Graphics Reference
In-Depth Information
10.2.2 The Hessian Normal Form
Figure 10.19 shows a line whose orientation is controlled by a normal unit
vector n =[ ab ] T .If P ( x , y ) is any point on the line, then p is a position
vector where p =[ xy ] T
and d is the perpendicular distance from the origin
to the line.
Therefore
d
=cos( α )
p
and
d =
p
cos( α )
(10.28)
But the dot product n · p is given by
n · p =
n p
cos( α )= ax + by
(10.29)
which implies that
ax + by = d
n
(10.30)
and because
n
= 1 we can write
ax + by
d = 0
(10.31)
where ( x , y ) is a point on the line, a and b are the components of a unit vector
normal to the line and d is the perpendicular distance from the origin to the
line. The distance d is positive when the normal vector points away from the
origin, otherwise it is negative.
Let's consider two examples.
Example 1 . Find the equation of a line whose normal vector is [3 4] T
and
the perpendicular distance from the origin to the line is 1.
To begin, we normalize the normal vector to its unit form.
Y
n
d
P ( x, y )
a
X
P
Fig. 10.19. The orientation of a line can be controlled by a normal vector n and
distance d .
 
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