CSCI 420: Computer Graphics

Hao Lihttp://cs420.hao-li.com

Fall 2018

3.1 Viewing and Projection

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Recall: Affine Transformations

• Given a point

• form homogeneous coordinates

• The transformed point is

[x y z]>

[x y z 1]>

[x0 y0 z0]>

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Transformation Matrices in OpenGL

• Transformation matrices in OpenGL are vectors of 16 values (column-major matrices)

• In glLoadMatrixf(GLfloat *m);

• Some books transpose all matrices!

represents

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m> = [m1,m2, . . . ,m16]>

Shear Transformations

• x-shear scales proportional to

• Leaves and values fixed

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x y

y z

Specification via Shear Angle

= shear angle

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cot(✓) = (x0 � x)/y

x0 = x+ y cot(✓)y0 = y

z0 = z

✓

[x, y] [x0, y0]x0 � x

x

y

y

✓

✓

Specification via Ratios

• For example, shear in both and direction

• Leave fixed

• Slope for -shear, for -shear

• Solve

• Yields

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y

x z

↵ �x z

Composing Transformations

• Let , and

• Then

matrix multiplication

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p = Aq q = Bs

p = (AB)s

AB

AB

s q p

Composing Transformations

• Every affine transformation is a composition of rotations, scalings, and translations

• So, how do we compose these to form an x-shear?

• Exercise!

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Outline

• Shear Transformation

• Camera Positioning

• Simple Parallel Projections

• Simple Perspective Projections

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Transform Camera = Transform Scene

• Camera position is identified with a frame

• Either move and rotate the objects

• Or move and rotate the camera

• Initially, camera at origin, pointing in negative z-direction

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The Look-At Function

• Convenient way to position camera

• gluLookAt(ex, ey, ez, fx, fy, fz, ux, uy, uz);

• e = eye point

• f = focus point

• u = up vector

u

e

ue

f

f

view plane

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OpenGL code

void display() { glClear (GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT); glMatrixMode (GL_MODELVIEW); glLoadIdentity();

gluLookAt (ex, ey, ez, fx, fy, fz, ux, uy, uz);

glTranslatef(x, y, z); ... renderBunny();

glutSwapBuffers(); }

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Implementing the Look-At Function

1. Transform world frame to camera frame - Compose a rotation with translation -

2. Invert to obtain viewing transformation - - Derive , then , then

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R T

W = TR

W VV = W�1 = (TR)�1 = R�1T�1

R T R�1T�1

World Frame to Camera Frame I

• Camera points in negative direction

• is unit normal to view plane

• Therefore, maps to

view plane

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z

n = (f � e)/kf � ek

R [0 0 � 1]> [nx ny nz]>

e n

u

f

World Frame to Camera Frame II

• maps to projection of u onto view plane

• This projection equals: - - -

view plane

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e n

u

f↵

v0

R [0 1 0]>

↵ = u>n/knk = u>n

v0 = u� ↵n

v = v0/kv0k

v

World Frame to Camera Frame III

• Set to be orthogonal to and ,

• ,

• is right-handed

view plane

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e n f

v

w

w vn

w = n⇥ v

[w v � n]>

Summary of Rotation

• gluLookAt(ex, ey, ez, fx, fy, fz, ux, uy, uz);

• ,

• ,

• .

• Rotation must map: - to - to - to

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n = (f � e)/kf � ek

v = (u� (u>n)n)/ku� (u>n)nk

w = n⇥ v

[1 0 0][0 1 0]

[0 0 � 1]

wv

n

World Frame to Camera Frame IV

• Translation of origin to

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e> = [ex ey ez 1]>

Camera Frame to Rendering Frame

• ,

• is rotation, so

• is translation, so negates displacement

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V = W�1 = (TR)�1 = R�1T�1

R R�1 = R>

T T�1

Putting it Together

• Calculate

• This is different from book [Angel, Ch. 5.3.2]

• There, are right-handed (here: )

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V = R�1T�1

u,v,n u,v,�n

Other Viewing Functions

• Roll (about z), pitch (about x), yaw (about y)

• Assignment 2 poses a related problem

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Outline

• Shear Transformation

• Camera Positioning

• Simple Parallel Projections

• Simple Perspective Projections

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Projection Matrices

• Recall geometric pipeline

• Projection takes 3D to 2D

• Projections are not invertible

• Projections are described by a 4x4 matrix

• Homogenous coordinates crucial

• Parallel and perspective projections 23

Parallel Projection

• Project 3D object to 2D via parallel lines

• The lines are not necessarily orthogonalto projection plane

source:Wikipedia

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Parallel Projection

• Problem: objects far away do not appear smaller

• Can lead to “impossible objects” :

Penrose stairs source:Wikipedia

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Orthographic Projection

• A special kind of parallel projection: projectors perpendicular to projection plane

• Simple, but not realistic

• Used in blueprints (multiview projections)

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Orthographic Projection Matrix

• Project onto

• , ,

• In homogenous coordinates

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z = 0

xp = x yp = y zp = 0

Perspective

• Perspective characterized by foreshortening

• More distant objects appear smaller

• Parallel lines appear to converge

• Rudimentary perspective in cave drawings:

Lascaux, France source: Wikipedia

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Discovery of Perspective

• Foundation in geometry (Euclid)

Mural from Pompeii, Italy

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Middle Ages

• Art in the service of religion

• Perspective abandoned or forgotten

Ottonian manuscript, ca. 1000

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Renaissance

• Rediscovery, systematic study of perspective

Filippo Brunelleschi Florence, 1415

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Projection (Viewing) in OpenGL

• Remember: camera is pointing in the negative z direction

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Orthographic Viewing in OpenGL

• glOrtho(xmin, xmax, ymin, ymax, near, far)

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zmin = near, zmax = far

Perspective Viewing in OpenGL

• Two interfaces: glFrustum and gluPerspective

• glFrustum(xmin, xmax, ymin, ymax, near, far);

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zmin = near, zmax = far

Field of View Interface

• gluPerspective(fovy, aspectRatio, near, far);

• and as before

• aspectRatio =

• Fovy specifies field of view as height ( ) angle

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w/h

y

near far

OpenGL code

void reshape(int x, int y) { glViewport(0, 0, x, y);

glMatrixMode(GL_PROJECTION); glLoadIdentity();

gluPerspective(60.0, 1.0 * x / y, 0.01, 10.0);

glMatrixMode(GL_MODELVIEW); }

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Perspective Viewing Mathematically

• = focal length

• so

• Note that is non-linear in the depth !

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d

y/z = yp/d yp = y/(z/d) = yd/z

yp z

Exploiting the 4th Dimension

Perspective projection is not affine:

Idea: exploit homogeneous coordinates

has no solution for

for arbitrary

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M

w 6= 0

Perspective Projection Matrix

• Use multiple of point

• Solve

with

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Projection Algorithm

• Input: 3D point to project

• Form

• Multiply with ; obtaining

• Perform perspective division: , ,

• Output:

• (last coordinate will be ) 40

[x y z]>

[x y z 1]>

[x y z 1]> [X Y Z W ]>M

X/W Y/W Z/W

[X/W,Y/W,Z/W ]>

d

Perspective Division

• Normalize to

• Perform perspective division after projection

• Projection in OpenGL is more complex(includes clipping)

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[X Y Z W ]> [X/W, Y/W, Z/W, 1]>

http://cs420.hao-li.com

Thanks!

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