1 Screen Coordinate System W orld Coordinate System Problem s - - PDF document

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CS 4 7 3 1 : Com put e r Gr a phics Le ct ur e 4 : 2 D Gr a phic Syst e m s Em m anuel Agu

Announcem ents

• Room: if few people drop class, I will request room change
• Project 1 should work on any of the CCC unix/ Linux

m achines

• Sim ply let TA know which m achine you worked on
• m yWPI:

2 D Graphics: Coordinate System s

• Screen coordinate system
• World coordinate system
• World window
• Viewport
• Window to Viewport m apping

Screen Coordinate System

• Screen: 2D coordinate

system (WxH)

• 2D Regular Cartesian Grid
• Origin (0,0) at lower left

corner (OpenGL convention)

• Horizontal axis – x
• Vertical axis – y
• Pixels: grid intersections

(0,0) y x (2,2)

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Screen Coordinate System Insert screen dump from OHI O (0,0) W orld Coordinate System

• Problem s with drawing in screen coordinates:
• I nflexible
• Difficult t o use
• One m apping: not applicat ion specific
• World Coordinate system : application- specific
• Exam ple: drawing dim ensions m ay be in m eters, km , feet,

etc.

Definition: W orld W indow

• World Window: rectangular region of drawing (in world

coordinates) to be drawn

• Defined by W.L, W.R, W.B, W.T

W.L W.R W.B W.T

Definition: View port

• Rectangular region in the screen used to display drawing
• Defined in screen coordinate system

V.L V.R V.B V.T

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W indow to View port Mapping

• Would like to:

• Need som e sort of m apping
• Called Window- to- viewport m apping
• Basic W- to- V m apping steps:

W indow to View port Mapping ( OpenGL W ay)

• Define window (world coordinates):

• Define Viewport (screen coordinates):

glViewport(left, bottom, right -left, top -bottom)

• All subsequent drawings are autom atically m apped
• Do m apping before any drawing (glBegin( ), glEnd( ))
• Two m ore calls you will encounter to set up m atrices:

• Type in as above for now, will explain later
• Ref: Hill Practice exercise 3.2.1, pg 86

W indow to View port Mapping ( Our W ay)

• How is window- to- viewport m apping done?
• Trigonom etry: derive Window- to- Viewport m apping
• Basic principles:

• You are given:

• Required: Calculate corresponding point (s.x, s.y) in screen

coordinates

W indow to View port Mapping ( Our W ay)

(x,y) (sx,sy)

L V R V L V Sx L W R W L W x . . . . . ) . ( − − = − − B V T V B V Sy B W T W B W y . . . . . ) . ( − − = − −

W.R-W.L W.R-W.L V.R-V.L V.R-V.L

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W indow to View port Mapping ( Our W ay)

L V R V L V Sx L W R W L W x . . . . . ) . ( − − = − −

B V T V B V Sy B W T W B W y . . . . . ) . ( − − = − −

Solve for S x, Sy in t er m s of x , y :

      − − − −       − − = L V L W L W R W L V R V x L W R W L V R V Sx . . . . . . . . . .       − − − −       − − = B V B W B W T W B V T V y B W T W B V T V Sy . . . . . . . . . .

W indow to View port Mapping ( Our W ay)

Solve, given t he form ulas:

      − − − − − − = L V L W L W R W L V R V x L W R W L V R V Sx . . . . . . . . . .       − − − − − − = B V B W B W T W B V T V y B W T W B V T V Sy . . . . . . . . . . ) 2 , , 4 , ( ) . , . , . , . ( = = T W B W R W L W W

What is ( S x, Sy) for point ( 3.4,1.2) in world coordinat es if:

) 240 , 80 , 380 , 60 ( ) . , . , . , . ( = = T V B V R V L V V

W indow to View port Mapping ( Our W ay)

Solution:

      − − − − − − = L V L W L W R W L V R V x L W R W L V R V Sx . . . . . . . . . .       − − − − − − = B V B W B W T W B V T V y B W T W B V T V Sy . . . . . . . . . . 332 60 80 = + = x Sx 176 80 80 = + = x Sy

Hence, point ( 3.4,1.2) in world = point ( 332,176) on screen

More W - t o-V Mapping

• W- to- V Applications:

• Mapping different window and viewport aspect ratios

( W/ H)

• Exam ple:

Window Viewport I m portant: Please read on your own, section 3.2.2 on pg. 92 of Hill

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Tiling: Exam ple 3 .2 .4 of Hill ( pg. 8 8 )

• Problem : want to tile dino.dat in 5x5 across screen
• Code:

gluOrtho2D(0, 640.0, 0, 440.0); for(int i=0;i < 5;i++) { for( int j = 0;j < 5; j++) { glViewport(i * 64, j * 44; 64, 44); drawPolylineFile(dino .dat); } }

Zoom ing

• Problem:
• dino. dat is current ly drawn on ent ire screen.

corners

• Solution:

Using m ouse to select screen rectangle for zoom ing ( Exam ple 2 .4 .2 , pg 6 4 ) for zoom ing

void myMouse(int button, int state, int x, int y) { static GLint corner[2]; static int numCorners = 0; // initialize if(button == GLUT_LEFT_BUTTON && state == GLUT_DOWN) { corner[ numCorners ].x = x; corner[ numCorners ].y = y; numCorners++; if(numCorners == 2) { glViewport(corner[0], corner[1]); numCorners = 0; } } }

Cohen- Sutherland Clipping

• Frequently want to view only a portion of the picture
• For instance, in dino.d at, you can select to view/ zoom in
• n only the dinosaur’s head
• Clipping: elim inate portions not selected
• OpenGL autom atically clips for you
• We want algorithm for clipping
• Classical algorithm : Cohen- Sutherland Clipping
• Picture has 1000s of segm ents : efficiency is im portant

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Clipping Points

(xmin, ym in) (xmax, ymax)

• Det erm ine whet her a point

( x,y) is inside or out side of t he world window? If ( xmin < = x < = xm ax ) and (ymin < = y < = ymax) then the point (x,y) is inside else the point is outside

Clipping Lines

• 3 cases:

(xmin, ym in) (xmax, ymax) 1 2 3

Clipping Lines: Trivial Accept

• Case 1: All of line in
• Test line endpoint s:
• N ot e : sim ply com paring x,y

values of endpoint s t o x,y values of rect angle

• Result : t rivially accept .
• Draw line in com plet ely

(Xmin, Ymin) (Xmax, Ymax) p1 p2 Xmin < = P1.x, P2.x < = Xmax and Ymin < = P1.y, P2.y < = Ymax

Clipping Lines: Trivial Reject

• Case 2: All of line out
• Test line endpoint s:
• N ot e : sim ply com paring x,y

values of endpoint s t o x,y values of rect angle

• Result : t rivially rej ect .
• Don’t dr aw line in

p1 p2

p1.x, p2.x < = Xmin

OR p1.x, p2.x > = Xmax OR p1.y, p2.y < = ymin OR p1.y, p2.y > = ym ax

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Clipping Lines: Non- Trivial Cases

• Case 3: Part in, part out
• Two variat ions:

line cuts through viewport

• Need t o find inside segm ent s
• Use sim ilar t riangles t o figure
• ut lengt h of inside segm ent s

e p2 p1 d delx dely

delx e dely d = Cohen- Sutherland pseudocode ( fig. 3 .2 3 )

int clipSegment (Point2& p1, Point2& p2, RealRect W) { do{ if(trivial accept) return 1; // whole line survives if(trivial reject) return 0; // no portion survives // now chop if(p1 is outside) // find surviving segment else(p2 is outside) // find surviving segment }while(1) }

Cohen- Sutherland I m plem entation

• Breaks space int o 4-bit words

• Syst em at ically chops against

four edges

• Can use C/ C+ + bit operat ions
• I m port ant : read Hill 3.3

FFFF TFFT FFFT FFTT TFFF TTFF FTFF FTTF FFTF

Param etric Equations

• I m plicit form
• Param etric form s:

• Som e algorithm s work in param etric form

) , ( = y x F t P P P t P * ) ( ) (

1

− + = 1 ≤ ≤ t

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References

• Hill, 3.1 – 3.3, 3.8