Graphics Programs Reference
In-Depth Information
Exercise 1.36: Calculate the expression of the plane containing the z axis and passing
through the point (1 , 1 , 0).
Exercise 1.37: In the plane equation Ax + By + Cz + D =0if D = 0, then the
plane passes through the origin. Assuming D
= 0, we can write the same equation as
x/a + y/b + z/c =1,where a =
D/A , b =
D/B ,and c =
D/C . What is the
geometrical interpretation of a , b ,and c ?
We operate with nothing but things which do not exist, with lines, planes, bodies,
atoms, divisible time, divisible space—how should explanation even be possible when
we first make everything into an image, into our own image!
—Friedrich Nietzsche
In some practical applications, the normal to the plane and one point on the plane
areknown. Itiseasytoderivetheplaneequationinsuchacase.
We assume that N is the (known) normal vector to the plane, P 1 is a known
point on the plane, and P is an arbitrary point in the plane. The vector P P 1 is
perpendicular to N , so their dot product N
( P P 1 ) equals zero. Since the dot
product is associative, we can write N P = N P 1 . The dot product N P 1 is just a
number, to be denoted by s ,soweget
N
P = s
or
N x x + N y y + N z z
s =0 .
(1.25)
Equation (1.25) can now be written as Ax + By + Cz + D =0,where A = N x ,
B = N y , C = N z ,and D =
P 1 . The three unknowns A , B ,and C are the
components of the normal vector, and D can be calculated from any known point P 1
on the plane. The expression N
s =
N
P = s is a useful equation of the plane and is used in
many applications.
Exercise 1.38: Given N ( u, w )=(1 , 1 , 1) and P 1 =(1 , 1 , 1), calculate the plane equa-
tion.
z
r
(0,0,3)
P 1
P 2
w s
y
u r
s
P
P 3
x
(3,0,0)
(a)
(b)
Figure 1.20: (a). A Plane. (b) Three Points on a Plane.
 
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