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1Computer Graphics
Computer Graphics
Homogeneous Coordinates & Transformations
Lecture 11/12
John Shearer Culture Lab – space 2
[email protected]://di.ncl.ac.uk/teaching/csc3201
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2Computer Graphics
Homogeneous CoordinatesThe homogeneous coordinates form for a three dimensional point [x y
z] is given as
p =[x’ y’ z’ w] T =[wx wy wz w] T
We return to a three dimensional point (for w0) byxx’/wyy’/wzz’/wIf w=0, the representation is that of a vectorNote that homogeneous coordinates replaces points in three
dimensions by lines through the origin in four dimensionsFor w=1, the representation of a point is [x y z 1]
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3Computer Graphics
Homogeneous Coordinates and Computer Graphics
• Homogeneous coordinates are key to all computer graphics systems– All standard transformations (rotation, translation, scaling)
can be implemented with matrix multiplications using 4 x 4 matrices
– Hardware pipeline works with 4 dimensional representations
– For orthographic viewing, we can maintain w=0 for vectors and w=1 for points
– For perspective we need a perspective division
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4Computer Graphics
Affine Transformations
• Every linear transformation is equivalent to a change in reference frame
• Every affine transformation preserves lines• However, an affine transformation has only 12
degrees of freedom because 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations
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5Computer Graphics
General Transformations
A transformation maps points to other points and/or vectors to other vectors
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6Computer Graphics
Affine Transformations
• Line preserving• Characteristic of many physically important
transformations– Rigid body transformations: rotation, translation– Scaling, shear
• Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints
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7Computer Graphics
Pipeline Implementation
transformation rasterizer
u
v
u
v
T
T(u)
T(v)
T(u)T(u)
T(v)
T(v)
vertices vertices pixels
framebuffer
(from application program)
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8Computer Graphics
Translation
• Move (translate, displace) a point to a new location
• Displacement determined by a vector d– Three degrees of freedom– P’=P+d
P
P’
d
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9Computer Graphics
How many ways?
Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way
object translation: every point displaced by same vector
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10Computer Graphics
Translation Using Representations
Using the homogeneous coordinate representation in some frame
p=[ x y z 1]T
p’=[x’ y’ z’ 1]T
d=[dx dy dz 0]T
Hence p’ = p + d or
x’=x+dx
y’=y+dy
z’=z+dz
note that this expression is in four dimensions and expressespoint = vector + point
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11Computer Graphics
Translation MatrixWe can also express translation using a 4 x 4 matrix T in homogeneous coordinatesp’=Tp where
T = T(dx, dy, dz) =
This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together
1000
d100
d010
d001
z
y
x
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12Computer Graphics
Translation Matrix• glTranslate produces a translation
by (x,y,z)• The current matrix is multiplied by
this translation matrix, with the product replacing the current matrix, as if glMultMatrix were called with that same matrix for its argument
void glTranslatef (GLfloat x , GLfloat y , GLfloat z );
1 0 0 x
0 1 0 y
0 0 1 z
0 0 0 1
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13Computer Graphics
Rotation (2D)Consider rotation about the origin by degrees
– radius stays the same, angle increases by
x’=x cos –y sin y’ = x sin + y cos
x = r cos y = r sin
x = r cos (y = r sin (
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14Computer Graphics
Rotation about the z axis
• Rotation about z axis in three dimensions leaves all points with the same z– Equivalent to rotation in two dimensions in planes of
constant z
x’=x cos –y sin y’ = x sin + y cos z’ =z
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15Computer Graphics
Rotation Matrix
1000
0100
00 cossin
00sin cos
R = Rz() =
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16Computer Graphics
Rotation about x and y axes
• Same argument as for rotation about z axis– For rotation about x axis, x is unchanged– For rotation about y axis, y is unchanged
R = Rx() =R = Ry() =
1000
0 cos sin0
0 sin- cos0
0001
1000
0 cos0 sin-
0010
0 sin0 cos
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17Computer Graphics
General Rotation About the Origin
x
z
yv
A rotation by about an arbitrary axiscan be decomposed into the concatenationof rotations about the x, y, and z axes
R() = Rz(z) Ry(y) Rx(x)
x y z are called the Euler angles
Note that rotations do not commuteWe can use rotations in another order butwith different angles
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18Computer Graphics
Rotation Matrix• glRotate produces a rotation of angle
degrees around the vector (xyz)• The current matrix is multiplied by this
rotation matrix with the product replacing the current matrix, as if glMultMatrix were called with that same matrix for its argument
void glRotatef (GLfloat angle , GLfloat x , GLfloat y , GLfloat z );
Where c=cos(angle), s=sin(angle)and ||(xyz)||=1 (if not, the GL will
normalize this vector).
x2(1−c)+c xy(1−c)−zs xz(1−c)+ys x
yx(1−c)+zs y2(1−c)+c yz(1−c)−xs y
xz(1−c)−ys yz(1−c)+xs z2(1−c)+c z
0 0 0 1
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19Computer Graphics
Rotation About a Fixed Point other than the Origin
Move fixed point to originRotateMove fixed point backM = T(pf) R() T(-pf)
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20Computer Graphics
Scaling
1000
000
000
000
z
y
x
s
s
s
S = S(sx, sy, sz) =
x’=sxxy’=syxz’=szx
p’=Sp
Expand or contract along each axis (fixed point of origin)
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21Computer Graphics
Scale Matrix• glScale produces a nonuniform
scaling along the x, y, and z axes. The three parameters indicate the desired scale factor along each of the three axes.
• The current matrix is multiplied by this scale matrix, with the product replacing the current matrix as if glScale were called with that same matrix as its argument
void glScalef (GLfloat x , GLfloat y , GLfloat z );
x 0 0 0
0 y 0 0
0 0 z 0
0 0 0 1
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22Computer Graphics
Reflection
corresponds to negative scale factors
originalsx = -1 sy = 1
sx = -1 sy = -1 sx = 1 sy = -1
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23Computer Graphics
Inverses
• Although we could compute inverse matrices by general formulas, we can use simple geometric observations– Translation: T-1(dx, dy, dz) = T(-dx, -dy, -dz)
– Rotation: R -1() = R(-)• Holds for any rotation matrix• Note that since cos(-) = cos() and sin(-)=-sin()R -1() = R T()
– Scaling: S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz)
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24Computer Graphics
Concatenation
• We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices
• Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p
• The difficult part is how to form a desired transformation from the specifications in the application
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25Computer Graphics
Order of Transformations
• Note that matrix on the right is the first applied
• Mathematically, the following are equivalent
p’ = ABCp = A(B(Cp))
• Note many references use column matrices to represent points. In terms of column matrices
p’T = pTCTBTAT