OpenGL7

In Homogeneous coordinate system
x' S x y' 0 = z' 0 w' 0 0 Sy 0 0 0 0 Sz 0 0 x 3 0 0 y . Eg, 0 0 z 1 w 0 0 2 0 0 0 0
1 2
0
0 1 3 0 1 2 = 0 2 1 1 1 1
Chapter 7 3-D Transformations & Hierarchical models
Transformations
Translation Scaling Rotation Reflection Change of coordinate systems
Hierarchical modeling
Scale 2
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Rotation about the origin in 2-D
(x’,y’)
r θ r φ
(x,y) x=r cosφ
In Cartesian, x’ = r cos(φ+θ) = r cosφ cosθ + r sinφ sinθ = x cosθ - y sinθ y’ = r sin(φ+θ) = r sinφ cosθ + r cosφ sinθ = x sinθ + y cosθ
In homogeneous coordinates
x' cosθ y ' sin θ = z' 0 w' 0 sin θ cosθ 0 0 0 0 x 0 0 y 1 0 z 0 1 w
y
In Opengl:
glRotatef( degree, 0., 0., 1. ); z
In Opengl: glTranslatef( Tx, Ty, Tz)
Translate
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Scaling relative to the origin The new position: x’ = Sx × x, y’ = Sy × y, z’ = Sz × z
(1, 1, 2)
(3, 2, 1)
(6, 8, 2)
In general, x’ = x + Tx, y’ = y + Ty, z’ = z + Tz In Homogeneous coordinate system
x' 1 y ' 0 = z ' 0 w' 0 0 0 Tx x 1 0 1 0 Ty y . Eg, 0 0 1 T z z ' 0 0 1 w 0 0 0 5 1 6 1 0 7 1 8 = 0 1 0 2 2 0 0 1 1 1
To rotate about an arbitrary axis
Let the axis passes through (x1, y1, z1), and (x2, y2, z2)
z
(x2, y2, z2)
(x1, y1, z1) y
Step 1 Translate (x1, y1, z1) to the origin
z c
(a, b,c)
α
v y b
Step 3 Rotate about the y-axis by β so that the axis coincides with the z-axis
β = tan 1 (a / v)
x z v y
a (a, 0,v)
β
x
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Step 4 Rotate about the z-axis Step 5 (Reverse of Step 3) Rotate about the y-axis by β Step 6 (Reverse of Step 2) Rotate about x-axis by -α Step 7 (Reverse of Step 1) Translate from the origin to (x1, y1, z1)
1
3-D Transformation
Right-handed 3-D Cartesian coordinate system
1 2
y
(3,2,1) 3
x
The position of a vertex is represented by a 3-tupleIn homogeneous coordinates, a 3-D point is represented by
By replacing x, y, z with y, z, x, respectively in previous formulas, we have y’ = y cosθ - z sinθ z’ = y sinθ + z cosθ x’ = x
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In homogeneous coordinates
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Rotation about xaxis in 3D
y
z
x z x
y
Redraw the coordinate axes so that the case looks like the preceding one.
1. Draw x-axis in the conventional z-axis direction 2. Draw y-axis in the conventional x-axis direction. 3. Determine the z-axis using the right-handed rule.
Rotation about z-axis in 3-D (Same as the 2-D case)
y r θ z r φ
(x,y) x=r cosφ (x’,y’)
x
In Cartesian, x’ = x cosθ - y sinθ y’ = x sinθ + y cosθ z’ = z
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(1, 1, 2)
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3. Backward translation Tx = 1, Ty = 1, Tz = 2 Note that the transformations are specified in reverse order in Opengl
glTranslatef( 1., 1., 2.); glScalef( Sx, Sy, Sz); glTranslatef( -1., -1., -2.); pyramid();
x' cos θ y' 0 = z ' sin θ w' 0 0 sin θ 1 0 0 0 0 cos θ 0 x 0 y 0 z 1 w
y
In Opengl:
glRotatef( degree, 0., 1., 0. ); z x
RotateY
By replacing x, y, z with z, x, y, respectively in previous formulas, we have z’ = z cosθ - x sinθ x’ = z sinθ + x cosθ y’ = y
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In homogeneous coordinates
x y z w
, where w ≠ 0.
x y in z w
x' The relations between y ' in Cartesian and z' homogeneous are
x’ = x/ w, y’ = x / w, z’ = z / w
Eg, the point (3, 2,1) of Cartesian corresponds to (3, 2, 1, 1) and (6, 4, 2, 2) of homogeneous. The point (1, 2, 3, 0.5) in homogeneous corresponds to (2, 4, 6) in Cartesian.
In Homogeneous coordinate system
x' cosθ y ' = sin θ w' 0 sin θ cosθ 0 0 x .707 .707 0 5 2.8 0 y . Eg, .707 .707 0 1 = 4.2 0 1 w 0 1 1 1 10
In Opengl: glScalef( Sx, Sy, Sz)
Scale
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Scaling relative to a fixed point, eg, (1, 1, 2) 1. Transform the point to origin Tx = -1, Ty = -1, Tz = -2 2. Conduct the scaling
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Eg, to rotate 45o
(2.8,4.2) (1,3) (x,y) (-1.4,2.8) (x’, y’) (5,1) (1,1) (0,1.4)
sin 45o = cos 45o = 0.7071 x’ = x cosθ - y sinθ = 0.7071 (x - y) y’ = x sinθ + y cosθ = 0.7071 (x + y)
0 x ' 1 y ' 0 cosθ = z ' 0 sin θ 0 w' 0 0 sin θ cosθ 0 0 x 0 y 0 z 1 w
y
In Opengl:
x glRotatef( degree, 1., 0., 0. ); z
RotateX
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//Rotate deg degree along the axis that passes through // (x1, y1, z1) and (x2, y2, z2)
double double double double a = x2 - x1, b = y2 - y1, c = z2 - z1; v = sqrt(b * b + c * c); alpha = atan( b/c)*180./3.1416; beta = -atan(a/v)*180./3.1416; //Step 1 //Step 2 //Step 3 //Step 4 //Step 5 //Step 6