Polygon Clipping - cs.gettysburg.educs.gettysburg.edu/~ilinkin/courses/Fall-2014/cs373/slides/clipping.pdfPolygon Clipping Examine the edges of given poly relative to current clipper

Polygon ClippingA polygon may be outside the view area (e.g. zoom). Determine what to draw

Polygon ClippingA polygon may be outside the view area (e.g. zoom). Determine what to draw

Polygon ClippingThe Sutherland-Hodgman algorithm uses 4 clippers (for each side of view area)

Polygon ClippingBottom Clipper


Polygon ClippingRight Clipper


Polygon ClippingTop Clipper

Polygon ClippingTop Clipper

Polygon ClippingLeft Clipper

Polygon ClippingLeft Clipper

Polygon ClippingNow we can use the polygon filling algorithm to draw the polygon

Polygon ClippingExamine the edges of given poly relative to current clipper and add point to new poly

s

p

outside viewnothing added


s

p

entering viewoutput i and p

i


s

p

inside viewoutput p only


s

p

exiting viewoutput i only

i


s

p

outside viewnothing added

s

p

entering viewoutput i and p

i

s

p

inside viewoutput p only

s

p

exiting viewoutput i only

i



outside clippernothing added


entering clipperadd i and p


inside clipperadd p only






and so on ...


last edge exiting clipperadd i only




exiting clipperadd i only


outside clipperadd i only




exiting clipperadd i only










and so on ...


last edge inside clipperadd p only


Polygon ClippingThe algorithm works for view area that is any convex polygon (e.g. cockpit view)

VectorsDescribe a direction in space

VectorsDescribe a direction in space

all vectors below represent the same direction

represented by the (x, y) coords of the vector anchored at the origin

VectorsVector (direction) defined by two points

vx = P

1(x) – P

0(x) different from v

x = P

0(x) - P

1(x)

vy = P

1(y) – P

0(y) v

y = P

0(y) - P

1(y)

P0

P1v

v

VectorsVector (direction) defined by two points

vx = P

1(x) – P

0(x)

vy = P

1(y) – P

0(y)

In vector form

v = P0P1 = P

1 - P

0P0

P1v

vP0

P1

Vector OperationsLength

|v| = sqrt(vx·v

x + v

y·v

y)

Addition/Subtraction

w = u ± v wx = v

x ± u

x, w

y = v

y ± u

y

Scalar multiplication

u = c·v ux = c·v

x, u

y = c·v

does not change the direction, just the magnitude

Dot product (vector multiplication)

c = u·v = vx·u

x + v

y·u

y = |v|·|u|·cos a

dot product is a scalar (number), not a vector

Vector OperationsDot product is useful for determining angle between vectors

u·v > 0 if angle < 90

u·v = 0 if angle == 90

u·v < 0 if angle > 90

Line / Edge IntersectionParametric line equation given two points P

0 and P

1 on the line

P = P0 + P

0P1·t , t in (-∞ , +∞)

x = P0(x) + P

0P1(x)·t

y = P0(y) + P

0P1(x)·t

t = 0

P0

P1

t = 1

t < 0

for points

behind P0

t > 0

for points

after P1

Line / Edge Intersection

P0

P1

Pi

Given a clipper and an edge (P0,P

1)

decide if P1 is inside the visible area

find the intersection Pi


P0

P1

Pe

Pi


1)

decide if P1 is inside the visible area – N·P

eP1 < 0 means P

1 is inside

pick a point Pe on the clipper

pick a vector perpendicular to clipper

use dot product

N·PeP1 < 0 means P

1 is inside N


P0

P1

Pe

Pi


1)

decide if P1 is inside the visible area – N·P

eP1 < 0 means P

1 is inside


N·PePi = 0 since N

perp. to clipper

N·(Pi-P

e) = 0 rewritten

P = P0 + P

0P1·t line equation of edge

Pi = P

0 + P

0P1·t

isome t

i will give P

i

N·(P0 + P

0P1·t

i - P

e) = 0 replace P

i

N·(P0-P

e + P

0P1·t

i) = 0 rearrange

N·(PeP0 + P

0P1·t

i) = 0

N·PeP0 + N·P

0P1·t

i) = 0

ti = –N·P

eP0 / N·P

0P1

N

PeP1

PePi

Line / Edge IntersectionGiven a clipper and an edge (P

0,P

1)

decide if P1 is inside the visible area

N·PeP1 < 0 means P

1 is inside


Pi = P

0 + P

0P1·t

iwhere t

i = –N·P

eP0 / N·P

0P1

but since we are interested in P1

Pi = P

1 + P

1P0·t

iwhere t

i = –N·P

eP1 / N·P

1P0

Documents

Polygon Clipping - cs.gettysburg.educs.gettysburg.edu/~ilinkin/courses/Fall-2014/cs373/slides/clipping.pdfPolygon Clipping Examine the edges of given poly relative to current clipper