Geometric Transformation

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Geometric Transformation 3-D Viewing
ECS175 Discussion Session 3

• Hiroshi Akiba Jan 30, 2004

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Outline

• Geometric Transformations
• Translation
• Scaling
• Rotations
• Problem1
• 3-D Viewing
• Perspective Viewing
• Problem2
• GLUT Callback functions

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Geometric Transformation

• Affine Transformation
• Preserve parallel lines and finite points.
• Includes Translation, Rotation, Scaling,
  Reflection, Shear

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3-D Translation

• Translate point P(x,y,z) to P(x,y,z) by
  distance d(tx,ty,tz)

P
P
d
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3-D Scaling

• Scale point P(x,y,z) to P(x,y,z) by a factor
  of s(sx,sy,sz)

P
P
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3-D Rotations (x-axis)

• Rotate point P(x,y,z) to P(x,y,z) by

y
z
x
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3-D Rotations (y-axis)

• Rotate point P(x,y,z) to P(x,y,z) by

y
z
x
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3-D Rotations (z-axis)

• Rotate point P(x,y,z) to P(x,y,z) by

y
z
x
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Transformation Matrices

• Complex transformations are obtained by the
  product of these matrices

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Problem1 Rotation About a Fixed Point

• Q Give a 4x4 matrix M for rotating a cube ?
  about the axis passing through P(-1,2,1) and
  parallel to z axis. Here, P is a center of the
  cube.

y
x
z
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Problem1 Rotation About a Fixed Point

• Solution
• M is the matrix to translate P to somewhere in
  the z axis such as origin, rotate ?about z axis,
  and translate back to P. Let the first
  translation matrix T and the rotation matrix R,
  then
• where
• the answer is verified by

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3-D Viewing

• Viewing(camera) coordinates
• Define position and orientation of a camera
• Object need to be described by this coordinates
  ?superimpose the viewing coordinates onto the
  world coordinates

Yw
Yw
gluLookAt(eye, at, up)

Xv
Yvup
Yv
d
eye
at
Zv
Xv
Zv
Zw
Zw
Xw
Xw
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Perspective Projection

• Projection reference point(center of projection)

Yw
Projection reference point
View Plane
Zw
Xw
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Problem2 Perspective Projection

• Q What are view coordinates of the perspective
  projection of the point P(3,3,-5) onto the view
  plane z-2. Here, the projection reference point
  is at (0,0,1) in the view coordinates.

P
Yw
?
Projection reference point
Zw
Xw
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Problem2 Perspective Projection

• Solution
• Since the point is on the z-2 plane, z
  coordinate is 2
• x and y coordinates are found by following graph
• Hence, the answer is (3/2, 3/2, -2)

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GLUT callback functions

• glutDisplayFunc(disp)
• void disp(void)
• glutKeyboardFunc(key)
• void key(unsigned char key, int x, int y)
• glutMouseFunc(mouse)
• void mouse(int button, int state, int x, int y)
• glutReshapeFunc(reshape)
• void reshape(int width, int height)

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Examples
Void mouse(int button, int state, int x, int y)
if(state GLUT_DOWN buttonGLUT_LEFT_BUTTO
N) exit(1) if(state GLUT_DOWN
buttonGLUT_RIGHT_BUTTON)
glutPostRedisplay()
Void key(char key) if(keyq keyQ
key \27) exit(0)
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References

• 1.Hearn baker, Computer Graphics width OpenGL,
  2004
• 2.Edward Angel, Interactive Computer Graphics A
  Top-Down Approach with OpenGL, 2000
• 3. Edward Angle, OpenGL A PRIMER, 2002
• 4.OpenGL Red Book, http//www.opengl.org/documenta
  tion/red_book_1.0/