# Mapping a generic quadrilateral onto a rectangle (in 2 dimensions)

1. May 24, 2005

### HenryHallam

Problem:
I have a computer image of a grid "square" from an aviation chart. The "square" is actually approximately rectangular, but the left and right sides aren't quite parallel and the top and bottom sides are parallel but very slightly curved.
I will assume/pretend that the top and bottom sides are not curved but that they may not necessarily be horizontal w.r.t. the computer image.

The corners of the "square" have known latitude and longtitude. i.e. top left corner has (lat,long) (a,c), top right has (a,d), bottom left has (b,c), bottom right has (b,d) so if you drew their latitudes and longtitudes as cartesian co-ordinates on a plane they would form a rectangle.

I wish to convert a given latitude and longtitude (from a GPS receiver) to a set of pixel co-ordinates on the image, and also vice versa.

The type of projection used by the chart is unknown, and even if I were to work it out I would like to be able to use the same technique for different charts which use different projections.

So what I think I am looking for basically is a way to determine the transformation that maps a generic quadrilateral onto a rectangle, given the co-ordinates of all four corners in both planes.

Is this what I need? How would I go about doing this? I understand the basics of matrix algebra.

Thanks very much for your time!

Henry Hallam