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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

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viewing parameters
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

240-422 Computer Graphics : Viewing in 3D_4

viewing parameters2
Viewing Parameters

240-422 Computer Graphics : Viewing in 3D_4

conversion to view plane coordinates
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

240-422 Computer Graphics : Viewing in 3D_4

3d rotation about an arbitrary axis
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

240-422 Computer Graphics : Viewing in 3D_4

graphical illustrations
Graphical Illustrations

240-422 Computer Graphics : Viewing in 3D_4

mathematical illustrations
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

240-422 Computer Graphics : Viewing in 3D_4

mathematical illustrations7
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

240-422 Computer Graphics : Viewing in 3D_4

mathematical illustrations8
Mathematical Illustrations
  • Rotation matrix, Rx
  • Reverse matrix, Rx-1

240-422 Computer Graphics : Viewing in 3D_4

mathematical illustrations9
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

240-422 Computer Graphics : Viewing in 3D_4

mathematical illustrations10
Mathematical Illustrations
  • Rotation matrix, Ry
  • Reverse matrix, Ry-1

240-422 Computer Graphics : Viewing in 3D_4

mathematical illustrations11
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

240-422 Computer Graphics : Viewing in 3D_4

back to view plane coordinates conversion
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

240-422 Computer Graphics : Viewing in 3D_4

mathematical illustrations13
Mathematical Illustrations

V=(Ny2 + Nz2)1/2

240-422 Computer Graphics : Viewing in 3D_4

mathematical illustrations14
Mathematical Illustrations

240-422 Computer Graphics : Viewing in 3D_4

clipping in 3d
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

240-422 Computer Graphics : Viewing in 3D_4

3d clipping
3D Clipping
  • Fig 8-38, 8-39, 8-40

240-422 Computer Graphics : Viewing in 3D_4

3d clipping17
3D Clipping

240-422 Computer Graphics : Viewing in 3D_4

front and back clipping
Front and Back Clipping
  • for the point (x1, y1, z1) to be visible:

z1<= FRONT-Z and z1 >= BACK-Z

240-422 Computer Graphics : Viewing in 3D_4

3d viewing transformation summary
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

240-422 Computer Graphics : Viewing in 3D_4

3d application flight simulator
3D Application: Flight Simulator
  • Fig 8-45, 8-46

240-422 Computer Graphics : Viewing in 3D_4