θ j)

= (X' cos θ-y' sin θ) I + (x' sin θ + y' cos θ) j

That is

Use x, y to represent x', y',

Three general coordinate transformations:

The transformation from the coordinate system {O; I, j} to the coordinate system {O '; I', j '} is calledGeneral Coordinate Transformation.

Note: Generally, the coordinate transformation can be completed in two steps. First, the coordinate system {O; I, j} is translated

{O '; I', j '}, and then tie the coordinates around the O' rotation θ = cosine (I, I '), that is

{O '; I', j ′}.

General transformation formula:

Set the coordinates of the old series {O; I, j} and new series {O '; I', j '} to (x, y) respectively)

(X', y'), the coordinates of {O '; I, j} Are (x ", y"), while O' is in {O; I, the coordinates under j} are (a, B), then

While

Bytes

Use x, y to represent x', y',

Note: The preceding coordinate transformation can also be completed by first rotating and then translating.

Example: It has a binary coordinate system {O; I, j} and {O '; I', j '}, and the straight lines of zhii' and J' Are in the coordinate system {O; I, the equation under j} is x + y + = 0, and x + y + = 0. Try to find the coordinate transformation formula.

Solution: Set the coordinates of P at any point in the plane under the old and new series to (x, y) (x', y ′)

The distance between P and I 'is expressed as a new coordinate.

Required y' Limit =

Thus y' = ±

Similarly, x' = ±

That is

Note:Notes should be noted when selecting formula ±

± = ±

If the line where I 'is located is 2x-y + 3 = 0, and the line where j' is located is x + 2y-2 = 0, the coordinate transformation formula is

Or

ThuThe coefficient variation law of the quadratic curve equation under Coordinate Transformation:

1. Pan

If the coordinate origin is translated to O '(,), the translation formula is

Then in the new system {O '; I, j} ≡ (x +) ² + 2 (+ x') (+ y') + (+ y ') ²

+ 2 (x' +) + 2 (y' +) + = 0

Note (x', y') When F (x' +, y' +)

= 'X' 2 'x 'y' + y '² + 2' X '+ 2' + ', then' = + + = F1 (,)

'= A21 ++ = F2 (,)

'= ² + 2 + ² + 2 + 2 +

= F (,)

Visible: Under the translation transformation, the quadratic curve equation

(1) The Quadratic coefficient remains unchanged;

(2) The coefficient of one item changes (,),(,);

(3) constant changes to F (,).

So if (,) is the center of the quadratic curve F (x, y) = 0, there will be no entry in the equation under the new system.

2. If the rotation angle is set to θ under the Rotation Transformation, the point on the plane is satisfied between the coordinates (x, y) (x', y') of the old and new series.

The equation of the ∴ quadratic curve in the new system is

F' (x', y') = f (x' cos θ-y' sin θ, + x' sin θ + y' cos θ)

= (X' cos θ-y' sin θ) ² + 2 (x' cos θ-y' sin θ) (+ x' sin θ + y' cos θ) +

(+ X' sin θ + y' cos θ) ² + 2 (x' cos θ-y' sin θ)

+ 2 (+ x 'sin θ + y 'cos θ) + = 0

Note f' (x', y ′) comment 'X' ² + 2' x'y' + 'y' ² + 2' X '+ 2' y' +'

It can be seen that, under the Rotation Transformation, the quadratic curve equation

1) The Quadratic coefficient is generally variable, but the quadratic coefficient of the new system is only related to the quadratic coefficient and rotation angle θ of the old system;

2) an item coefficient can also be an edge, but there is an item in the new process;

3) constant.

It can be seen from the formula expression. If the Alpha angle is selected

(

Ceg2 θ =

For Rotation Transformation, the new equation will not cross the product items.