Visual Imaging in the

Electronic Age

Perspective Transformations

Lecture # 3

August 30, 2016

Prof. Donald P. Greenberg

• We have seen how a pinhole camera

(or camera obscura) works

• We have also seen Brunelleschi’s

experiment with his perspective panels

• Both are obviously two dimensional

and exist on a planar (flat) surface

Perspective Depth

• We now know how to mathematically (and

geometrically) construct the perspective image.

(e.g., xs, ys)

• But how do we find the third dimension (zs)?

How a Camera Lens Works

• Rays which pass through the focal point are bent

to leave the lens perpendicular to its principal

axis.

® Donald P. Greenberg - Cornell Program of Computer Graphics

® Donald P. Greenberg - Cornell Program of Computer Graphics

Railroad Tracks in Perspective

elevation

How a Camera Lens Works

• Rays which pass through the focal point are bent

to leave the lens perpendicular to its principal

axis.

• Rays which are perpendicular to the principal

axis of the lens are bent through the focal point.

How a Camera Lens Works

f

P(xe, ye, ze)

P(xp, yp, zp)

-zp - 2f

Ze

Xe

Ye

A

A

B

B

f

-yp

e

ep

e

ep

z

x

f

x

z

y

f

y

,Using triangles A and A':

fz

y

f

y

p

pe

2

Using triangles B and B':

e

ep

e

ep

z

yfy

z

xfx ,

How a Camera Lens Works

f

fz

y

y

z

f

y

y p

e

p

ee

p 2 ,

We know that:

e

p

e

p

z

ffz

z

f

f

fz 2

2 ,2

e

pz

ffz

2

2

How a Camera Lens Works

How a Camera Lens Works

Camera

Film

Plane

Camera

Film

Plane

® Donald P. Greenberg - Cornell Program of Computer Graphics

http://upload.wikimedia.org/wikipedia/en/5/5f/DoF-sym.png

Perspective Projections in

Computer Graphics

Eye coordinate systemPlan or elevation view

Object Space

(One point

perspective)

Image Space

(One point

perspective)

Object Space

(Two point

perspective)

Image Space

(Two point

perspective)

Object Space

(Three point

perspective)

Image Space

(Three point

perspective)

3D Perspective Image

(x & y coordinates in virtual image

space)

e

e

pz

xfx

e

e

pz

yfy

e

e

pz

xx constant

e

e

pz

yy constant

e

pz

ffz

2

2

e

pz

z1

21constantconstant

3D Perspective Image

(z depth coordinate in virtual image

space)

Equivalent Distorted Geometries

in Virtual Image Space

® Donald P. Greenberg - Cornell Program of Computer Graphics

Mapping a Viewing Frustum

to a Standard Viewbox

11

ze = DXe

Frustum of vision Screen coordinate system

Ze

Ye

ze = F

Xs

Ys Zs

1

Viewing Frustum - User Parameters

Location and Orientation (vectors):

Eye location of eye point

At a point along view direction vector

Up a vector defining the Y axis on the hither plane

Shape (scalars):

fov field of view angle

aspect view plane aspect ratio

hither perpendicular distance to hither plane

yon perpendicular distance to yon plane

At

Up

Eye

aspect =

width / height

hitheryon

fov

Uses of Perspective in Enlarging

Architectural Space (3D)

Palazzo Spada

Borromini

Imss,Firenze 2004 - 3 Scenografia prospettica

Imss,Firenze 2004 - 3 Scenografia prospettica

Imss,Firenze 2004 - 3 Scenografia prospettica

Imss,Firenze 2004 - 3 Scenografia prospettica

Imss,Firenze 2004 - 3 Scenografia prospettica

Imss,Firenze 2004 - 3 Scenografia prospettica

Imss,Firenze 2004 - 3 Scenografia prospettica

Borromini

Palazzo Spada

Borromini

Palazzo Spada

Video - Palazzo Spada Video(spada.wmv)

® Donald P. Greenberg - Cornell Program of Computer Graphics

Probe Demo

® Donald P. Greenberg - Cornell Program of Computer Graphics

End. . .