1Computer Graphics

Computer Graphics

Homogeneous Coordinates & Transformations

Lecture 11/12

John Shearer Culture Lab – space 2

john.shearer@ncl.ac.ukhttp://di.ncl.ac.uk/teaching/csc3201

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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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

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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General Transformations

A transformation maps points to other points and/or vectors to other vectors

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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

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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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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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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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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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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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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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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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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Rotation Matrix

1000

0100

00 cossin

00sin cos

R = Rz() =

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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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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

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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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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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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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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Reflection

corresponds to negative scale factors

originalsx = -1 sy = 1

sx = -1 sy = -1 sx = 1 sy = -1

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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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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

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