Line and Polygon ClippingFoley & Van Dam, Chapter 3

Topics• Viewing Transformation Pipeline in 2D• Line and polygon clipping• Brute force analytic solution• Cohen-Sutherland Line Clipping Algorithm• Cyrus-Beck Line Clipping Algorithm• Sutherland-Hodgman Polygon Clipping• Sampling Theorem (Nyquist Frequency)

Viewing Transformation in 2D

x world

y worldx view

y view

World and Viewing Coordinates

Normalized Device Coordinates

1

1

x view

y view

Clipping

Device Coordinates

Viewing Transformation in 2D

A scene in World-Coordinates

World-Coordinates to Viewing Coordinates

Viewing Coordinates to Normalized Device Coordinates

Normalized Device Coordinates to Device Coordinates

WC

VC

NDCDC

Clipping

• Objects are given in world coordinates• The world is viewed through a window• The window is mapped onto a device viewport

Line and Polygon Clipping

The problem:Given a set of 2D lines or polygons and a window, clip the lines or polygons to their regions that are inside the window

Motivations

• Efficiency• Display in portion of a screen• Occlusions

clip rectangle

Line Clipping

• We will deal only with lines (segments)

• Our window can be described by two extreme points:

(xmin,ymin) and (xmax,ymax)

• A point (x,y) is in the window iff:xmin ≤ x ≤ xmax and ymin ≤ y ≤ ymax

Brute Force Analytic Solution

• The intersection of convex regions is always convex

• Since both W and S are convex, their intersection is convex, i.e a single connected segment of S

Question: Can the boundary of two convex shapes intersect more than twice?

0, 1, or 2 intersections between a line and a window

W W W

S S S

Pseudo Code for Midpoint Line Drawing

Assume x1>x0 and 0 < slope ≤ 1

Line(x0,y0,x1,y1)begin

int dx, dy, x, y, d, ∆E, ∆ΝE ;x:= x0; y=y0;dx := x1-x0; dy := y1-y0;d := 2*dy-dx;∆E := 2*dy; ∆ΝΕ := 2*(dy-dx); PlotPixel(x,y);while(x < x1) do

if (d < 0) then d:=d+ ∆E;x:=x+1;

end;else

d:=d+ ∆ΝE;x:=x+1;y:=y+1;

end;PlotPixel(x,y);

end;end;

Line Clipping

Q E

NEM

y = ymin

x = xmin

(xmin,Round(mxmin+B))

(xmin, mxmin+B)

Midpoint Algorithm: Intersection with a vertical edge

Line Clipping

Midpoint Algorithm: Intersection with a horizontal edge

AB y = ymin

x = xmin

y = ymin-1/2

Cohen-Sutherland for Line Clipping• Clipping is performed by computing intersections with four

boundary segments of the window:

Li, i=1,2,3,4

• Purpose: Fast treatment of lines that are trivially inside/outside the window

• Let P=(x,y) be a point to be classified against window W

• Idea: Assign P a binary code consisting of a bit for each edge of W. The bit is 1 if the pixel is in the half-plane that does not contain W

Cohen-Sutherland for Line Clipping

bit 1 0123

y < yminy > ymaxx > xmax

y ≥ yminy ≤ ymaxx ≤ xmax

4 x < xmin x ≥ xmin

0101

0001

01100100

1001

0010

1010

0000

1000ymin

ymax

xmin xmax

1

234

Cohen-Sutherland for Line Clipping

Given a line segment S from p0=(x0,y0) to p1=(x1,y1) to be clipped against a window W

If code(p0) AND code(p1) is not zero, then S istrivially rejected

If code(p0) OR code(p1) is zero, then S is trivially accepted

0101

0001

01100100

1001

0010

1010

0000

1000

Cohen-Sutherland for Line ClippingOtherwise: let assume w.l.o.g. that p0 is outside W

• Find the intersection of S with the edge corresponding to the MSB in code(p0) that is equal to 1. Call the intersection point p2.

• Run the procedure for the new segment (p1 , p2).

A

BC

D

EF

GH

I

1

2

34

Cyrus-Beck Line Clipping

V0

V1

inside

N

U

Inside/Outside Test:• Assume WLOG that V=(V1-V0) is the border vector where

"inside" is to its right

• If V=(Vx,Vy), N is the normal to V, pointing outside, defined by N=(-Vy,Vx)

• Vector U points "outside" if N·U > 0

• Otherwise U points "inside"

Cyrus-Beck Line Clipping

P0

insideN

Q

P1V(t)

L

The parametric line P(t)=P0+(P1-P0)t

The parametric vector V(t)=P(t)-Q

The segment P0P1 intersects the line L at t0 satisfying V(t0)·N=0

The intersection point is P(t0)

∆=P1-P0 points inside if (P1-P0)·N<0. Otherwise it points outside

If L is vertical, intersection can be computed using the explicit equation

Cyrus-Beck Line Clipping

p0p1

Ni Qi

• Denote p(t)=p0+(p1-p0)t t∈[0..1]

• Let Qi be a point on the edge Li with outside pointing normal Ni

• V(t) = p(t)-Qi is a parameterized vector from Qi to the segment P(t)

• Ni· V(t) = 0 iff V(t) ⊥ Ni

• We are looking for t satisfying Ni· V(t) = 0

Cyrus-Beck Line Clipping0 = Ni· V(t)

= Ni· (p(t)-Qi)= Ni· (p0+(p1-p0)t-Qi)= Ni· (p0-Qi) + Ni· (p1-p0)t

Solving for t we get:

where ∆=(p1-p0)

Comment: If Ni·∆=0, t has no solution (V(t) ⊥ Ni)

Ni·(p0-Qi) -Ni·(p1-p0)

t = Ni·(p0-Qi) -Ni·∆

=

Cyrus-Beck Line Clippingp1

p0

PL

PE

p0

p1

PEPE

PL

PL

p1

PL

PEp0

Cyrus-Beck Line Clipping

• The intersection of p(t) with all four edges Li is computed, resulting in up to four ti values

• If ti<0 or ti>1 , ti can be discarded• Based on the sign of Ni·∆, each intersection

point is classified as PE (potentially entering) or PL (potentially leaving)

• PE with the largest t and PL with the smallest tprovide the domain of p(t) inside W

• The domain, if inverted, signals that p(t) is totally outside

Sutherland-HodgmanPolygon-Clipping Algorithm

Sutherland-HodgmanPolygon-Clipping Algorithm

right clip boundary bottom clip boundary

left clip boundary top clip boundary

Idea: Clip a polygon by successively clipping against each (infinite) clip edge

After each clipping a new set of vertices is produced.

Sutherland-HodgmanPolygon-Clipping Algorithm

clip boundary

inside outsides

clip boundary

inside outside

s

p

clip boundary

inside outsidep si

sclip boundary

inside outside

pi

p added tooutput list

i added tooutput list no output i and p added to

output list

p

For each clip edge - scan the polygon and consider the relation between successive vertices of the polygon

Each iteration adds 0, 1 or 2 new vertices

Assume vertex s has been dealt with, vertex p follows:

Sutherland-HodgmanPolygon-Clipping Algorithm

WindowV3

V'2

V”2

V1

V’3

V2

V’1

LeftClipping

RightClipping

BottomClipping

TopClipping

V1V2V3

V1V2V’2V’3

V1V2V’2V’3

V’1V2V’2V”2

Sampling TheoremQuestion: How dense should be the pixel grid in order to draw properly a drawn object?

Given a sampling at intervals equal to d then one may recover frequencies of wavelength > 2d

Aliasing: If the sampling interval is more than 1/2 the wavelength, erroneous frequencies may be produced

Sampling Theorem

1D Example:

0 π 2π 3π 4π 5π 6π

Rule of Thumb: To observe details of size d one must sample at d/2 intervals

To observe details at frequency f (=1/d) one must sample at frequency 2f. The Frequency 2f is the NYQUIST frequency

Sampling Theorem2D Example: Moire’ Effect