3D Coordinates & Transformations · 2020. 7. 31. · 3D Coordinates & Transformations Prof. Aaron...

3D Coordinates & Transformations

Prof. Aaron Lanterman (Based on slides by Prof. Hsien-Hsin Sean Lee) School of Electrical and Computer Engineering

Georgia Institute of Technology

2

3D graphics rendering pipeline (1)

•  Geometry Pipeline –  Processing Vertices –  Mainly floating-point operations

•  Rasterization Pipeline –  Processing Pixels –  Mainly dealing with Integer operations

Geometry Processing

x y

z

x y

z

x y

z

x y

z

x y

z

x

y

Rasterization Processing

x

y

3


•  Geometry Pipeline –  Processing Vertices –  Mainly floating-point operations

•  Rasterization Pipeline –  Processing Pixels –  Mainly dealing with Integer operations –  MMX was originally designed to accelerate

this type of functionality

Geometry Processing

x y

z

x y

z

x y

z

x y

z

x y

z

x

y


x

y

MMX ISA

4


•  Geometry Pipeline –  Processing Vertices –  Mainly floating-point operations –  SSE/SSE2 were designed for this part



Geometry Processing

x y

z

x y

z

x y

z

x y

z

x y

z

x

y


x

y

MMX ISA SSE/SSE2 ISA

5

Fixed-function 3D graphics pipeline

•  Geometry Pipeline –  Processing Vertices –  Mainly floating-point operations –  SSE/SSE2 were designed for this part



Geometry Processing

x y

z

x y

z

x y

z

x y

z

x y

z

x

y


x

y Performed by GPU

6

3D Coord: Math textbooks use z-up

+y

+x

+z

Z-up, Right-Handed System

7

3D Coord: Real games tend to use y-up

+z

+x

+y

Left-Handed System

+z

+x

+y

Right-Handed System • OpenGL •  XNA

• Direct3D • Unity3D

8

X-Y natural for screen coordinates

+z (away from the viewer)

+x

+y

Left-Handed System

+z (toward the viewer)

+x

+y

Right-Handed System • OpenGL •  XNA

• Direct3D • Unity3D

9

Some use Z-up for world coordinates

+y

+x

+z

Left-Handed System

+y

+x

+z

Right-Handed System

•  Z-up, LHS: Unreal •  Z-up, RHS: Quake/Radiant, Source/Hammer, C4 Engine • Nearly everything still uses Y-up for screen coordinates!

10

Another view

+y

+x

+z

Left-Handed System Right-Handed System

+x

+y

+z

• Quake/Radiant • Source/Hammer • C4 Engine

• Unreal

11

3D “object” modeling software

3D Studio Max, Blender

+z

+x

+y

Right-Handed System

Maya, Milkshape

+y

+x

+z

Right-Handed System

12

Geometry format – vertex coordinates

(X1, Y1, Z1)

(X2, Y2, Z2)

(X3, Y3, Z3)

+Z

+X

+Y

X1

Z1

Y1

13

Geometry format ─ vertex normals

(NX1, NY1, NZ1)

(NX2, NY2, NZ2)

(NX3, NY3, NZ3)

+Z

+X

+Y

14

Geometry format ─ vertex colors

(R1, G1, B1, A1)

(R2, G2, B2, A2)

(R3, B3, B3, G3)

+Z

+X

+Y

15

Triangle-based geometry representation

V1 V2

V3 V4

V5

V6 V7

V8

V9

Triangle List (note the vertex order)

V5 V4

V3

V2

V1

Triangle Strip

V5 V4

V1

V3

V2

Triangle Fan

Careful!

16

Specifying a 3D object (1)

V1

V2 V3

V4

V5

V6 V7 Triangle list {v1, v3, v2}, {v1, v5, v3}, {v5, v6, v3}, {v4, v3, v6}, {v1, v7, v6}, {v1, v6, v5}

Triangle strip {v5, v3, v1, v2}, {v5, v6, v3, v4}, {v7, v6, v1, v5}

•  Vertex ordering is critical when culling mode enabled •  We will discuss normal computation later

17

V8

Specifying a 3D object (2)

•  Vertex ordering is critical when culling mode enabled •  We will discuss normal computation later

V1

V2 V3

V4

V6 V7 Triangle list {v1, v2, v7}, {v2, v8, v7}, {v2, v3, v4}, {v2, v4, v8}, {v4, v7, v8}, {v4, v6, v7}

Triangle strip {v1, v2, v7, v8}, {v3, v4, v2, v8}, {v6, v7, v4, v8}

18

3D rendering pipeline

Clipp

ing

Persp

ectiv

e Divi

de

View

port

Tran

sform

Raste

rizati

on

Proje

ction

Tran

sform

Wor

ld Tr

ansfo

rm

View

Tran

sform

Lighti

ng

Back

face C

ulling

19

Transformation pipeline • World Transformation

– Model coordinates à World coordinates

• View Transformation – World coordinates à Camera space

• Projection Transformation – Camera space à View plane

• These are a series of matrix multiplications

20

World transformation

•  Translation •  Rotation •  Scaling

+x

+z

+y

World origin

World Coordinates

Local model coordinates

Local model coordinates

21

View transformation

+x

+z

+y

World origin

World Coordinates

+y

+x

+z

•  Camera position •  Look vector

22

Projection transformation • Set up camera internals

• Set up – Field of View (FOV) – View frustum – View planes

• Will discuss in the next lecture

23

Homogeneous coordinates • Enable all transformations to be done by

“multiplication” – Primarily for translation (see next few slides)

• Add one coordinate (w) to a 3D vector

• Each vertex has [x, y, z, w] – w will be useful for perspective projection – w should be 1 in a Cartesian coordinate system

24

Transformation 1: translation (Offset)

+x

+z

+y

+x

+z

+y

(x, y, z)

(xt, yt, zt)

25

Translation matrix

[xt,yt,zt,1]= [x,y,z,1]⋅

1 0 0 00 1 0 00 0 1 0Tx Ty Tz 1

"

#

$$$$

%

&

''''

• Example of a row-coordinate convention • Direct3D, XNA, HLSL/Cg use row coordinates • OpenGL & non-graphics world uses column

coordinates

26

Transformation 2: scaling

+x

+z

+y

+x

+z

+y

27

Scaling matrix

[xs,ys,zs,1]= [x,y,z,1]⋅

Sx 0 0 00 Sy 0 00 0 Sz 00 0 0 1

"

#

$$$$

%

&

''''

28

Transformation 3: rotation

+x

+z

+y

+y

+x

+z

29

2D rotation

+x

+y

+x

+y

θ

(x, y)

(x’, y’)

+x

+y

(x, y) θ ϕ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−⋅=

1000cossin0sincos

]y,1x,[],1y',x'[ θθθθ

Rotate along which axis?

30

3D rotation matrix (LHS)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⋅=

1000010000cossin00sincos

]z,1y,x,[],1z',y',x'[θθθθ

Rotation along Z axis

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

⋅=

10000cos0sin00100sin0cos

]z,1y,x,[],1z',y',x'[θθ

θθ

Rotation along Y axis

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⋅=

10000cossin00sincos00001

]z,1y,x,[],1z',y',x'[θθθθ

Rotation along X axis

31

Non-commutative property (1)

1.  Counter-clockwise 90o along y 2.  Clockwise 90o along x

+x

+z

+y

�

� +x

+z

+y

1.  Clockwise 90o along x 2.  Counter-clockwise 90o along y

�

�

32


+x

+z

+y

�

�

+x

+z

+y

�

�

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−⋅

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−−

1000000101000010

1000

0)2

cos()2

sin(0

0)2

sin()2

cos(00001

1000

0)2

cos(0)2

sin(0010

0)2

sin(0)2

cos(

ππ

ππ

ππ

ππ

� �

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−−

⋅

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−1000001000010100

1000

0)2

cos(0)2

sin(0010

0)2

sin(0)2

cos(

1000

0)2

cos()2

sin(0

0)2

sin()2

cos(00001

ππ

ππ

ππ

ππ

� �

(x’’, y’’, z’’) = (-z, -x, y)

(x’’, y’’, z’’) = (-y, -z, x)

33


1.  Translation by (x, y, z) 2.  Scale by 2 times

+x

+z

+y

1.  Scale by 2 times 2.  Translation by (x, y, z)

�

�

+x

+z

+y

� �

34


+x

+z

+y

�

�

+x

+z

+y

� �

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1000000000

1000000000000

1010000100001

SzTzSyTySxTxSz

SySx

SzSy

Sx

TzTyTx

� �

(x’’, y’’, z’’) = (x*Sx+Sx*Tx, y*Sy+Sy*Ty, z*Sz+Sz*Tz)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1000000000

1010000100001

1000000000000

TzTyTxSz

SySx

TzTyTxSz

SySx

� �

(x’’, y’’, z’’) = (x*Sx+Tx, y*Sy+Ty, z*Sz+Tz)

Offsets were scaled as well

35


• Ordering matters !

• Be careful when performing matrix multiplication

36

View transformation revisited

+x

+z

+y

World origin

World Coordinates

+y

+x

+z

•  Camera position •  Look vector

37

Specifying the view transformation • Most commonly parameterized by:

–  Position of camera –  Position of point to look at –  Vector indicating “up” direction of camera

•  In Direct3D: D3DXMatrixLookAtLH!–  D3D uses a LHS, but also have D3DXMatrixLookAtRH

•  In XNA: Matrix.CreateLookAt (RHS) •  In OpenGL: gluLookAt (RHS) • Can also build a rotation+translation matrix as if the

camera was an object in scene, then take the inverse of that matrix

msdn.microsoft.com/en-us/library/bb205342(VS.85).aspx msdn.microsoft.com/en-us/library/bb205343(VS.85).aspx

3D Coordinates & Transformations · 2020. 7. 31. · 3D Coordinates & Transformations Prof. Aaron...

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