Cartesian Coordinate problem

[TMan]

I have two sets of five points defined in two different Cartesian coordinate systems. The points are in (almost exactly) the same relative position as the points in the other system though they may be uniformly offset. Each set of points is on the same plane. How do I find a delta x and delta y to transform any point from one system to the other with these 10 (or fewer) reference points?

Regards,

TMan

 

[mmm4444bot]

find a delta x and delta y to transform any point from one system to the other

Are the scales the same on both coordinate systems?

What are the coordinates of the five points on each?

 

[TMan]

Follow-up info

Are the scales the same on both coordinate systems?

What are the coordinates of the five points on each?

The scales may be different between the two systems and I don't have test points yet. I'm attempting to make a simple application that examines sample points on two different maps and comes up with a way to plot any point from one map onto another. Each map uses it's own X and Y coordinate system (as opposed to lat/lon) and the upper left of each map is (0,0). I imagine I should have stated that in my original question. Sorry.

Regards,

TMan

 

[mmm4444bot]

I think that you need more information about how the coordinates are related.

Do you know in advance that there exists a simple mapping (i.e., relationship that lends itself to one formula fits all)?

Otherwise, what precision do you require?

 

[TMan]

I think that you need more information about how the coordinates are related.

Do you know in advance that there exists a simple mapping (i.e., relationship that lends itself to one formula fits all)?

Otherwise, what precision do you require?

Unfortunately no. The only thing I know is that the two sets of points will be in the same relative positions to each other and each coordinate system is (0,0) at the upper left hand corner. The differences will be that the scale between the two coordinate systems may be different (though the scaling factor will be identical along the x and y axis) and the sets of points will likely be offset by a constant x and y. The precision doesn't have to be super-close as I believe that the relative differences between related points will be quite small.

Regards,

TMan

 

[mmm4444bot]

If the delta-x and delta-y are fixed (i.e., the offset is constant), then writing the conversion formula is easy enough, but perhaps I misinterpreted your original question.

If you don't have any information about the offset, and you don't have a pair of coordinates from which to calculate the deltas, in what way could you expect to determine the deltas?

I think this is a good point for you to explain the application. Perhaps, that will help me to understand what you're asking.

By the way, the positive direction on the vertical axis is downward in your coordinate system, yes?

PS: I may be delayed in responding further; I have a train to board.

 

[TMan]

If the delta-x and delta-y are fixed (i.e., the offset is constant), then writing the conversion formula is easy enough, but perhaps I misinterpreted your original question.

If you don't have any information about the offset, and you don't have a pair of coordinates from which to calculate the deltas, in what way could you expect to determine the deltas?

I think this is a good point for you to explain the application. Perhaps, that will help me to understand what you're asking.

By the way, the positive direction on the vertical axis is downward in your coordinate system, yes?

PS: I may be delayed in responding further; I have a train to board.

A explanation of what I'm attempting to do is a great idea so here it is:
I have two maps drawn with vector graphics. They may be scaled differently and offset somewhat from each other. I have the coordinates of everything on the map, lots of geometrical objects and a set of five reference points on each map. Additionally I am not guaranteed to be able to programatically match the geometries between the sets, but I can match the reference points. For that reason I thought it might be easier to use five reference points that I know exist on each map. All of the geometries and points are in the same relative position on each map. The point sets will not be perfectly related to each other but will be very close. Also each map is on a single plane with the (0,0) coordinate in the upper-left corner so that x increases to the right and y increases down.

What I wanted to derive from this data was a formula that would allow me to take any arbitrary point on one map and derive a good approximation of the same relative point on the other. And yes, the delta x and delta y on points between maps should be constant. I thought that since the relative position of each reference point was (very nearly) the same that there would be some way to derive an offset that could be applied to each map that would allow me to take a point on the first map, add a delta-x and delta-y to the coordinates and then use that as the same point on the second.

Thanks for your time.

Regards,

TMan

 

[JeffM]

I am not sure I understand the question.

There are two co-ordinate planes with different co-ordinate systems and at least three distinct points in each plane that define triangles that are similar, identically oriented, and both in the fourth quadrant.

If the axes of one plane are v, w and the axes of the other plane are x, y, then there exists a >0 such that av + b = x and aw + c = y. The co-ordinates of the two planes are related by a scalar and two offsets. We must find the scalar and the offsets.

The first thing to do is to find the corresponding vertices of the two triangles. If the two triangles have the same orientation, this is easy to do. One way is to look at the vertices of the two triangles. There are three possibilities. One vertex is closer to the horizontal axis than the other two, one of which is closer to the vertical axis. Two vertices are equally close to the horizontal axis but differ in their distance from the vertical axis. Two vertices are equally close to the vertical axis but differ in their distance from the horizontal axis.

Let's consider the lengths of the three sides. Consider two corresponding sides. The length of that side in the v-w plane is

\(\displaystyle \sqrt{(v_2 - v_1)^2 + (w_2 - w_1)^2}.\)

The length of the corresponding side in the x-y plane is

\(\displaystyle \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{[(av_2 + b) - (av_1 + b)]^2 + [(aw_2 + c) - (aw_1 + c)]^2} =\)

\(\displaystyle \sqrt{(av_2 - av_1)^2 + (aw_2 - aw_1)^2} = \sqrt{a^2[(v_2 - v_1)^2 + (w_2 - w_1)^2]} = a\sqrt{(v_2 - v_1)^2 + (w_2 - w_1)^2}.\)

So taking the ratio of the lengths of any pair of corresponding sides will give you a.

Consider any corresponding vertex for the two triangles, say \(\displaystyle (v_1,\ w_1)\ and\ (x_1,\ y_1.)\)

\(\displaystyle b = x_1 - av_1\ and\ c = y_1 - aw_1.\)

Has this answered your question? Have I even uderstood your question?