Transformations in 3D Right-handed coordinate system z-axis points - - PowerPoint PPT Presentation

Transformations in 3D

z x y Right-handed coordinate system – z-axis points toward viewer

Translation in 3D

What are the coordinates of P after translating by d=(dx,dy,dz)

P(x,y,z) P'(x',y', z')

x' = ? y' = ? z' = ?

Translation in 3D

What are the coordinates of P after translating by d=(dx,dy,dz)

P(x,y,z) P'(x',y', z')

x' = x + dx = 1*x + 0*y + 0*z + 1*dx y' = y + dy = 0*x + 1*y + 0*z + 1*dy z' = z + dz = 0*x + 0*y + 1*z + 1*dz in matrix form

x' y' z' 1 x y z 1

= *

1 dx 1 dy 1 dz 1

Scaling in 3D

What are the coordinates of P after scaling by sx,sy,sz

P(x,y,z) P'(x',y',z')

x' = ? y' = ? z' = ?

Scaling in 3D

What are the coordinates of P after scaling by sx,sy,sz

P(x,y,z) P'(x',y',z')

x' = sx * dx = sx*x + 0*y + 0*z y' = sy * dy = 0*x + sy*y + 0*z z' = sz * dz = 0*x + 0*y + sz*z in matrix form

x' y' z' 1 x y z 1

= *

sx sy sz 1

Rotation in 3D around Z-axis

Rotating in a plane parallel to the XY-plane Same as rotating in 2D keep the Z-coordinate the same

P(x,y,z) a r r P'(x',y',z') P(x,y,z) a r r P'(x',y',z')

Rotation in 3D around Z-axis

Rotating in a plane parallel to the XY-plane Same as rotating in 2D keep the Z-coordinate the same Same matrix with extra row, column

cosα

• sinα

sinα cosα 1 1

x' y' z' 1 x y z 1

= *

P(x,y,z) a r r P'(x',y',z') P(x,y,z) a r r P'(x',y',z')

Rotation in 3D around Y-axis

P'(x',y',z') P(x,y,z)

Rotating in a plane parallel to the XY-plane Same as rotating in 2D keep the Y-coordinate the same

a P'(x',y',z') P(x,y,z) a

Rotation in 3D around Y-axis

cosα sinα 1

• sinα

cosα 1

P'(x',y',z') P(x,y,z)

Rotating in a plane parallel to the XY-plane Same as rotating in 2D keep the Y-coordinate the same

x' y' z' 1 x y z 1

= *

a P'(x',y',z') P(x,y,z) a

Rotation in 3D around X-axis

P(x,y,z) P'(x',y',z') α P(x,y,z) P'(x',y',z') α

Rotating in a plane parallel to the YZ-plane Same as rotating in 2D keep the X-coordinate the same

Rotation in 3D around X-axis

1 cosα

• sinα

sinα cosα 1

P(x,y,z) P'(x',y',z') α P(x,y,z) P'(x',y',z') α

Rotating in a plane parallel to the YZ-plane Same as rotating in 2D keep the X-coordinate the same

x' y' z' 1 x y z 1

= *

Rotation around arbitrary axis

P(x,y,z) α P'(x',y',z') A(ax,ay,az)

can reduce to rotation around Z-axis by realignment of coordinate system

Rotation around arbitrary axis

P(x,y,z) A(ax,ay,az)

Rotation around arbitrary axis

P(x,y,z) A(ax,ay,az) P(x,y,z) A(ax,ay,az) A'(ax',ay',az') A”(ax”,ay”,az”)

compute A',A” s.t. A,A',A” perpend.

Rotation around arbitrary axis

P(x,y,z) A(ax,ay,az) P(x,y,z) A(ax,ay,az) A'(ax',ay',az') A”(ax”,ay”,az”) P(x,y,z) A(ax,ay,az) A'(ax',ay',az') A”(ax”,ay”,az”)

compute A',A” s.t. A,A',A” perpend. align A',A”,A with main axes X,Y,Z

Rotation around arbitrary axis

P(x,y,z) A(ax,ay,az) P(x,y,z) A(ax,ay,az) A'(ax',ay',az') A”(ax”,ay”,az”) P(x,y,z) A(ax,ay,az) A'(ax',ay',az') A”(ax”,ay”,az”)

compute A',A” s.t. A,A',A” perpend. align A',A”,A with main axes X,Y,Z

ax' ay' az' ax” ay” az” ax ay az 1

M

Rotation around arbitrary axis

P(x,y,z) A(ax,ay,az) P(x,y,z) A(ax,ay,az) A'(ax',ay',az') A”(ax”,ay”,az”) P(x,y,z) P'(x',y',z') A(ax,ay,az) A'(ax',ay',az') A”(ax”,ay”,az”) α

compute A',A” s.t. A,A',A” perpend. align A',A”,A with main axes X,Y,Z rotate around z

ax' ay' az' ax” ay” az” ax ay az 1

M

ca

• sa

sa ca 1 1

R

Rotation around arbitrary axis

P(x,y,z) α P'(x',y',z') A(ax,ay,az) P(x,y,z) A(ax,ay,az) A'(ax',ay',az') A”(ax”,ay”,az”) P(x,y,z) P'(x',y',z') A(ax,ay,az) A'(ax',ay',az') A”(ax”,ay”,az”) α

compute A',A” s.t. A,A',A” perpend. align A',A”,A with main axes X,Y,Z r e s t

• r

e a l i g n m e n t rotate around z

ax' ay' az' ax” ay” az” ax ay az 1

M

ax' ax” ax ay' ay” ay az' az” az 1

M'

ca

• sa

sa ca 1 1

R

Rotation around arbitrary axis

P(x,y,z) α P'(x',y',z') A(ax,ay,az)

can reduce to rotation around Z-axis by realignment of coordinate system R = M' * Rz * M multiply top of stack by R

Rendering Pipeline

Mmodel Mview*Mproj World space View space Screen space

P’ = Mproj Mview Mmodel * P

http://www.songho.ca/opengl/gl_transform.html

The Camera Model (2D)

v u e camera parameters: e – eye point coordinates v – the directions of the view u – up vector e – what is the position (ex,ey)

• f the camera in the world

v – what are the coordinates (vx,vy) of the viewing direction in the world u – a vector perpendicular to v v u e The camera parameters and the coordinates of the model are specifjed in world space

The Camera Model (2D)

v u e To render the model need to: represent it from world space to camera view space coordinates – Mview project it on the camera film – Mproj v u e x y x' y'

The Camera Model (2D)

World to View space transformation (Mview) – align camera axes with the world axes u v e

The Camera Model (2D)

T R

World to View space transformation (Mview) – align camera axes with the world axes u v e

The Camera Model (2D)

T R

1

• ex

1

• ey

1

World to View space transformation (Mview) – align camera axes with the world axes u v e

The Camera Model (2D)

T R

1

• ex

1

• ey

1 vx vy ux uy 1

World to View space transformation (Mview) – align camera axes with the world axes u v e

The Camera Model (2D)

T R

1

• ex

1

• ey

1 vx vy ux uy 1

Mview = R*T

World to View space transformation (Mview) – align camera axes with the world axes u v e