Quickly derive the elliptical rotation coordinate conversion formula and derivation process

Rapid elliptical rotation coordinate transformation formula and derivation process

To see the elliptical rotation coordinate transformation formula and derivation process, we must first look at the rotation transformation and translation transformation relationship between the two rectangular coordinate systems.

Let’s look at the rotation transformation first.

There are two right-handed spiral plane rectangular coordinate systems, UOV and XOY.

2 The coordinate system has a common origin O.

The angle between the positive direction of the U-axis of U0V and the positive direction of the X-axis of X0Y is W.

[You can draw an XOY coordinate system on paper, and then let the U axis be in the first quadrant of XOY to draw a UOV coordinate system. 0

but,

If the coordinates of a point P on the plane in the XOY coordinate system are (X, Y), and the coordinates in the UOV coordinate system are (U, V).

[Draw a point P on the common part of the first quadrant of XOY and UOV, and then draw vertical lines from P to X, Y, U, and V respectively]

but

X = U*COS(W) - V*SIN(W)

Y = U*SIN(W) V*COS(W)

U = X*COS(W) Y*SIN(W)

V = X*SIN(W) - Y*COS(W)

so,

A standard ellipse in XOY X^2/A^2 Y^2/B^2 = 1 satisfies the equation in UOV becomes

[U*COS(W) - V*SIN(W)]^2/A^2 [U*SIN(W) V*COS(W)]/B^2 = 1

U^2{[BCOS(W)]^2 [ASIN(W)]^2} V^2{[BSIN(W)]^2 [ACOS(W)]^2} 2UV[COS(W )SIN(W)][A^2 B^2] - (AB)^2 = 0,

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Look at translation transformation again.

There are two right-handed spiral plane rectangular coordinate systems, UO'V and XOY.

2 The U and X coordinate axes of the 2 coordinate system are parallel to each other, and the V and Y coordinate axes are also parallel to each other.

The coordinates of the origin O' of UO'Y in XOY are (S, T).

but,

If the coordinates of a point P on the plane in the XOY coordinate system are (X, Y), and the coordinates in the UO'V coordinate system are (U, V).

X = U S

Y = V T

U = X - S

V = Y - T

so,

A standard ellipse in XOY X^2/A^2 Y^2/B^2 = 1 The equation satisfied in UO'V becomes

[U S]^2/A^2 [V T]^2/B^2 = 1.

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Combine translation and rotation,

There are two right-handed spiral plane rectangular coordinate systems, UO'V and XOY.

The coordinates of the origin O' of UO'Y in XOY are (S, T).

U0'The angle between the positive direction of the U axis of V and the positive direction of the X axis of X0Y is W.

but,

If the coordinates of a point P on the plane in the XOY coordinate system are (X, Y), and the coordinates in the UO'V coordinate system are (U, V).

X = U*COS(W) - V*SIN(W) S

Y = U*SIN(W) V*COS(W) T

U = (X-S)*COS(W) (Y-T)*SIN(W)

V = (X-S)*SIN(W) - (Y-T)*COS(W)

so,

A standard ellipse in XOY X^2/A^2 Y^2/B^2 = 1 The equation satisfied in UO'V becomes

[U*COS(W) - V*SIN(W) S]^2/A^2 [U*SIN(W) V*COS(W) T]/B^2 = 1

The last relationship should be what you want. . .

Write coordinate conversion formula in excel

Excel coordinate conversion:

In work, we often encounter situations where we need to convert a table with the amount in yuan into an amount in 10,000 yuan. It is very troublesome to modify it item by item, and even using the formula is inconvenient. You can use Excel’s Paste Special function to batch process data:

First enter 10000 in a blank cell outside the business table in the same Excel worksheet, select this cell, and select "Copy" in the "Edit" menu;

Then, select the cell range where the data needs to be modified, select "Paste Special" in the "Edit" menu, select "Divide" under the "Operation Bar" in the "Paste Special" dialog box, and click "OK" ;

Finally, format the modified cell range and delete the 10000 originally entered in a blank cell.

In order to avoid differences caused by the mantissa after conversion, when selecting the cell range where data needs to be modified, cells with calculation formulas set, such as subtotals, totals, etc., should not be included. After the above processing, pay attention to checking the relevant data relationships in the table and correct the errors found

Coordinate conversion instructions:

Coordinate transformation is the position description of a spatial entity and is the process of transforming from one coordinate system to another. This is achieved by establishing a one-to-one correspondence between the two coordinate systems. It is an indispensable step in establishing the mathematical foundation of maps in the measurement and compilation of maps of various scales. Then the desired coordinates must be transformed in the same way as the original coordinates to find the corresponding position in the new coordinates.

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