About Coordinate Spaces and Transformations (1.2)
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About Coordinate Spaces and Transformations
This topic describes coordinate spaces and transformations. You should also
be familiar with the following topics:
◄Presentation spaces and device contexts►
◄Segments and retained graphics►
◄Clipping►
A coordinate space is a two-dimensional set of points. A point is the
smallest definable location in a coordinate space. There are four coordinate
spaces in Microsoft Operating System/2 Presentation Manager:
♦ World coordinate space
♦ Model coordinate space
♦ Page coordinate space
♦ Device coordinate space
A coordinate system is a means of specifying the location of a point in a
coordinate space. For each of the four coordinate spaces in MS OS/2, there
is a corresponding coordinate system. The four coordinate systems for MS
OS/2 are two-dimensional systems with x- and y-axes, an origin, and
dimensions. The x-axis extends horizontally across the system; the y-axis,
vertically across the system. The origin is located at the intersection of
the x- and y-axes and lies at the center of the system. The world, model,
and page coordinate systems' x- and y-axes extend from 0 to 134,217,727 in
the positive direction and from 0 to -134,217,728 in the negative direction.
The device coordinate system's x- and y-axes extend from 0 to 32,767 in the
positive direction and from 0 to -32,768 in the negative direction.
When you specify a coordinate in MS OS/2, the values you use are long
integers (32-bits).
Single-Coordinate-Space Systems
All graphics systems use at least one coordinate space and system to
generate output on a video display or printer. The simplest systems use a
single coordinate space, whose points are the pels on the display. In
single-coordinate-space, or single-space, systems, one corner of the display
(usually the lower-left) corresponds to the origin of the coordinate system;
the positive x-axis extends from the origin to the lower-right corner of the
display; and the positive y-axis extends from the origin to the upper-left
corner of the display.
There are two fundamental drawbacks to a graphics system with a single
coordinate space:
♦ Any application written for the graphics system is device dependent.
♦ It is difficult to draw output in convenient units like centimeters,
feet, inches, or twips.
Since the coordinate space in a simple system is made up of pels, and since
the shape of pels varies from device to device, the output from an
application written for one device may look different on another device.
Single-coordinate-space systems are difficult to use for drawing or
computer-aided-design (CAD) applications since these applications work in
units of inches, feet, meters, and so on. For example, to draw a simple
horizontal line that is 1 inch long, you must perform the following tasks:
1 Determine the width (in inches) of the video display or printer paper.
2 Determine the width (in pels) of the video display.
3 Divide the width in pels by the width in inches to determine the ratio of
pels to inches.
4 Using this ratio, perform the line-drawing command.
Multiple-Coordinate-Space Systems
Single-coordinate-space, or single-space, systems do not provide the
flexibility that most applications and application developers need. An
application written for a single-coordinate-space system will create
meaningful output only on a device on which the pels are a certain width and
height. Since spreadsheet, word-processing, desktop-publishing, and CAD
applications normally draw output on video displays and printers or
plotters, it is obvious that a single-coordinate-space system is
impractical. In Presentation Manager, however, you can avoid the limitations
of a single-coordinate-space system by using additional coordinate spaces
and other operations, called transformations. Transformations require two
coordinate spaces──a source coordinate space and a target coordinate space.
Transformations
When copying objects from one coordinate space to another, MS OS/2 applies
one or more of five operations, called transformations, to the points that
define the object. These five transformations──scaling, rotation,
translation, shear, and reflection──enable you to write applications that
can do the following tasks:
♦ Run on a variety of video displays, printers, and plotters.
♦ Use convenient units of measurement (centimeters, inches, feet,
kilometers, miles, etc.).
♦ Scroll and zoom pictures.
♦ Rotate and scale objects in pictures.
♦ Shift the positions of objects in pictures.
♦ Display mirror images of objects in pictures.
♦ Display sheared objects in pictures.
When your application copies an object from the source coordinate space to
the target coordinate space, the five transformations work as follows:
♦ The scaling transformation makes the object look bigger or smaller.
♦ The rotation transformation rotates the object.
♦ The translation transformation shifts the object with respect to the
origin of the coordinate system.
♦ The shear transformation rotates either all the vertical or all the
horizontal lines in an object.
♦ The reflection transformation creates a mirror image of an object with
respect to the x- or y-axis.
Presentation Manager uses three-by-three matrices to represent the three
transformations. For any point (x,y) in a source coordinate space, you can
determine the corresponding point (x',y') in the target coordinate space by
using the following matrix multiplication:
|M11 M12 M13|
[x y 1] * |M21 M22 M23| = [x' y' 1]
|M31 M32 M33|
In the following example, the transformations are set to unity (represented
by the identity matrix). In such cases, the original point in the source
coordinate space is always identical to its corresponding point in the
target coordinate space.
|1 0 0|
[3 100 1] * |0 1 0| = [3 100 1]
|0 0 1|
In the next example, the identity matrix is replaced with a new matrix that
translates the point when the point is copied from the source coordinate
space to the target coordinate space.
|1 0 0|
[3 100 1] * |0 1 0| = [100 3 1]
|97 -97 1|
In this example, the point (3,100) was translated to (100,3) by changing the
values of M31 and M32.
The MATRIXLF Structure and Fixed Values
A special data structure called the MATRIXLF structure contains nine fields
that correspond to the nine elements in a three-by-three transformation
matrix. Five of these fields are 32-bit long integer values; the remaining
four are special 32-bit fixed values.
A fixed value is a binary representation of a floating-point number. A fixed
value has two parts: the high-order 16-bits and the low-order 16-bits. The
high-order 16-bits contain a signed integer in the range -32,768 through
32,767; the low-order 16-bits contain the numerator of a fraction, in the
range 0 through 65,535 (the denominator for this fraction is 65,536).
You use fixed values to store the sines and cosines of angles for the
rotation transformation. You also use fixed values to store scaling factors
for the scaling transformation. To store the cosine of 60 degrees in a fixed
value, you would multiply 0.5 by 65,536. The result, 32,768, is the value
you would assign to a fixed value in the MATRIXLF structure. To store a
scaling factor of, for example, 3 in a fixed value, you would multiply 3 by
65,536. Again, the result, 196,608, is the value you would assign to a fixed
value in the MATRIXLF structure.
The MAKEFIXED macro provides a quick and convenient method for determining
fixed values. This macro requires two arguments: The first is the integer
part of the fixed value, and the second is the fraction part of the fixed
value. In the following example, MAKEFIXED is used to determine the
fixed-value equivalent of 1-and-1/8:
matlf.fxM11 = MAKEFIXED(1, 8192)
The first argument, 1, is the integer part of the fixed value. The second
argument, 8192, is the result of multiplying 65,536 by 1/8.
The following type definition shows each of the fields in the MATRIXLF
structure. Fields with an fx prefix contain fixed values; fields with an l
prefix contain long values.
typedef struct _MATRIXLF { /* matlf */
FIXED fxM11; /* M11 */
FIXED fxM12; /* M12 */
LONG lM13; /* M13 */
FIXED fxM21; /* M21 */
FIXED fxM22; /* M22 */
LONG lM23; /* M23 */
LONG lM31; /* M31 */
LONG lM32; /* M32 */
LONG lM33; /* M33 */
} MATRIXLF;
Scaling Transformations
When you scale an object by using the scaling transformation, the matrix
element M11 contains the horizontal scaling component (horz), and the matrix
element M22 contains the vertical scaling component (vert):
|horz 0 0|
| 0 vert 0|
| 0 0 1|
Rotation Transformations
When you rotate an object clockwise by using the rotation transformation,
the matrix element M11 contains the cosine of the rotation angle; M12, the
negative sine of the rotation angle; M21, the sine of the rotation angle;
and M22, the cosine of the rotation angle:
|cos -sin 0|
|sin cos 0|
|0 0 1|
Remember that MS OS/2 applies a transformation to all points in the source
coordinate space. This means that unless an object is drawn about the origin
of the source coordinate space, translation will occur when the object is
rotated or scaled.
Translation Transformations
When you translate an object by using the translation transformation, the
matrix element M31 contains the horizontal translation component, and the
matrix element M32 contains the vertical translation component, as follows:
|1 0 0|
|0 1 0|
|horz vert 1|
Shear Transformations
There are two shear transformations: vertical and horizontal. When an
application uses the vertical shear transformation, it affects only the
vertical lines in the object. In the matrix of the vertical shear
transformation, the element M21 contains a horizontal shear component, and
the element M22 contains a vertical shear component, as follows:
|1 0 0|
|horz vert 0|
|0 0 1|
When an application uses the horizontal shear transformation, it affects
only the horizontal lines in the object. In the matrix of the horizontal
shear transformation, the element M11 contains a horizontal shear component,
and the matrix element M21 contains a vertical shear component, as folows:
|horz vert 0|
|0 1 0|
|0 0 1|
If an application shears an object that contains two orthogonal vectors (two
perpendicular lines), the vectors will no longer be orthogonal.
Reflection Transformations
The reflection transformation creates a mirror image of an object with
respect to the x- or the y-axis. The matrix element M11 contains the
horizontal reflection component, which causes reflection about the y-axis.
The matrix element M22 contains the vertical reflection component, which
causes reflection about the x-axis. The reflection components are always
negative.
|horz 0 0|
|0 vert 0|
|0 0 1|
Combining Transformations
Each time you call a Gpi function that sets a transformation, you can
specify whether you want to combine it with existing transformations and, if
so, whether you want MS OS/2 to apply the existing transformation before or
after the new transformation.
If you want MS OS/2 to replace any existing transformations with the new
transformation, use the TRANSFORM_REPLACE flag when you set the
transformation. If you want MS OS/2 to apply the new transformation before
it applies the existing transformation, use the TRANSFORM_PREEMPT flag when
you set the new transformation. If you want MS OS/2 to apply the new
transformation after it applies the existing transformation, use the
TRANSFORM_ADD flag when you set the new transformation.
Round-off Error
Whenever you use transformations in your application, you should check for
any round-off error that occurs after multiple scaling, rotation, shear, or
reflection transformations. For example, if your application uses a rotation
transformation to rotate the hands of a clock, the accuracy of the clock
will diminish due to rounding off after the transformation. In order to
prevent the loss of accuracy, your application should check the results of
each transformation.
World Coordinate Space
A world coordinate space, or world space, is where all Presentation Manager
drawing operations begin. Whenever you draw with one of the line, arc,
character, area, or image primitives, the coordinates that you specify are
world coordinates. World-coordinate units can be inches, fractions of an
inch, meters, centimeters, miles, kilometers, yards, and so on.
If your application uses the retained-drawing mode to store subpictures in
chained segments, MS OS/2 assigns a new world space to each segment. In
other words, if the segment attribute is set to ATTR_CHAINED and you call
the GpiOpenSegment function, the subpicture associated with that segment is
drawn in a new world space.
There is a special clipping area called a clip path that you can use to
define a part of a world space that you want to copy into the next
coordinate space (the model space). The advantage of a clip path is that it
is the only clipping region that can be nonrectangular. Its edges can
include arcs, curves, and straight lines. The coordinates that define the
dimensions and shape of a clip path are always world coordinates. You can
create a clip path by calling the GpiBeginPath, GpiEndPath, and
GpiSetClipPath functions and one or more Gpi primitives that define the
dimensions and location of the path. For more information, see the topic
◄Clipping►.
Model Coordinate Space
A model coordinate space, or model space, is where you build a model, or a
complete picture, from subpictures in one or more world spaces. You can have
more than one model space, each corresponding to a particular model.
There is a special clipping area called a viewing limit that you can use to
define a part of a model space that you want to copy into the next
coordinate space (the page space). A viewing limit is always rectangular,
and the coordinates that define its location and dimensions are always model
coordinates. You can set the viewing limit by calling the
GpiSetViewingLimits function.
World-to-Model-Space Transformations
There are three transformations that operate between the world space and the
model space:
♦ The model transformation
♦ The segment transformation
♦ The instance transformation
Model Transformations
Model transformations are used for retained- or nonretained-drawing
operations. This means model transformations affect the output from the
GpiDrawChain, GpiDrawFrom, and GpiDrawSegment functions and any of the Gpi
functions that occur outside of a retained-drawing segment.
You can determine the values for the current model transformation by calling
the GpiQueryModelTransformMatrix function, which returns the model
transformation's scaling, rotation, and translation values in a
three-by-three matrix. Similarly, you can set these values by calling the
GpiSetModelTransformMatrix function, passing it the scaling, rotation, and
translation values in a three-by-three matrix.
In a retained-graphics application, you could use a model transformation to
rotate, scale, or translate parts of a subpicture. You can set and reset the
model transformation any number of times within a retained-drawing segment.
Segment Transformations
Segment transformations alter retained-drawing output. Unlike a model
transformation, which can be set and reset within a segment bracket, a
segment transformation must be set outside of a segment bracket.
You can determine the values for the current segment transformation by
calling the GpiQuerySegmentTransformMatrix function, which returns the
scaling, rotation, and translation values in a three-by-three matrix and
passes a segment identifier. Similarly, you can set the values for the
segment transformation by calling the GpiSetSegmentTransformMatrix function,
passing it the necessary values (in a three-by-three matrix) and a segment
identifier.
The user of a CAD or drafting application may need to zoom in on part of a
subpicture that was clipped in world space by a clip path. The application
could perform this operation by setting the segment transformation for that
particular subpicture after setting the clip path.
Instance Transformations
Instance transformations alter the retained-drawing output from special
segments referred to as "called" segments. A called segment usually contains
a subpicture that is duplicated several times in other subpictures. The
instance transformation positions, sizes, and rotates the subpicture each
time it is duplicated. You can set the values for the instance
transformation by calling the GpiCallSegmentMatrix function, passing it the
transformation values (in a three-by-three matrix) and a segment
identifier.
Page Coordinate Space
A page coordinate space, or page space, is where a complete picture is
prepared for viewing in a window on a video display, printing on a page of
printer paper, or plotting on a page of plotter paper. Page-coordinate units
can be increments of an inch, a meter, several pels, a twip, or some
arbitrary value. (A twip is a standard unit in the typesetting industry that
measures 1/1440 of an inch.) You specify the units used for page coordinates
when you call the GpiCreatePS function and create a presentation space.
If your application uses retained-drawing mode, the picture in page space
will contain parts of models (or pictures) from each of the model spaces
that have not been clipped; the picture will also contain any additional
graphics-primitive output (that has not been clipped) that the application
generated in nonretained-drawing mode. If your application uses
nonretained-drawing mode, the picture in page space will contain all of the
graphics-primitive output that has not been clipped.
In the page space, there is a special clipping area called a graphics field
that you can use to define the part of the page space that you want to copy
into the next coordinate space (the device space). The graphics field is
always rectangular. The coordinates that define the location and dimensions
of the graphics field are always page coordinates.
Model-to-Page-Space Transformations
There are two transformations that operate between the model space and the
page space:
♦ The default viewing transformation
♦ The viewing transformation
Default Viewing Transformations
Default viewing transformations scroll or zoom pictures in a window on a
video display or on a page of printer or plotter paper. You can determine
the current values for the default viewing transformation by calling the
GpiQueryDefaultViewMatrix function, which returns the
default-viewing-transformation values in a three-by-three matrix. You can
set these values by calling the GpiSetDefaultViewMatrix function and passing
it the transformation values (in a three-by-three matrix).
Viewing Transformations
Viewing transformations create pictures from multiple model spaces. For
example, you could copy a picture from a model space to a page space,
leaving all parts of the picture intact. Then you could copy a part of
another picture from a second model space by using a viewing limit to define
the part of the picture that you want to copy and then using the viewing
transformation to scale and translate that part. You can determine the
current values for the viewing transformation by calling the
GpiQueryViewingTransformMatrix function, which returns the values for the
viewing transformation in a three-by-three matrix. You can set the values by
calling the GpiSetViewingTransformMatrix function and passing it the
transformation values (in a three-by-three matrix).
Device Coordinate Space
A device coordinate space, or device space, is the final space in which a
picture is drawn before it appears in a window or on the page of a printer
or plotter. Device-coordinate units are pels if the page units are pels,
twips, increments of an inch, or increments of a meter. Device-coordinate
units are arbitrary if the page units are arbitrary.
Page-to-Device-Space Transformations
There is one transformation between the page space and the device space──it
is called the device transformation. Unlike the other transformations that
rotate, scale, and translate objects, the device transformation only scales
and translates objects; also, instead of a three-by-three matrix, the device
transformation uses two rectangles. (The location of the first rectangle is
fixed; the location of the second rectangle is movable.) These two
rectangles are the presentation page and the page viewport.
Presentation Pages
A presentation page is a rectangle in a page space. Its lower-left corner is
always positioned at the origin of the page space. You can determine the
dimensions of the presentation page by calling the GpiQueryPS function,
which returns a pointer to a SIZEL structure that contains the page
dimensions. If you specify arbitrary page units when you create a
presentation space, you must also specify the dimensions of the presentation
page. If you specify any other page unit, MS OS/2 automatically sets the
dimensions of the presentation page for you.
Page Viewports
A page viewport is a rectangle in a device space. MS OS/2 always copies the
presentation-page rectangle into the page-viewport rectangle. You can
determine the current dimensions of the page viewport by calling the
GpiQueryPageViewport function, which returns a pointer to a RECTL structure
that contains the coordinates of the viewport. You can set the location and
dimensions of the page viewport by calling the GpiSetPageViewport function
and passing it a pointer to a RECTL structure that contains the new values.
The ratio of the page width to the page-viewport width defines a horizontal
scaling factor, and the ratio of the page height to the viewport height
defines a vertical scaling factor.
♦