computer graphics (cs 543) 5 (part projection (part...

Computer Graphics (CS 543) Lecture 5 (part 2): Projection (Part 2):

Derivation

Prof Emmanuel Agu

Computer Science Dept.Worcester Polytechnic Institute (WPI)

Parallel Projection normalization find 4x4 matrix to transform user‐specified

view volume to canonical view volume (cube)

glOrtho(left, right, bottom, top,near, far)

Canonical View Volume

User‐specifiedView Volume

Parallel Projection: Ortho

Parallel projection: 2 parts1. Translation: centers view volume at origin


2. Scaling: reduces user‐selected cuboid to canonical cube (dimension 2, centered at origin)


Translation lines up midpoints: E.g. midpoint of x = (right + left)/2

Thus translation factors:‐(right + left)/2, ‐(top + bottom)/2, ‐(far+near)/2

Translation matrix:

10002/)(1002/)(010

2/)(001

nearfarbottomtop

leftright


Scaling factor: ratio of ortho view volume to cube dimensions Scaling factors: 2/(right ‐ left), 2/(top ‐ bottom), 2/(far ‐ near) Scaling Matrix M2:

1000

0200

0020

0002

nearfar

bottomtop

leftright


Concatenating Translation x Scaling, we get Ortho Projection matrix

X

1000

0200

0020

0002

nearfar

bottomtop

leftright

10002/)(1002/)(010

2/)(001

nearfarbottomtop

leftright

1000

200

020

002

nearfarnearfar

farnear

bottomtopbottomtop

bottomtop

leftrightleftright

leftright

P = ST =

Final Ortho Projection

Set z =0 Equivalent to the homogeneous coordinate transformation

Hence, general orthogonal projection in 4D is

1000000000100001

Morth =

P = MorthST

Perspective Projection

Projection – map the object from 3D space to 2D screen

x

y

z

Perspective()Frustrum( )

Perspective Projection: Classical

(0,0,0)

-N

Projection plane

Eye (COP)

(x,y,z)

(x’,y’,z’)

-z

- z

yBased on similar triangles:

y’ N y -z

Ny’ = y x

-z

=

VRP

COP

Object in 3 space

Projectors

Projected image

Near Plane(VOP)

+ z

Perspective Projection: Classical

So (x*,y*) projection of point, (x,y,z) unto near plane N is given as:

Numerical example:Q. Where on the viewplane does P = (1, 0.5, ‐1.5) lie for a

near plane at N = 1?

zNy

zNxyx ,**,

)333.0,666.0(5.1

15.0,5.1

11,**,

zNy

zNxyx

VRP

COP

Object in 3 space

Projectors

Projected image

Pseudodepth

Classical perspective projection projects (x,y) coordinates to (x*, y*), drops z coordinates

But we need z to find closest object (depth testing)!!!

VRP

COP

Object in 3 space

Projectors

Projected image

(0,0,0)z

Map to same (x*,y*)Compare their z values?

Perspective Transformation

Perspective transformationmaps actual z distance of perspective view volume to range [ –1 to 1] (Pseudodepth) for canonical view volume

NearFar

1 -1

Canonical view volume

Actual view volume

Pseudodepth

Actual depth We want perspectiveTransformation and NOT classical projection!!

Set scaling zPseudodepth = az + bNext solve for a and b


We want to transform viewing frustum volume into canonical view volume

(-1, -1, 1)

(1, 1, -1)


x

y

z

Perspective Transformation using Pseudodepth

Choose a, b so as z varies from Near to Far, pseudodepthvaries from –1 to 1 (canonical cube)

Boundary conditions z* = ‐1 when z = ‐N z* = 1 when z = ‐F

zbaz

zNy

zN

xzyx ,,**,*,

Near Far


Actual view volume

Pseudodepth

Actual depth

-1 1

Z*

Z

Transformation of z: Solve for a and b

Solving:

Use boundary conditions z* = ‐1 when z = ‐N………(1) z* = 1 when z = ‐F………..(2)

Set up simultaneous equations

zbazz

*

)1........(1 baNNN

baN

)2........(1 baFFF

baF


Multiply both sides of (1) by ‐1

Add eqns (2) and (3)

Now put (4) back into (3)

)1........(baNN

)2........(baFF

)3........(baNN

aFaNNF

)4.........()(NFNF

FNNFa


Put solution for a back into eqn (3)

So

bNF

NFNN

)(

)3........(baNN

NFNFNNb

)(

NFNF

NFNNFNNF

NFNFNNFNb

2)()( 22

NFNFa

)(

NFFNb

2

What does this mean?

Original point z in original view volume, transformed into z* in canonical view volume

whereNear Far


Actual view volume

-1 1

Original vertex z value

Transformed vertex z* value

zbazz

*

NFNFa

)(

NFFNb

2

Homogenous Coordinates

Want to express projection transform as 4x4 matrix Previously, homogeneous coordinates of

P = (Px,Py,Pz) => (Px,Py,Pz,1) Introduce arbitrary scaling factor, w, so that

P = (wPx, wPy, wPz, w) (Note: w is non‐zero) For example, the point P = (2,4,6) can be expressed as (2,4,6,1) or (4,8,12,2) where w=2 or (6,12,18,3) where w = 3, or….

To convert from homogeneous back to ordinary coordinates, first divide all four terms by w and discard 4th term

Perspective Projection Matrix

Recall Perspective Transform

We have:

In matrix form:

1

)(0100

00000000

zbazz

Nyz

Nx

wzbazw

wNywNx

wwzwywx

baN

N

zbaz

zNy

zN

xzyx ,,**,*,

zNxx

*z

Nyy

*zbazz

*

Perspective Transform Matrix

Original vertex

TransformedVertex

Transformed Vertex after dividing by 4th term


In perspective transform matrix, already solved for a and b:

So, we have transform matrix to transform z values

1

)(0100

00000000

zbazz

Nyz

Nx

wPbaPw

wNPwNP

wwPwPwP

baN

N

z

z

y

x

z

y

x

NFNFa

)(

NFFNb

2


Not done yet!! Can now transform z! Also need to transform the x = (left, right) and y = (bottom, top)

ranges of viewing frustum to [‐1, 1] Similar to glOrtho, we need to translate and scale previous matrix

along x and y to get final projection transform matrix we translate by

–(right + left)/2 in x ‐(top + bottom)/2 in y

Scale by: 2/(right – left) in x 2/(top – bottom) in y

1 -1

x

y

left rightbottom

top


Translate along x and y to line up center with origin of CVV –(right + left)/2 in x ‐(top + bottom)/2 in y

Multiply by translation matrix:

1 -1

x

y

left rightbottom

top

10000100

2/)(0102/)(001

bottomtopleftright

Line up centersAlong x and y


To bring view volume size down to size of of CVV, scale by 2/(right – left) in x 2/(top – bottom) in y

Multiply by scale matrix:

1 -1

x

y

left rightbottom

top

Scale size downalong x and y

10000100

0020

0002

bottomtop

leftright


glFrustum(left, right, bottom, top, N, F) N = near plane, F = far plane

0100

2)(00

020

00minmax

2

NFFN

NFNF

bottomtopbottomtop

bottomtopN

leftrightleftright

xxN

010000

000000

10000100

2/)(0102/)(001

10000100

0020

0002

baN

Nbottomtop

leftright

bottomtop

leftright

Scale

Final PerspectiveTransform Matrix

Translate

Previous PerspectiveTransform Matrix


After perspective transformation, viewing frustum volume is transformed into canonical view volume

(-1, -1, 1)

(1, 1, -1)


x

y

z

Geometric Nature of Perspective Transform

a) Lines through eye map into lines parallel to z axis after transformb) Lines perpendicular to z axis map to lines perp to z axis after transform

Normalization Transformation

original clippingvolume original object new clipping

volume

distorted objectprojects correctly

Implementation

Set modelview and projection matrices in application program Pass matrices to shader

void display( ){.....model_view = LookAt(eye, at, up);projection = Ortho(left, right, bottom,top, near, far);

// pass model_view and projection matrices to shaderglUniformMatrix4fv(matrix_loc, 1, GL_TRUE, model_view);glUniformMatrix4fv(projection_loc, 1, GL_TRUE, projection);

.....}

Build 4x4 projection matrix

Implementation

And the corresponding shader

in vec4 vPosition;in vec4 vColor;Out vec4 color;uniform mat4 model_view;Uniform mat4 projection;

void main( ){

gl_Position = projection*model_view*vPosition;color = vColor;

}

References Interactive Computer Graphics (6th edition), Angel and

Shreiner Computer Graphics using OpenGL (3rd edition), Hill and Kelley

computer graphics (cs 543) 5 (part projection (part...

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