Clipping

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Chapter 12
Clipping

12.1 Finite View Volume

Add the following planes.

Hither--A front clipping plane. Objects close to the eye contribute few seemingly unrelated components.
Yon--A back clipping plane. Distant objects contribute meaningless “blobs”.

---
--------
-|------ --- |
-|------| --- |
---|||||||||||||||||||||||||||
-||||--------------- ---
----- ---- | ---
hither ------ | ---
--------
window ---
yon

yon --- ---
--------------------|--
-----------------------| |
hither------------- | |
| | | |
| | ||||||||---------|----
| |||||||||-------------
----||----------

Equations

Lines

x = x₁ + t(x₂ - x₁)
y = y₁ + t(y₂ - y₁)
z = z₁ + t(z₂ - z₁)
Planes Ax + By + Cz + D = 0
Plane/line Intersection: 4 equations, 4 unknowns

12.2 Clipping After Projecting

Against viewing box

12.3 Cohen-Sutherland Cubic 3D Line Clipper

||| |||
zmax|||||------||||||--
zmin |||||| |||||| |
ymax ||||-------||||-- | |
| | ||---
| | |||||
ymin |||--------|||||-

xmin xmax

-----------------------------------------
---- |--- |
---- ----| |
----- ---|- | |
|------------|-----------|---------------- | | |
| | | | | |01 |
1001**1000**1010**--------------------- 00 | |
| | | | 0 | | ----
0001**0000**0010** | |1 | -----
--------------------------------------- -|---
0101**0100**0110** | | ----
----------------------------------------

Each region assumes a 6-bit outcode

|-----|--------|---------|--------|----------
updown###|##right##|###left###|#front##|##back###|
|#####|########|#########|########|#########|
---------------------------------------------

B1 = outcode(P1)
B2 = outcode(P2)
REPEAT
   CASE                      trivial test
     ( B1 OR  B2 )  = 000000 : accept
     ( B1 AND B2 ) <> 000000 : reject
     ELSE                     intersect
       IF B1 <> 000000 THEN
         P1 := (P1,P2) intersect
                side in B1
         B1 = outcode(P1)
       ELSE
         P2 := (P1,P2) intersect
                side in B2
         B2 = outcode(P2)
   END
UNTIL clipping is impossible

12.4 Liang-Barsky Cubic 3D Line Clipper

Parametric formulas

x = x₁ + (x₂ - x₁)t
y = y₁ + (y₂ - y₁)t
z = z₁ + (z₂ - z₁)t
Inequalities
0 <t <1 x_min<x₁ + (x₂ - x₁)t<x_max y_min<y₁ + (y₂ - y₁)t<y_max z_min<y₁ + (z₂ - z₁)t<z_max

| |
| |
| |
t=1o |
o--------------------
| |
| |
Tmax•|| |
--|•||---------------
Tmin ||| |
| ||||
| |o|
|||
||
o t = 0

z -||||-------||||-
max|||| ||||||
zmin|-------||||| |
ymax ||| |||
| | ||||---
| ||||||
ymin------||||--
x x
min max
Tmin := 0
Tmax := 1
dX := X2-X1
adjust(Xmin-X1, dX)
adjust(X1-Xmax,-dX)
dY := Y2-Y1
adjust(Ymin-Y1, dY)
adjust(Y1-Ymax,-dY)
dZ := Z2-Z1
adjust(Zmin-Z1, dZ)
adjust(Z1-Zmax,-dZ)
IF Tmin > 0 THEN
   X1 := X1 + round(dX * Tmin)
   Y1 := Y1 + round(dY * Tmin)
   Z1 := Z1 + round(dZ * Tmin)
IF Tmax < 1 THEN
   X2 := X2 + round(dX * Tmax)
   Y2 := Y2 + round(dY * Tmax)
   Z2 := Z2 + round(dZ * Tmax)

12.5 Sutherland-Hodgman Polygon Clipper

Clip against 6 sides in succession

--| -------| ---------| ---------|
clipclipclip | |clip | |clip | |clip | -||||||||
in --||| --||| --||| --|||out |--
updownright | |left | front | back | |||||||
--- ------- ---------- ---------

FOR clip_side := up, down,
right, left, front, back DO
CLIP polygon AGAINST clip_side

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