Week 5 - Monday What did we talk about last time? Lines and planes - - PowerPoint PPT Presentation

Week 5 - Monday

 What did we talk about last time?  Lines and planes  Trigonometry  Transforms

 A transform is an operation that changes points, vectors, or

colors

 We can use them to position and animate objects, lights, and

cameras

 A linear transform is one that holds over vector addition and

scalar multiplication

• Rotation
• Scaling
• Can be represented by a 3 x 3 matrix

 Adding a vector after a linear transform makes an affine

transform

 Affine transforms can be stored in a 4 x 4 matrix using

homogeneous notation

 Affine transforms:

• Translation
• Rotation
• Scaling
• Reflection
• Shearing

Notation Name Characteristics T(t) Translation matrix Moves a point (affine) R Rotation matrix Rotates points (orthogonal and affine) S(s) Scaling matrix Scales along x, y, and z axes according to s (affine) Hij(s) Shear matrix Shears component i by factor s with respect to component j E(h,p,r) Euler transform Orients by Euler angles head (yaw), pitch, and roll (orthogonal and affine) Po(s) Orthographic projection Parallel projects onto a plane or volume (affine) Pp(s) Perspective projection Project with perspective onto a plane or a volume slerp(q,r,t) Slerp transform Interpolates quaternions q and r with parameter t

 Move a point from one place to another by vector t = (tx, ty, tz)  We can represent this with translation matrix T               = = 1 1 1 1 ) , , ( ) (

z y x z y x

t t t t t t T t T

 Rotation, like translation, is a rigid-body transform (points

don't change distance from each other and handedness doesn't change)

 An orientation matrix is used to define up and forward

(usually for a camera)

 We often express rotation in terms of 3 separate x, y, and z

rotation matrices

              − = 1 cos sin sin cos 1 ) ( φ φ φ φ φ

x

R               − = 1 cos sin 1 sin cos ) ( φ φ φ φ φ

y

R               − = 1 1 cos sin sin cos ) ( φ φ φ φ φ

z

R

 Usually all the rotations are multiplied together before

translations

 But if you want to rotate around a point

• Translate so that that point lies at the origin
• Perform rotations
• Translate back

 Scaling is easy and can be done for all axes at the same time

with matrix S

 If sx = sy = sz, the scaling is called uniform or isotropic and

nonuniform or anisotropic otherwise

              = 1 ) (

z y x

s s s s S

 A shearing transform distorts one dimension in terms of

another with parameter s

 Thus, there are six shearing matrices Hxy(s), Hxz(s), Hyx(s),

Hyz(s), Hzx(s), and Hzy(s)

 Here's an example of Hxz(s):

              = 1 1 1 1 ) ( s s

xz

H

 To make Hij(s), start with the identity matrix and put s in row i,

column j

 See how the only the affected dimension changes

 Because matrix multiplications are not commutative, order of

transforms matters

 Still, parts (or the entirety) of the transform can be

precomputed and stored as a single matrix

 In math world, matrices are applied from right to left, thus

TRSp scales point p, rotates it, then translates it

 In the MonoGame world, it's the opposite!

• How does that work?!

 This example from the book shows how the same sets of

transforms, applied in different orders, can have different

• utcomes

 A rigid-body transform preserves lengths, angles, and

handedness

 We can write any rigid-body transform X as a rotation matrix

R multiplied by translation matrix T(t)               = = 1 ) (

22 21 20 12 11 10 02 01 00 z y x

t r r r t r r r t r r r R t T X

 Because R is orthogonal, its inverse is its

transpose

 Because of the nature of a translation, its inverse

is just its negative

 Thus, the inverse of X is

• X-1 = (T(t)R)-1 = R-1T(t)-1 = RTT(-t)

 The matrix used to transform points will not always work on surface

normals

 Rotation is fine

• Uniform scaling can stretch the normal (which should be unit)
• Non-uniform scaling distorts the normal

 Transforming by the transpose of the adjoint always gives the correct

answer

 In practice, the transpose of the inverse is usually used

 Because of the singular value theorem, we can write any square, real-

valued matrix M with positive determinant as:

• M = R1SR2
• where R1 and R2 are rotation matrices and S is a scaling matrix

 (M-1)T

= ((R1SR2)-1)T = (R2

• 1S-1R1
• 1)T

= (R1

• 1)T(S-1)T(R2
• 1)T

= R1S-1 R2

 Rotations are fine for the normals, but non-uniform scaling will distort

them

 The transpose of the inverse distorts the scale in the opposite direction

• 1. With homogeneous notation, translations do not affect normals

at all

• 2. If only using rotations, you can use the regular world transform

for normals

• 3. If using rotations and uniform scaling, you can use the world

transform for normals

• However, you'll need to normalize your normals so they are unit
• 4. If using rotations and non-uniform scaling, use the transpose of

the inverse or the transpose of the adjoint

• They only differ by a factor of the determinant, and you'll have to

normalize your normals anyway

 Euler angles  Quaternions  Vertex blending  Morphing  Projections

 Keep reading Chapter 4  Project 1 due Friday  Assignment 2 due next Friday