15-462: Computer Graphics

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!" Transformations

Translation,Translation, rotation,rotation, scalingscaling 2D2D 3D3D HomogeneousHomogeneous coordinatescoordinates TransformingTransforming normalsnormals ExamplesExamples

Shirley Chapter 6 Uses of Transformations • Modeling – build complex models by positioning simple components – transform from object coordinates to world coordinates • Viewing – placing the virtual camera in the world – specifying transformation from world coordinates to camera coordinates • Animation – vary transformations over time to create motion

CAMERA OBJECT

WORLD

Computer Graphics 15-462 3 Rigid Body Transformations

Rotation angle and line about which to rotate

Computer Graphics 15-462 4 Non-rigid Body Transformations

Distance between points on object do not remain constant

Computer Graphics 15-462 5 Basic 2D Transformations Scale

Shear chalkboard

Rotate

Computer Graphics 15-462 6 Composition of Transformations

• Created by stringing basic ones together, e.g. – “translate p to the origin, rotate, then translate back” can also be described as a rotation about p • Any sequence of linear transformations can be collapsed into a single matrix formed by multiplying the individual matrices together • Order matters! • Can apply a whole sequence of transformations at once

chalkboard

0 1 2 3 Translate to the origin, rotate, then translate back.

Computer Graphics 15-462 7 3D Transformations

• 3-D transformations are very similar to the 2-D case • Scale • Shear • Rotation is a bit more complicated in 3-D – different rotation axes

chalkboard

Computer Graphics 15-462 8 Fixed Euler Angles for 3-D Rotations • Independent rotations about each coordinate axis – angle interpolation for animation generates bizarre motions – rotations are order-dependent, and there are no conventions about the order to use • Widely used anyway, because they're “simple”

Computer Graphics 15-462 9 Euler Angles for 3-D Rotations

Computer Graphics 15-462 10 Other representations of 3D orientation • Relative rather than fixed Euler angles • Axis/Angle: rotate by ! about axis V • Quaternions: – a generalization of complex numbers –3 give the rotation axis - magnitude is sin !/2 –1 gives cos !/2 (the amount of rotation) –unit magnitude - points on a 4-D unit sphere

Computer Graphics 15-462 11 But what about translation? •Translation is not linear--how to represent as a matrix?

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Computer Graphics 15-462 12 But what about translation? •Translation is not linear--how to represent as a matrix?

chalkboard

•Trick: add extra coordinate to each vector •This extra coordinate is the homogeneous coordinate, or w •When extra coordinate is used, vector is said to be represented in homogeneous coordinates •We call these matrices Homogeneous Transformations

Computer Graphics 15-462 13 W? Where did that come from? • Practical answer: –W is a clever algebraic trick. –Don’t worry about it too much. The w value will be 1.0 for the time being (until we get to perspective viewing transformations)

• More complete answer: –(x,y,w) coordinates form a 3D projective space. –All nonzero scalar multiples of (x,y,1) form an equivalence class of points that project to the same 2D Cartesian point (x,y). –For 3-D graphics, the 4D projective space point (x,y,z,w) maps to the 3D point (x,y,z) in the same way.

Computer Graphics 15-462 14 Homogeneous 2D Transformations

The basic 2D transformations become Translate: Scale: Rotate: !" %" !" %" 10tx sx 00 !cos" ( )sin( 0%" $" '" $" '" 01ty 0 sy 0 $"sin( cos( 0'" $" '" $" '" $" 0 0 1'" #"0 0 1&" #"0 0 1&" #" &"

Now any sequence of translate/scale/rotate operations can be combined into a single homogeneous matrix by multiplication. 3D transforms are modified similarly

Computer Graphics 15-462 15 Rigid Body Transformations

Rotation angle and line about which to rotate

Computer Graphics 15-462 16 Rigid Body Transformations

•A transformation matrix of the form

xx xy tx

yx yy ty 001

where the upper 2x2 submatrix is a rotation matrix and column 3 is a translation vector, is a rigid body transformation. •Any series of rotations and translations results in a rotation and translation of this form (and no change in the distance between vertices)

Computer Graphics 15-462 17 Sequences of Transformations

x' x' x' x' x' x' x' x' x' TRANSFORMED POINTS M

PARAMETERS Often the same MATRICES transformations are applied M to many points Calculation time for the

M matrices and combination is negligible compared to that of transforming the points x x x x x x x x x x UNTRANSFORMED Reduce the sequence to a POINTS single matrix, then transform

Computer Graphics 15-462 18 Collapsing a Chain of Matrices. • Consider the composite function ABCD, i.e. p’ = ABCDp • Matrix multiplication isn’t commutative - the order is important • But matrix multiplication is associative, so can calculate from right to left or left to right: ABCD = (((AB) C) D) = (A (B (CD))). • Iteratively replace either the leading or the trailing pair by its product

M ! D M ! A

M ! CM or M ! MB both give the same result. M ! BM M ! MC

M ! AM M ! MD Premultiply Postmultiply

Computer Graphics 15-462 19 What is a Normal?– refresher Indication of outward facing direction for lighting and shading

Order of definition of vertices in OpenGL

Right hand rule

Computer Graphics 15-462 20 Transforming Normals • It’s tempting to think of normal vectors as being like porcupine quills, so they would transform like points • Alas, it’s not so. • We need a different rule to transform normals.

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Computer Graphics 15-462 21 Examples

• Modeling with primitive shapes

Computer Graphics 15-462 22 Announcements

• Reading for Thursday: Shirley Ch: 2,6, & 7 • Project 1 due next Tuesday... • Style: 10% • Any questions? • Bulletin Board Mini-Demonstration