Topic 6: 3D Transformations Homogeneous 3D transformations Scene - PowerPoint PPT Presentation

Topic 6: 3D Transformations • Homogeneous 3D transformations • Scene Hierarchies • Change of basis and rotations in 3D

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Logitics • Assignment 1 Due Tomorrow • Assignment 2 available today/tomorrow • For assignment questions use the bulletin board or email: • csc418tas@cs.toronto.edu • “When will your slides be online ?” • Today J

Representing 2D transforms as a 3x3 matrix Translate a point [x y] T by [t x t y ] T : x’ = 1 0 t x x y’ 0 1 t y y 1 0 0 1 1 Rotate a point [x y] T by an angle t : x’ = cost -sint 0 x y’ sint cost 0 y 1 0 0 1 1 Scale a point [x y] T by a factor [s x s y ] T x’ = s x 0 0 x y’ 0 s y 0 y 1 0 0 1 1

Representing 3D transforms as a 4x4 matrix Translate a point [x y z] T by [t x t y t z ] T : x’ = 1 0 0 t x x y’ 0 1 0 t y y z’ 0 0 1 t z z 1 0 0 0 1 1 Rotate a point [x y z] T by an angle t around z axis: x’ = cost -sint 0 0 x y’ sint cost 0 0 y z’ 0 0 1 0 z 1 0 0 0 1 1 Scale a point [x y z] T by a factor [s x s y s z ] T x’ = s x 0 0 0 x y’ 0 s y 0 0 y z’ 0 0 s z 0 z 1 0 0 0 1 1

Elementary Rotations in 3D

Rotation About Arbitrary Vector?

Rotation About Arbitrary Vector?

Rotation About Arbitrary Vector: Construction

Scene Hierarchies

Change of reference frame/basis matrix p c y b o x a z p = ap x ’ + bp y ’ + cp z ’ + o p = a b c o p’ 0 0 0 1 -1 p’= a b c o p 0 0 0 1

Viewing Pipeline object world camera modeling viewing projection transform transform transform cannonical view vol. 4D cannonical screen viewport cartesianize 2D transform perspective divide

Topic 7: 3D Viewing • Camera Model • Orthographic projection • The world-to-camera transformation • Perspective projection • The transformation chain for 3D viewing

The Pinhole Camera

Camera model virtual image Ideal pinhole camera image pinhole object Real pinhole camera image aperture object

Camera model Real pinhole camera image aperture object Camera with a lens aperture object

Camera model Camera with a lens aperture object Depth of Field

The Pinhole Camera: Basic Geometry in 2D

Camera model v W u

Viewing Pipeline object world camera modeling viewing projection transform transform transform cannonical view vol. 4D cannonical screen viewport cartesianize 2D transform perspective divide

Viewing Transform V up P eye P ref

Camera model v W u

Viewing Transform V up w=(P eye -P ref )/||P eye -P ref || P eye P ref

Camera model v W u

Viewing Transform V up w P eye u=(V up xw)/|| V up xw || P ref

Viewing Transform V up w v=wxu P eye u P ref

Change-of-basis Matrix V up v=wxu P eye w u P ref

Change-of-basis Matrix V up v=wxu P eye w u P ref

Viewing Pipeline object world camera modeling viewing projection transform transform transform canonical view vol. 4D canonical screen viewport cartesianize 2D transform perspective divide

Camera model What is the difference between these images?

Camera model What is the difference between these images? Perspective Orthographic

Parallel vs Perspective projection

Orthographic projection p’=[x y 1] T p=[x y z 1] T

Orthographic projection Is |p-q| = |p’-q’| ? q’ If m= (p+q)/2, Is m’ = (p’+q’)/2? m’ p’=[x y 1] T q m p=[x y z 1] T

Parallel vs Perspective projection

Camera model Perspective Projection

Perspective projection w v u

View frustum

Perspective projection w v P’ P d u

Simple Perspective w v P’ P d u P(x,y,z) y P’(x’,y’,z’) z (0,0,d) Image plane

Simple Perspective X Z

X Z

Using Homogenous Coordinates

X Z

X Z

Homogenous Coordinate after Orthographic Projection Homogenous Coordinate after Perspective Projection

Viewing volumes Projected image

Transforming the View Frustim

Cannonical view volume t Map 3D to a cube centered y f at the origin of side length 2! r x z l n p’ b p

The orthographic view volume

The orthographic view volume

The orthographic view volume

The orthographic view volume

Homogeonous Coords and Perspective View Plane View Plane Warped to look like Camera Space Orthographic Projection

Perspective Matrix

Perspective Matrix

Verify

Verify

Verify

Viewing Pipeline object world camera modeling viewing projection transform transform transform canonical view vol. 4D canonical screen viewport cartesianize 2D transform perspective divide

Viewport Transform

Viewport Transform

Viewing Pipeline object world camera modeling viewing projection transform transform transform canonical view vol. 4D canonical screen viewport cartesianize 2D transform perspective divide

Viewport Transform