Two Dimensional Viewing

Viewing Pipeline

Window – world coordinate area for display

Viewport – area on display device to which window is mapped

Window – what is viewed

Viewport – where it is displayed

Viewing Transformations – mapping part of a world coordinate scene to device coordinates

aka – Window-Viewport Mapping

2D Viewing-Transformation Pipeline

Viewing Coordinate Reference Frame

Procedure:

1. Set up viewing-coordinate origin at some world position Po(xo,yo)
2. Set up orientation of reference frame

e.g. could set up view-up vector

can compute component of u = (ux, uy) & v = (vx,vy)

3. obtain matrix for converting world coordinates to viewing coordinates

i. translate viewing origin to world origin

ii. rotate to align two coordiate reference frame

Window-to-Viewport Coordinate Transformation

• Device transformations using:
• Xwl, Xwh, Ywl, Ywh
• Xvl, Xvh, Yvl, Yvh

• Use Transformation:

1. Set up windowvl

2. Translate window

3. Scale to normalize

4. Scale to viewport

5. Translate to Viewport

What are the final formulas?

Xv = [(Xvh - Xvl)/(Xwh - Xwl)] (X - Xwl) + Xvl

Yv = [(Yvh - Yvl) / (Ywh - Ywl)] (Yw - Ywl) + Yvl

Window-Viewport Mapping
Xwh	Xwl	Ywh	Ywl	Xwh-Xwl	Ywh-Ywl
7.5	2.5	9	2.5	5	6.5
Xvh	Xvl	Yvh	Yvl	Xvh-Xvl	Yvh-Yvl
900	400	700	400	500	300
No.	x	y	X-Viewport	Y-Viewport
1	1.0	4.0	250.000	469.231
2	4.0	10.0	550.000	746.154
3	8.0	8.0	950.000	653.846
4	8.0	4.0	950.000	469.231
1	1.0	4.0	250.000	469.231
Window Coordinates			Viewport Coordinates
x	y		x	y
2.5	2.5		400	400	Window/Viewport
2.5	9		400	700	Outline
7.5	9		900	700
7.5	2.5		900	400
2.5	2.5		400	400
1.0	4.0		250	469	Data Points
4.0	10.0		550	746
8.0	8.0		950	654
8.0	4.0		950	469
1.0	4.0		250	469

Clipping Operations

Clipping Algorithms -> procedure that identifies portions of picture either inside or outside specified regions

Clip Window -> region against which an object is clipped

Applications of Clipping:

• Extracting part of defined scene for viewing
• Identifying visible surfaces in 3D Views
• Anti-aliasing line segments or object boundaries
• Creating objects using solid modelling procedures
• Drawing and painting operations

· Clipping operations eliminate everything outside window

· Two Approaches:

1. Clip word coordinates -> only contents of window interior mapped to device coordinates
2. Map all world coordinates to device coordinates -> then clip against viewport boundaries

· Raster systems -> clipping part of scan conversion

Point Clipping

Rectangular clipping window…

Point P(x,y) saved for display IF

Xwl <= x <= Xwh

Ywl <= y <= Ywh

Line Clipping

Line Clipping Procedure:

1. Test line segment to determine if it resides completely inside clipping window

2. Test to see if it is completely outside

3. It we are unable to tell, perform intersection tests

Inside-Outside Test

· Check both endpoints -> if both P1 and P2 are within boundaries – line is saved

· If both endpoints outside – drop line segment

· Remaining may require calculation of multiple intersections

Sutherland-Cohen Line Clipping

· Every line endpoint is assigned a 4 bit Region code.

· The bit is set depending on the location of the endpoint with respect to that window component as shown below:

Bit 1: Left of window Bit 2: Right of window Bit 3: Below window Bit 4: Above window
Example: P1 -> 0001, P 2 -> 1000 P3 -> 0001, P4 -> 0100 P5 -> 0000, P6 -> 0010 P7 -> 0001, P8 -> 0001

Determine the bit code by testing the endpoints with window as follows:

o If x is less than Xwl then set bit 1

o If x is greater than Xwh then set bit 2

o If y is less than Ywl then set bit 3

o If y is greater than Ywh then set bit 4

§ Possible Algorithm

o Calculate difference between endpoint coordinates and clipping boundary

o Use sign bit of difference to set region code

x – Xwl bit 1

Xwh – x bit 2

y – Ywl bit 3

Ywh – y bit 4

o Determine Visibility

· Any line whose endpoints have region codes of 0000 is visible

o Any lines that have 1 at same bit position for each endpoint are outside

· Use AND operation as test

1001

0101

0001

· If result is not zero – line is completely outside

o All other lines must be tested against boundaries -> one at a time

§ Procedure:

o Begin with P1 and lower boundary and check against left, right, and bottom boundaries

o Find intersection point P1’ with bottom boundary and replace P1 with P1’

o Do same with P2 but we find two intersections p2’ and p2’’

o P2’ is above window but P2’’ is on window boundary

Computing line intersections

§ Need to fined (x,y) position alone a line segment

§ We know …

y = mx + b

m = ( y2 – y1) / ( x2 – x1 )

b = y1 – m x1

so…

y = m x + y1 – m x1

y = y1 + m ( x – x1)

§ substitute for x => Xwh or Xwl to calculate y coordinate of intersection

also

x = x1 + ( y – y1) / m

§ substitute for y => Ywh or Ywl to calculate x coordinate of intersection

§ Example:

P1 : (1, 4 ) and P2 : ( 4, 10 )

Xwh	Xwl	Ywh	Ywl
7.5	2.5	9	2.5

Answer:

Intersection with Ywh == x = 3.5

Intersection with Xwl == y = 7

So…

P1’ = ( Xwl, 7) = ( 2.5, 7 )

P2’ = ( 3.5, Ywh ) = ( 3.5, 9 )

3D Display Methods

Synthetic Camera Model

• Position and orientation
• Angle of View
• Depth of Field

Projection Methods

• Parallel

Perspective

Depth Cueing

Visible Line and Surface Identification

Surface Rendering

Wireframe => Surface Rendering => Light, Texture, etc.

Exploded View

Cutaway View

3D and Stereoscopic Views

Rendering Pipeline

Transformation Pipeline

3D Viewing

Projections

• Parallel Projection

Coordinate positions transformed to viewplane with parallel lines

• Perspective Projection

Coordinate positions transformed to viewplane with converging lines

Parallel Projections

• Orthographic Projection => used for front, side, back, top, bottom views

Transformation equations for parallel projection

Orthographic Projection:

x_p = x

y_p = y

Oblique Projection

Perspective Projections – general case

…where ( x_prp, y_prp, z_prp ) is the perspective reference point

Begin with parametric equation for line…

x_p = x - (x – x_prp)u

y_p = y - (y – y_prp)u

z_p = z - (z – z_prp)u

u = 0 -> 1

solve for u where z_vp = z_p (viewplane)

substitute in x_p, y_p equations

multiplying through and rearranging…

Special Case -- z_vp = z_p = 0

Example: Special Case where z_vp = z_p = 0

Perspective Projection				Xprp	Yprp	Zprp
				0.5	0.5	-1.5
Point	x	y	z	x'	y'	z'
1	0.30	0.30	0.30	0.33	0.33	0
2	0.30	0.70	0.30	0.34	0.67	0
3	0.30	0.30	0.70	0.36	0.36	0
4	0.30	0.70	0.70	0.37	0.64	0
5	0.70	0.70	0.70	0.64	0.64	0
6	0.70	0.30	0.70	0.64	0.36	0
7	0.70	0.70	0.30	0.67	0.67	0
8	0.70	0.30	0.30	0.67	0.33	0

Example Too:

Surface of Revolution
	x1	y1	z1	x2	y2	z2
0	0.750	0.250	0.000	0.25	0.750	0.000
60	0.375	0.250	-0.650	0.125	0.750	-0.217
120	-0.375	0.250	-0.650	-0.125	0.750	-0.217
180	-0.750	0.250	0.000	-0.25	0.750	0.000
240	-0.375	0.250	0.650	-0.125	0.750	0.217
300	0.375	0.250	0.650	0.125	0.750	0.217
360	0.750	0.250	0.000	0.25	0.750	0.000

		xc		yc		zc
Perspective Projection		0.1		3		-10
X1p	Y1p		X2p		Y2p
0.750	0.250		0.250		0.750
0.387	0.059		0.123		0.700
-0.415	0.059		-0.132		0.700
-0.750	0.250		-0.250		0.750
-0.340	0.418		-0.118		0.798
0.364	0.418		0.127		0.798
0.750	0.250		0.250		0.750

12.4 View Volumes and General Projections

• View window or projection window used to set up telephoto or wide angle scene

Focal Lengths and Angles of View

35mm Camera	Focal Length(mm)	Angle of View (Degrees)
Extreme Telephoto	800	3.5
	400	6.0
	200	12.5
Moderate Telephoto	135	18.0
	85	29.0
	50	46.0
Normal	43	53.0
Moderate Wide Angle	24	84.0
Wide Angle	18	94.0

Viewing

• Edges of window parallel to x.y axes

a – parallelepiped b – frustum

Symmetric Perspective-Projection Frustum

Window dimensions specify field of view

• Second equation used by graphics libraries (eg OpenGL)
• These formulas are substituted into the perspective projection formulas above.
• The perspective equations map the symmetric frustum into a parallelepiped view volume

Normalized Perspective Projection View Coordfinates

Three Dimensional Clipping

3D Clipping Planes for Symmetric Cube :