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Mapping Problem diagram/description

The screen size and coordinates (a,b,c,d) are known, and the Area of Interest (AOI) size and coordinates (x,y,w,z) are known.

If an object/coordinate/pixel, xx(1,1) is detected in AOI, how do I find or map the coordinates to xx(2,3)?

Note The AOI size/coordinates, and the location of the object/coordinate/pixel/point, even if changed, should result in the correct mapping of the detected object on the screen.

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3 Answers 3

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If the units are the same in both axis systems, the transformation equations is just that of a translation

$$(x_s,y_s)=(x_a,y_a)+(1,2)$$ (where $(1,2)$ are the coordinates of the top-left corner of the AOI wrt the screen).

Reciprocally,

$$(x_a,y_a)=(x_s,y_s)-(1,2).$$

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Combine a polar coordinate system with the line slope formula and the line distance formula.

Or use a land surveying system:

Any y-coordinate that is South of zero is negative. Any x-coordinate that is West of zero is negative.

Point A is (y1, x1) . Point B is (y2, x2) .

The direction of A to B is:

InvTan((x2 - x1)/(y2 - y1)) .

If (x2 - x1) is positive that is East else West. If (y2 - y1) is positive that is North else South. This procedure allows a quadrant direction to be determined between any two points.

The distance of A to B is:

SquareRoot of ((x2 - x1)^2 + (y2 - y1)^2) .

The combination of direction and distance is a vector but here as an inverse. The coordinates of a point is the location of the point.

Forwarding a point (or setting a point) from A to B is:

y2 = y1 + (Cos(Direction) * Distance) . x2 = x1 + (Sin(Direction) * Distance) .

A North direction is a positive value added to y1 else negative. An East direction is a positive value added to x1 else negative.

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    $\begingroup$ There is absolutely no reason to use polar coordinates. This adds tons of confusion and complexity. $\endgroup$
    – user65203
    Commented Dec 2, 2018 at 14:45
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If the area-of-interest is a picture-in-picture feature then it is both scaled and translated.

Scaling multiplies all coordinates and both x and y coordinates by the same scalar. Scaling just makes the figure bigger or smaller but proportionally the same.

Translating adds the same x translation value to all x coordinates and adds the same y translation value to all y coordinates. Translating just moves the figure without changing its size.

Then the inverse operations are obvious.

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