How do I convert between two different 2D coordinate systems?

Question

I'm trying to convert a coordinate from one coordinate system to another, so that I can draw it on an image.

Basically the destination coordinate system is as follows:

X range: 0 to 1066
Y range: 0 to 1600

(just a standard image I'm drawing onto with the size of 1066 x 1600)

The position I'm trying to draw onto the image actually has the exact same size, but the coordinate system is different. The span all of the coordinates is 1066x1600.

But a coordinate example would be:

(111.33f, 1408.41f)
(-212.87f, 1225.16f)

The range of this coordinate system is:

X range: -533.333 to 533.333
Y range: 533.333 to 2133.333

I feel like this is VERY simple math, but for some reason I'm not getting it.

How can I convert the coordinates provided into the first coordinate system?

Answer 1

You can normalize the first value, this will give you a value in the [0,1] range. You can think of that as X percentage, the percentage the value maps to between the minimum and maximum values. Then you can find where that percentage belongs in your destination coordinate system by seeing what value is X percentage through the destination system. I'll use Java code as an example language, I'm sure the concepts are clear enough to translate to any language though.

So normalize:

public static float normalize(float value, float min, float max) {
    return Math.abs((value - min) / (max - min));
}

Using your example you'd input:

xPercent = normalize(x,0,1066);

Then find where it lays in the destination system. With something like

destX = xPercent*(Math.abs(max-min)) + min;

Or to use your values:

destX = xPercent*(Math.abs(533.33--533.33)) + -533.33;

So for example with an x value of 1000 you'd map that to your destination coordinate system to 467.29.

Alternatively, if the coordinate systems will always be the same, you can pre-compute the ratio between them.

So:

xRatio = (Math.abs(srcMax-srcMin))/(Math.abs(destMax-destMin));

destX = x*xRatio+destMin;

Answer 2

It is simple math:

res = ( src - src_min ) / ( src_max - src_min ) * ( res_max - res_min ) + res_min

src - source coordinate system

res - result coordination system

Edit - explanation of math

( src - src_min ) / ( src_max - src_min ) translates it to coordinate system starting at zero with equal length of source coordinate system (0.0, src_max - src_min ). Then it scales value to coordinate system (0.0, 1.0).

* ( res_max - res_min ) this scales value to coordinate system starting at zero with length of result coordination system (0.0, dst_max - dst_min)

+ res_min translates value to result coordinate system (dst_min, dst_max)

Answer 3

The basic equation for 2D coordinate tranformation (in algebra, without rotation involved) is:

TargetCoordinate = TranslateFactor + ScalingFactor*SourceCoordinate

given two points in TargetCoordinate (T1, T2) that corresponds to two points in SourceCoordinate(S1, S2), TranslateFactor and ScalingFactor is given by solving :

T1 = TranslateFactor + ScalingFactor*S1
T2 = TranslateFactor + ScalingFactor*S2

which result :

TranslateFactor = (T2*S1 - T1*S2) / (S1 - S2)
ScalingFactor   = (T2 - T1) / (S2 - S1)

In your case, for the x coordinate

S1 = 0    -> T1 = -533.333
S2 = 1066 -> T2 = 53.333

And thus,

TranslateFactor = -533.333
ScalingFactor   = 1.000625
=> TargetCoordinate = (-533.333) + (1.000625)*SourceCoordinate

y coordinate follow the same procedure

Answer 4

Making a few assumptions:

• You are (eventually) interested in a matrix impementation, for convenience and power; and
• You are familiar with homogeneous coordinates.

Then the question migrates to: What is the homgeneous transformation matrix for my basis change?

To answer this we need the answers to three subsidiary quesions first:

1. Where did my origin move to?
2. What happened to my X-axis? Let (M11,M12) be the coordinates of the point
3. What happened to my Y-axis?

Define the answers to these thre questions to be as follows:

1. (M31,M32) are the coordinates of the new origin under the original coordinate system.
2. (M11,M12) are the coordinates of the new unit x-vector in the original coordinate system.
3. (M21,M22) are the coordinates of the new unit y-vector in the original coordinate system.

Then the homogenous transformation matrix is:

( M11, M12,  0 )
( M21, M22,  0 )
( M31, M32,  1 )

My convention here is that points are represented by row vectors, which is the normal computer graphics convention; the mathematicians and physicists often use the oppsoite.