- ML - if you are using Moscow ML load "nj93"
- A means of looking at BMP bitmap files
- BMP is the standard used by MS Windows and others. If you are using Windows then PaintBrush will do. If you are using an X system then you may have access to xv which will process BMP files and will also write to other formats such as GIF used by WWW viewers. (Local users only: if xv does not work then your path is not set up properly, try the command /usr/local/X11R6/bin/xv at Craiglockhart or /apps/bin/xv from Merchiston)

- Some bitmaps to deform. You can use xv to grab any image that shows up on your screen or you can use one of mine: for socialists, for nationalists or for Baywatch fans. Save one of these to your own file space - I refer to labour.bmp throughout. Note that even if your viewer does not show these files as pictures you can still save them.

`width`, in pixels`height`in pixels- The colour map (this maps 1,4 or 8 bit numbers onto 24 bit colour values.) It takes the form of a string, each entry in the colour table occupies 4 bytes even though only the first three are significant.
- A function which gives the colour (index) at any pixel
(
`x`,`y`) where`x`runs from`0`to`width-1`and`y`runs from`0`to`height-1`.

fun swap(x,y)=(y,x); fun transpose(w,h,c,f) = (h, w, c, f o swap); writebmp (transpose(readbmp "labour.bmp")) "tmp.bmp";Use xv to take a look at the input file "labour.bmp" and the output file "tmp.bmp".

fun trans f (w,h,c,f') = let fun toSq (x,y)=(2.0*real x/real w - 1.0,2.0*real y/real h - 1.0) fun frmSq(x,y)=(floor((x+1.0)*real w/2.0),floor((y+1.0)*real h/2.0)) in (w,h,c,f' o frmSq o f o toSq)end; fun lookat f = writebmp (trans f (readbmp "labour.bmp")) "tmp.bmp";We can lookat any function which takes two real numbers and returns two real numbers try the function bigger as defined here:

fun bigger(x,y)=(2.0*x,2.0*y); lookat bigger;Use xv to look at the file "tmp.bmp" now.

Here are some more functions to try. Copy them all into ML then try some.

fun relf(x,y)=(~x:real,y); fun blow(x,y) = (x*0.5,y*0.5); fun fish(x,y)=let val r=sqrt(x*x+y*y) in (r*x,r*y) end; fun unfish p (x,y)=let val r=(sqrt(x*x+y*y)+p)/(1.0+p) in (x/r,y/r) end; fun wasp(x,y)=(x/(y*y+1.0)*2.0,y); fun fat p (x,y) = (x*(y*y+p)/p,y:real); fun rot a (x,y) = let val c=cos a val s=sin a in (x*c-y*s,x*s+y*c) end; fun whirl(x,y)=let val r=sqrt(x*x+y*y) in rot (1.0-r)(x,y) end; fun wave(x,y)=(x+sin(3.0*y)/4.0,y); fun shear(x,y)=(x+y/2.0,y); fun polo(x,y)=(2.0*arctan(x/y)/3.1415,sqrt(x*x+y*y)-0.2);

You can of course make up your own functions either by definition or by composition of some of the above.

The point (0.5,0.5) is halfway from the centre to the top right corner,
(1.0,1.0) is that top right corner.
The colour of the point (0.5,0.5) on the transformed image is taken from
the colour of the point (1.0,1.0) on the original.

Thus the image seen represents the inverse of the function applied. We
can apply the function forwards - by mapping each point of the original
onto a point on the new, but there might be gaps if the function is an
enlargement at any point.

Anyone wishing to pursue this might consider using a ByteArray to write
to. One might use a ByteArray to store a row of pixels and hold an Array
of these to represent the pixel plane. These structures are
"non-strict" - that is they do not have the property of referential
transparency and should be avoided wherever possible.

Rather than encourage such unfunctionally correct programming I shall merely
give a few pointers:

- open the structures ByteArray and Array to find out how they work
- for every pixel in the source, update the point corresponding to the transformed coordinates
- to obtain 3D transformations maintain a colour array and a z-buffer, the z-buffer holds the distance of the nearest point plotted, only over-plot points which are nearer
- to wrap a BMP onto a surface you will need a triple of parametric
equations for the surface, for example:
- sphere: (x,y) -> (cos x cos y, sin x cos y, sin y)
- toroid: (x,y) -> (cos x(R+rcos y),sin x(R+rcos y),r sin y)
- cone: (x,y) -> (y cos x,y sin x, y)

open_in: string -> instream input: instream * int -> stringThe header of a BMP file contains 54 bytes.

val fh = open_in "test.bmp"; val header = input(fh,54);The format of the BMP file is quite involved and you do need to know the details - skip this if you are not interested:

The header includes some numbers, some are stored as two byte values, with the most significant first, some are four byte values again with the MSB first. We use the functions get2 and get4 to convert two byte or four byte strings into integers:

fun get2 s = 256*ordof(s,0) + ordof(s,1) fun get4 s = 256*(256*(256*ordof(s,0)+ordof(s,1))+ordof(s,2))+ordof(s,3);Within the header there is the width and height at position 18 and 22 respectively, these are both 4 byte values:

fun width h = get4(substring(h,18,4)); fun height h = get4(substring(h,22,4));There is also a colour table, the size of which is in position 10, and the bit map itself, the size of which is in position 34. There may be either 1, 4 or 8 bits per pixel (there may even be 24 bits but I have not allowed for this). The function