1. 240-422 Computer Graphics : Viewing in 3D_4
Viewing Parameters
View plane: plane of our display surface
View reference point (VRP): center of attention, all other viewing parameters are expressed relative to this point
View plane normal (VPN): look direction
View distance: distance from camera to VRP
View-up direction: vector pointing to top of camera
View plane coordinates: film coordinates
object coordinates: coordinates that the objects lie
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Computer Graphics : Viewing in 3D_4

2. 240-422 Computer Graphics : Viewing in 3D_4
Viewing Parameters
15-Nov-18
Computer Graphics : Viewing in 3D_4

3. Conversion to View Plane Coordinates
We wish to perform a series of transformations which will change the object coordinates into the view plane coordinates.
First step: translate the origin to the correct position for the view plane coordinate system (shifting to VRP then shifting along the VPN by the VIEW-DISTANCE.
Second step: align the z axis
15-Nov-18
Computer Graphics : Viewing in 3D_4

4. 3D Rotation about an Arbitrary Axis
1. Translate the axis to origin
2. Rotate about x until the axis of rotation is in the xz plane
3. Rotate about y axis until the z axis corresponds to the axis of rotation
4. Rotate about z (axis of rotation)
5. Reverse the rotation about y
6. Reverse the rotation about x
7. Reverse the translation
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Computer Graphics : Viewing in 3D_4

5. Graphical Illustrations
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Computer Graphics : Viewing in 3D_4

6. Mathematical Illustrations
Suppose the rotation axis is defined by a point (x1, y1, z1) and a vector [A B C], so the line equations are
x = Au + x1
y = Bu + y1
z = Cu + z1
The initial translation matrix and its reverse translation are
15-Nov-18
Computer Graphics : Viewing in 3D_4

7. Mathematical Illustrations
Rotation about x axis
V = (B2+C2)1/2
sin(I) = B/V
cos(I) = C/V y y
(A, B, C)
(0, B, C)
(0, B, C) B V x x I C z z
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Computer Graphics : Viewing in 3D_4

8. Mathematical Illustrations
Rotation matrix, Rx
Reverse matrix, Rx-1
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Computer Graphics : Viewing in 3D_4

9. Mathematical Illustrations
Rotation about y y
L = (A2+B2+C2)1/2
V = (L2-A2)1/2=(B2+C2)1/2
sin(J) = A/L
cos(J) = V/L A x J L V z
Rotation axis
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Computer Graphics : Viewing in 3D_4

10. Mathematical Illustrations
Rotation matrix, Ry
Reverse matrix, Ry-1
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Computer Graphics : Viewing in 3D_4

11. Mathematical Illustrations
Rotation about the z axis
The actual transformation for a rotation about an arbitraty axis is given by
R = T-1 Rx-1 Ry-1 Rz Ry Rx T
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Computer Graphics : Viewing in 3D_4

12. Back to “View Plane Coordinates Conversion”
Parameters:
VPR = (xr, yr, zr)
VPN = [Nx, Ny, Nz]
View-up = [xup, yup, zup]
View-distance = VD
The entire transformation is
TMATRIX = Rz Ry Rx T
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Computer Graphics : Viewing in 3D_4

13. Mathematical Illustrations
V=(Ny2 + Nz2)1/2
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Computer Graphics : Viewing in 3D_4

14. Mathematical Illustrations
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Computer Graphics : Viewing in 3D_4

15. 240-422 Computer Graphics : Viewing in 3D_4
Clipping in 3D
Clipping against planes, not against lines as in 2D
Front plane clipping
Back plane clipping
Top plane clipping
Bottom plane clipping
Left plane clipping
Right plane clipping
Clipping Process
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Computer Graphics : Viewing in 3D_4

16. 240-422 Computer Graphics : Viewing in 3D_4
3D Clipping
Fig 8-38, 8-39, 8-40
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Computer Graphics : Viewing in 3D_4

17. 240-422 Computer Graphics : Viewing in 3D_4
3D Clipping
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Computer Graphics : Viewing in 3D_4

18. Front and Back Clipping
for the point (x1, y1, z1) to be visible:
z1<= FRONT-Z and z1 >= BACK-Z
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Computer Graphics : Viewing in 3D_4

19. 3D Viewing Transformation Summary
1. Draw object in the object coordinates
2. Specify viewing parameter (VPR, VPN, VD, etc.)
3. Convert object coordinates to view plane coordinates
4. Perform 3D clipping
5. Project the objects in viewing onto the view plane
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Computer Graphics : Viewing in 3D_4

20. 3D Application: Flight Simulator
Fig 8-45, 8-46
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Computer Graphics : Viewing in 3D_4