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Inside Macintosh: QuickDraw GX Environment and Utilities /
Chapter 8 - QuickDraw GX Mathematics / About QuickDraw GX Mathematics

Transformation Operations With Mappings

A mapping is a 3 3 perspective matrix that performs transformations of spatial locations in two dimensions. You can apply a mapping operation to a set of points either directly (as when directly modifying the geometry of a shape), or indirectly, by multiplying a mapping with another mapping (as when altering the mapping in the transform object associated with a shape).

QuickDraw GX uses mappings to perform the following transformations on shapes or other two-dimensional data:

When you multiply two or more matrices to obtain a cumulative result, you concatenate, or accumulate the transformations of, both mappings. Matrix multiplication is not commutative. This means that [A] [B] \x82 [B] [A]. As a result, the order that you concatenate is important. [A] is postmultiplied by [B] if [A] is replaced by [A] [B]. Conversely, [A] is premultiplied by [B] if [A] is replaced by [B] [A]. A mapping is applied to a point via postmultiplication (which is to say that points are row vectors); therefore, the default for applying one mapping to another is also postmultiplication.

Multiple concatenations can occur in QuickDraw GX, such as when drawing picture shapes or when drawing any shape through a hierarchy of view ports. If you are going to apply several mappings to a relatively large bitmap or other shape, it is advantageous to concatenate the mappings first (with the MapMapping function) and then apply the resultant mapping to the shape (with the GXMapShape function).

The motivation is speed. It is much faster to concatenate mappings than to apply a mapping to a large number of points. For bitmaps, an additional motivation is accuracy. Each time a shape is transformed, a certain amount of roundoff error is introduced. Because the pixels of a bitmap are at integral coordinates, the roundoff error is on the average of a quarter pixel, compared with thousandths of a pixel for fixed-point coordinates.

QuickDraw GX provides two groups of mapping functions. The first group allows you to copy and perform standard matrix operations on mappings. With these functions, you can

The second group allows you to modify how a mapping transforms the objects or coordinate space it is applied to. With these functions, you can

Figure 8-2 shows an example of how modifying a mapping can modify the scaling, rotation, skewing, and perspective of a shape.

Figure 8-2 Transformation operations with a mapping matrix

Characteristics of a Mapping

QuickDraw GX achieves these two-dimensional transformations of shapes and points on a plane by matrix multiplication of each Cartesian point P by the mapping matrix [T] to generate a transformed point P'.

P(x, y) [T] = P'(x', y')

To multiply a two-dimensional point by a three-dimensional matrix, we first expand it to a three-dimensional point (x, y, 1). After multiplication, the resulting point is (x', y', z'), which normalizes to (x'/z', y'/z, 1) or, in two dimensions, (x'/z', y'/z).

The QuickDraw GX mapping is defined as 

struct gxMapping { Fixed map[3][3];};
The mapping consists of linear elements a, b, c, and d; perspective elements u and v; translation elements h and k; and the scale factor w, which is commonly set to fract1. Although defined as containing only Fixed numbers, the rightmost column of the matrix--containing elements u, v, and w--consists of fract numbers. Figure 8-3 shows the elements of the matrix in place.

Figure 8-3 Mapping matrix elements

Point P(x, y) is transformed to point P'(x', y') by matrix multiplication of the row vector
[x y 1] by the mapping matrix to yield the expanded general expression shown in
Figure 8-4.

Figure 8-4 Applying a mapping matrix to a point

The x and y elements of the transformed vector can be mapped back to the x and y-coordinates by dividing each element by the term ux + vy + w. The resulting general expression for the transformation of point P(x, y) to P'(x', y') is shown in Figure 8-5.

Figure 8-5 The point (x, y) as transformed by the mapping matrix

A mapping is normalized whenever the transformation matrix element w has the value 1. Most QuickDraw GX mapping operations will be automatically normalized. However, mappings that an application generates itself might not be normalized. Subsequent operations with that mapping may be slow.

If a mapping does not specify perspective (that is, if its perspective elements u and v are zero), normalization of the transformation involves dividing the map by the absolute value of w, if possible. If this division is not possible (due to overflow) or if the mapping specifies perspective, normalization involves bit-shifting each element of the mapping to the left. The amount of shift provided by the minimum of the following two operations is selected:

The identity mapping, or identity matrix, has the unique characteristic that it maps points to the same point. The identity matrix has all diagonal elements equal to 1 and all other matrix elements have the value 0. The identity matrix is shown in Figure 8-6.

Figure 8-6 The identity matrix

The inverse of a mapping is the mathematical inverse of the matrix. This means that if you concatenate a mapping with its inverse, you will get the identity matrix.

The rest of this section discusses the use of the mapping functions in modifying the translation, scaling, rotation, and skewing factors in a mapping. It ends with a discussion of how to modify the perspective factors in a mapping. For additional information about the use of mappings in the transform object and in view port and view device objects, see the chapters "Transform Objects" and "View-Related Objects," respectively, in Inside Macintosh: QuickDraw GX Objects.

Translation by a Relative Amount

You can use the MoveMapping function to make a relative change (in both x and y) to the translation specified by a mapping. Matrix elements h and k control the amount of the translation. Figure 8-7 shows what happens to a mapping M when you call MoveMapping and specify horizontal and vertical offsets of hOffset and vOffset. A purely translational matrix is applied to the target mapping, so that the resultant mapping's translation is increased by the specified offsets.

Figure 8-7 Changing the translation specified by a mapping

Figure 8-8 shows the use of the MoveMapping function to provide translation of a mapping by the increments given by the hOffset and vOffset parameters. The MoveMapping function is described on page 8-67.

Figure 8-8 Translation by a relative amount with MoveMapping

Translation to a Specified Point

You can specify translation of the origin to a given point by using the MoveMappingTo function. Moving the origin means that the point (0, 0) will become the point (h, k) after the mapping is applied to it. Matrix elements h and k again control the amount of translation. Figure 8-9 shows what happens to a mapping M when you call MoveMappingTo and specify the desired location (hPosition, vPosition). A relative translation of (-h/w, -k/w) is applied to the target mapping to bring its origin to (0, 0), and then a relative translation of (hPosition, vPosition) is applied. The resultant mapping ends up with translational values of hPosition and vPosition.

Figure 8-9 Setting the origin specified by a mapping

Figure 8-10 shows the use of the MoveMappingTo function to move the origin to a specific location. Note that this figure assumes that the origin of the shape--point (0.0, 0.0) in its geometry--is at its upper left corner. The MoveMappingTo function is described on page 8-68.

Figure 8-10 Translation to a specific origin location

Scaling

You can use the ScaleMapping function to modify the scaling factors in a mapping. Matrix elements a and d in the mapping matrix control the degree of the scaling in the horizontal and vertical directions, respectively. Figure 8-11 shows what happens to a mapping M when you call ScaleMapping with horizontal and vertical scaling factors of hFactor and vFactor and a center of scaling at (xCenter, yCenter). First, a relative translation of -xCenter and -yCenter moves the center of scaling to (0, 0); then a purely scaling matrix multiplies the scaling by hFactor and vFactor; finally, another relative translation moves the center of scaling by +xCenter and +yCenter. In effect, the center of scaling is moved to (0, 0), the scaling is applied, and the scaling center is then moved back to where it was.

Figure 8-11 Changing the amount of scaling specified by a mapping

Figure 8-12 shows the use of the ScaleMapping function scale for various horizontal and vertical factors, in which the center of scaling corresponds to the center of the shape. The ScaleMapping function is described on page 8-69.

Figure 8-12 Scaling horizontally and vertically

Note that if vFactor equals hFactor, scaling is uniform in both directions. If vFactor is not equal to hFactor, distortion of the image occurs, as shown in Figure 8-12.

The mapping matrix also accommodates reflection transformations. If hFactor is negative, a reflection about the vertical axis occurs. If vFactor is negative, a reflection about the horizontal axis occurs. If both vFactor and hFactor are negative, a 180 rotation occurs.

Rotation

You can use the RotateMapping function to modify the rotation specified by a mapping. Matrix elements a, b, c, and d together specify the angle of rotation. Figure 8-13 shows what happens to a mapping M when you call RotateMapping to rotate by an angle b about a rotational origin of xCenter and yCenter. First, a relative translation of -xCenter and -yCenter moves the center of rotation to (0, 0); then a purely rotational matrix adds b to the amount of rotation already specified in the mapping; finally, another relative translation moves the center of rotation by +xCenter and +yCenter, back to where it was.

Figure 8-13 Changing the degree of rotation specified by a mapping

Figure 8-14 shows the use of the RotateMapping function to change the rotation of a mapping. Note that positive values of the angle parameter cause clockwise rotation (consistent with y values increasing downward), and note also that changing the center of rotation can significantly change the final position of the rotated objects. The RotateMapping function is described on page 8-70.

Figure 8-14 Rotating about different center points

Skewing

You can use the SkewMapping function to modify the skewing imposed by a mapping. Matrix elements b and c control the amount of the skew. Element b controls skew in the y direction and element c controls skew in the x direction. Figure 8-15 shows what happens to a mapping M when you call SkewMapping with x and y skew factors of xSkew and ySkew, and a skew origin (the point at which no shearing takes place) of xCenter and yCenter. First, a relative translation of -xCenter and -yCenter moves the center of skewing to (0, 0); then a purely skewing matrix modifies the amount of skew already specified in the mapping; finally, another relative translation moves the center of skewing by +xCenter and +yCenter, back to where it was.

Figure 8-15 Changing the amount of skew specified by a mapping

Figure 8-16 shows the use of the SkewMapping function to change the skew specified by a mapping. (Note that the skew in the x direction in Figure 8-16 is negative; as y decreases--upward--the amount of shear in the x direction increases.) The SkewMapping function is described on page 8-71.

Figure 8-16 Skewing a shape both horizontally and vertically

Perspective

You can manipulate the elements of a mapping to modify its specification of perspective. The matrix elements u, v, and w determine how the perspective will appear when the mapping is applied. The action performed on a point by a mapping whose perspective elements are nonzero is shown in Figure 8-18.

Figure 8-17 Changing the perspective specified by a mapping

There is currently no QuickDraw GX function that modifies the perspective-controlling elements of a mapping for you. If you wish to create perspective, you need to modify the individual matrix elements directly.

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© Apple Computer, Inc.
7 JUL 1996