Working with the Camera - Essentials of Interactive Computer Graphics - page 420

with a mouse-click position pt dc :

pt dc =( x dc , y dc ) ,

or expressed as a vector:

V dc = x dc y dc .

DC-to-NDC transform. Recall that in 2D, the NDC space is
NDC space: −

1 ≤ x ndc ≤ 1 ,

− 1 ≤ y ndc ≤ 1 .

×

To transform from DC to NDC space, we scale the DC space down to 2 2and

then translate the center of the region to the origin, or:

2
W dc ,

2
H dc )

V ndc = V dc S ( T ( − 1 ,− 1 ) .

Expanding the scale and translation operators, we get

= x ndc y ndc

V ndc

= x dc y dc S

2 2

(

W dc , H dc )

T ( − 1 ,− 1 )

2 x dc
W dc H dc T

2 y dc

= ( − 1 ,− 1 )

2 x dc

1

2 y dc

W dc − H dc −

= 1 ,

or

2 × x dc

x ndc =

W dc −

1 ,

(15.6)

2 × y dc

y ndc =

H dc −

1 .

Because the mouse-click position is in 2D, we cannot compute the z -component
for the transformed NDC result. In other words, any z value is valid, or

x ndc y ndc z ndc

V ndc = ,

where (from Equation (15.6)):
⎧
⎨

2 × x dc

x ndc =

W dc −

1 ,

2 × y dc

y ndc =

H dc −

1 ,

⎩

− 1 ≤ z ndc ≤ 0 .

As illustrated in Figure 15.12, V ndc actually defines a line in the NDC space.
The mouse-click position can be considered to be located on the near plane with
(

x ndc , y ndc , 0 ) , which defines a line that is parallel to the z -axis and exists from the

far plane at ( x ndc , y ndc ,− 1 ) .