CS361Week 5 - Wednesday

Last time

What did we talk about last time? Project 2 Normal transforms Euler angles Quaternions

Questions?

Project 2

Quaternions

Quaternions

Quaternions are a compact way to represent orientations

Pros: Compact (only four values needed) Do not suffer from gimbal lock Are easy to interpolate between

Cons: Are confusing Use three imaginary numbers Have their own set of operations

Definition

A quaternion has three imaginary parts and one real part

We use vector notation for quaternions but put a hat on them

Note that the three imaginary number dimensions do not behave the way you might expect

wvwzyxwv qqkqjqiqq qqq ),ˆ(ˆ

kjiij

jikki

ikjjk

kji

1222

Operations

Multiplication

Addition

Conjugate

Norm

Identity

),(ˆˆ vvwwvwvwvv rqqr rqrqrqrq

),(ˆˆ wwvv rq rqrq

),(ˆ*wv qqq

2222)̂( wzyx qqqqn q

)1,(ˆ 0i

More operations

Inverse

One (useful) conjugate rule: Note that scalar multiplication is just like

scalar vector multiplication for any vector Quaternion quaternion multiplication is

associative but not commutative For any unit vector u, note that the

following is a unit quaternion:

*2

1 ˆ)̂(

1ˆ qq

qn

)cos,(sinˆ φφuq

*** ˆˆ)̂ˆ( qrrq

Quaternion transforms

Take a vector or point p and pretend its four coordinates make a quaternion

If you have a unit quaternion the result of is p rotated around the u axis by 2

Note that, because it's a unit quaternion,

There are ways to convert between rotation matrices and quaternions

The details are in the book

p̂)cos,(sinˆ φφuq

1ˆˆˆ qpq

*1 ˆˆ qq

Quaternion rotation example Take vector (0,0,3), write it as p =

(0,0,3,0) Let's rotate it 90° around the x-axis using

a quaternion We want where u is the x-axis and is 45°

p = (0, -3 sin , 0, 3cos , 0) = (0, -6sincos,3cos2-3sincos, 0)= (0, -3, 0, 0)

Slerp

Short for spherical linear interpolation Using unit quaternions that represent

orientations, we can slerp between them to find a new orientation at time t = [0,1], tracing the path on a unit sphere between the orientations

To find the angle between the quaternions, you can use the fact that cos = qxrx + qyry + qzrz + qwrw

rqrqs ˆsin

)sin(ˆsin

))1(sin(),̂,ˆ(̂

φtφ

φtφ

t

Student Lecture: Vertex Blending and Morphing

Vertex blending

If we animate by moving rigid bodies around each other, joints won't look natural

To do so, we define bones and skin and have the rigid bone changes dictate blended changes in the skin

Vertex blending continued The following equation shows the effect that

each bone i (and its corresponding animation transform matrix Bi(t) and bone to world transform matrix Mi ) have on the p, the original location of a vertex

Vertex blending is popular in part because it can be computed in hardware with the vertex shader

1

0

1

0

1 1 where )()(n

ii

n

iiii wtwt pMBu

Morphing

Morphing is the technique for interpolating between two complete 3D models

It has two problems: Vertex correspondence▪ What if there is not a 1 to 1 correspondence

between vertices? Interpolation▪ How do we combine the two models?

Morphing continued

We're going to ignore the correspondence problem (because it's really hard)

If there's a 1 to 1 correspondence, we use parameter s[0,1] to indicate where we are between the models and then find the new location m based on the two locations p0 and p1

Morph targets is another technique that adds in weighted poses to a neutral model

10)1( ppm ss

Projections

Finally, we deal with the issue of projecting the points into view space

Since we only have a 2D screen, we need to map everything to x and y coordinates

Like other transforms, we can accomplish projection with a 4 x 4 matrix

Unlike affine transforms, projection transforms can affect the w components in homogeneous notation

Orthographic projections

An orthographic projection maintains the property that parallel lines are still parallel after projection

The most basic orthographic projection matrix simply removes all the z values

This projection is not ideal because z values are lost Things behind the camera are in front z-buffer algorithms don't work

1000

0000

0010

0001

0P

Canonical view volume

To maintain relative depths and allow for clipping, we usually set up a canonical view volume based on (l,r,b,t,n,f)

These letters simply refer to the six bounding planes of the cube Left Right Bottom Top Near Far

Here is the (OpenGL) matrix that translates all points and scales them into the canonical view volume

1000

200

02

0

002

0

nfnf

nf

btbt

bt

lrlr

lr

P

Stupid SharpDX

OpenGL normalizes to a canonical view volume from [-1,1] in x, [-1,1] in y, and [-1,1] in z

Just to be silly, SharpDX normalizes to a canonical view volume of [-1,1] in x, [-1,1] in y, and [0,1] in z

Thus, its projection matrix is:

1000

100

02

0

002

0

nfn

nf

btbt

bt

lrlr

lr

P

Perspective projection

A perspective projection does not preserve parallel lines

Lines that are farther from the camera will appear smaller

Thus, a view frustum must be normalized to a canonical view volume

Because points actually move (in x and y) based on their z distance, there is a distorting term in the w row of the projection matrix

Perspective projection matrix

Here is the SharpDX projection matrix

It is different from the OpenGL again because it only uses [0,1] for z

0100

00

02

0

002

0

nffn

nffbtbt

btn

lrlr

lrn

P

SharpDX Projection Examples

Quiz

Upcoming

Next time…

Light Materials Sensors

Reminders

Read Chapter 5 for FridayExam 1 next Friday Keep working on Project 2