Page 218 - DCAP507_SYSTEM_SOFTWARE

System Software

Notes 5. One common mistake is to "go all the way" and always use the ………………….. macro, or
its misnamed predecessor AC_CANONIC_SYSTEM.
6. Canonic syntax was introduced for ………………….. in Windows 7.

7. When coding queries in applications running on Windows 7 and later, you must use
canonic syntax to programmatically generate …………………. .

14.2 Specification Translation

Reduction of the general equation of the second degree to canonic form
Any equation of the second degree
2
2
2
1. f(x, y, z) = ax + by + cz + 2fyz + 2gxz + 2hxy + 2px + 2qy + 2rz + d = 0
Figure 14.1

can be reduced to one of 17 different canonic forms by a suitable translation and rotation. Each
canonic form represents a quadric surface. Figure 14.1 shows a quadric surface (an ellipsoid)
along with its canonic coordinate system x -y -z located at some point (x , y , z ) in space
c c c 0 0 0
(as referred to the X-Y-Z system). Reduction of a particular second degree equation to canonic
form involves the following steps:
1. Determining the location (x , y , z ) of the origin of the canonic system x-y -z of the surface.
0 0 0 c c c
2. Determining the orientation of the x -y -z system (as referred to the X-Y-Z system).
c c c
3. Determining the expression for our equation as expressed with respect to the x -y -z
c c c
system by performing those substitutions associated with a translation of the X-Y-Z system
to the point (x , y , z ) and then a rotation to the orientation to the x -y -z system.
0 0 0 c c c
Determining the Origin (x , y , z ) of the Canonic System
0 0 0
How is the point (x , y , z ) found? If the quadratic surface has a center, point (x , y , z ) corresponds
0 0 0 0 0 0
to a center. A quadric surface may have a single center, a line of centers, or a plane of centers. If
there is more than one center, translation to any center will do. Of the 17 quadric surfaces, 14
have centers. We compute the coordinates of a center using the equation:

x
éa h g ùé ù é ù p
ê h b f úê ú ê ú = 0
+ q
y
ê úê ú ê ú
z
ê g f c úê ú ê ú r
ë ûë û ë û
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