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Transformations		

	(see sample code in files xlib.c and xform.c on DISK 2)


Moving images from one device to another is an important part
of device independence. Image data should be stored using a coordinate 
space that is device independent. This coordinate space should be at a
higher resolution than any device, so there is always more information
in a image than can be displayed. Image data with less resolution
than the device will cause increased aliasing (jaggies). Storing images
as floating point offers the best resolution, but takes a tremendous toll
in computation time and storage space. Integer space can hold enough
resolution for most devices. 

For example, images might be stored at a resolution of 1000 pixels/inch. 
Only a few phototypesetters can better that resolution. A one inch circle 
centered on an 8 1/2 x 11 inch page would have its center at (4250, 6500)
and a radius length of 500.

Image coordinates are also referred to as "user" or "world" coordinates
in much of the literature on computer graphics.

"Transformation" is the process of converting image coordinates to 
coordinates for a specific device. For example, we can define the image
space described above described above as having (0,0) at the upper left
corner and (8500, 11000) at the lower left corner. If a certain device
has 72 dots per inch, those corners would be (0,0) and (576,792) in
device coordinates.

We use transformation equations to map any point from image coordinate
space to device space:

	dev_x = x_xscale * user_x + x_xtrans ;
	dev_y = x_yscale * user_y + x_ytrans ;

Since we are working in an integer space, the scaling values will round off,
so we expand the scaling values into an integer numerator and denominator,
and do a multiply and a divide to get the effect of a single floating point
multiply:

	dev_x = (x_xmul * user_x) / x_xdiv + x_xtrans ;
	dev_y = (x_ymul * user_y) / x_ydiv + x_ytrans ;

We use a function SMUL_DIV that holds the result of the multiply in a long
value (to avoid overflow and keep resolution) and then divides back to an
integer.

    dev_x = SMUL_DIV( user_x, x_xmul, x_xdiv ) + x_xtrans ;
    dev_y = SMUL_DIV( user_y, x_ymul, x_ydiv ) + x_ytrans ;

	(functions x_udx_xform, x_udy_xform, x_udx_scale, x_udy_scale)

To calculate the coefficients needed for the equations above, we need
the upper left corner and size of the user space and device space rectangles:

    x_xmul = dev_w ;
    x_xdiv = user_w ;
    x_ymul = dev_h ;
    x_ydiv = user_h ;
    x_xtrans = dev_x - SMUL_DIV( user_x, x_xmul, x_xdiv ) ;
    x_ytrans = dev_y - SMUL_DIV( user_y, x_ymul, x_ydiv ) ;

	(function x_sxform)

We can use these coefficients to go from device coordinates to
user coordinates:

    user_x = SMUL_DIV( x - x_xtrans, x_xdiv, x_xmul ) ;
    user_y = SMUL_DIV( y - x_ytrans, x_ydiv, x_ymul ) ;

	(functions x_dux_xform, x_duy_xform, x_dux_scale, x_duy_scale)

Another element that affects these equations is the aspect ratio of 
the pixels on the device. The IBM color monitor in monochrome mode, for
example, has pixels that are more than twice as tall as they are
wide. To make a square in image coordinates appear square on the
screen, we must calculate what rectangle in device space fits a 
rectangle in image coordinate space. 

The equation starts as:

	device area height	    user area height
if	------------------    <     ----------------
	device area width	    user area width

then 	the device area is wider than the user area and the device width must
	be reduced
else	the device height must be reduced.

We must take the pixel size into account, so the equation expands to:

	device area height in pixels * pixel height   user area height
	------------------------------------------- < ----------------	
	device area width  in pixels * pixel width    user area width 

or
	dev_h * dev_attrÆ 4 Å   user_h
	---------------------- < ------
	dev_w * dev_attrÆ 3 Å   user_w

To avoid any roundoff problems when working with integers, equation resolves to

	dev_h * dev_attrÆ 4 Å * user_w < dev_w * dev_attrÆ 3 Å * user_h

We use LONGS to avoid overflow. To calculate new width or height
the above equation is turned into an equality and solved for one or the other
of dev_h or dev_w. To find a new dev_h, for example:

	dev_h = dev_w * dev_attrÆ 3 Å * user_h / (dev_attrÆ 4 Å * user_w)

To avoid roundoff and overflow, turn this into a series of multiplications
and divisions: 

	dev_h = dev_w * (dev_attrÆ 3 Å / dev_attrÆ 4 Å) * (user_h / user_w)

and turned into a series of SMUL_DIVs.

	(function x_saspect)



The sample program "xform.c" demonstrates transformations from user to
device space and device to user space. Note that you can zoom in, draw a
sketch, zoom out, view the sketch and zoom back in without losing any
resolution in the sketch data.

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