Computer Viewing

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Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 1

Computer Viewing

Ed Angel

Professor of Computer Science, Electrical and Computer

Engineering, and Media Arts University of New Mexico

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Objectives

• Introduce the mathematics of projection

• Introduce OpenGL viewing functions

• Look at alternate viewing APIs

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Computer Viewing

• There are three aspects of the viewing process, all of which are implemented in the pipeline,

- Positioning the camera

• Setting the model-view matrix

- Selecting a lens

• Setting the projection matrix

- Clipping

• Setting the view volume

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The OpenGL Camera

• In OpenGL, initially the object and camera frames are the same

- Default model-view matrix is an identity

• The camera is located at origin and points in the negative z direction

• OpenGL also specifies a default view

volume that is a cube with sides of length 2 centered at the origin

- Default projection matrix is an identity

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Default Projection

Default projection is orthogonal

clipped out

z=0 2

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Moving the Camera Frame

• If we want to visualize object with both positive and negative z values we can either

- Move the camera in the positive z direction

• Translate the camera frame

- Move the objects in the negative z direction

• Translate the world frame

• Both of these views are equivalent and are determined by the model-view matrix

- Want a translation (glTranslatef(0.0,0.0,-d);)

-d > 0

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Moving Camera back from Origin

default frames

frames after translation by –d d > 0

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Moving the Camera

• We can move the camera to any desired position by a sequence of rotations and translations

• Example: side view

- Rotate the camera

- Move it away from origin - Model-view matrix C = TR

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OpenGL code

• Remember that last transformation specified is first to be applied

glMatrixMode(GL_MODELVIEW) glLoadIdentity();

glTranslatef(0.0, 0.0, -d);

glRotatef(90.0, 0.0, 1.0, 0.0);

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The LookAt Function

• The GLU library contains the function gluLookAt to form the required modelview matrix through a simple interface

• Note the need for setting an up direction

• Still need to initialize

- Can concatenate with modeling transformations

• Example: isometric view of cube aligned with axes

glMatrixMode(GL_MODELVIEW):

glLoadIdentity();

gluLookAt(1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0., 1.0. 0.0);

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gluLookAt

gluLookAt(eyex, eyey, eyez, atx, aty,

atz, upx, upy, upz)

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Other Viewing APIs

• The LookAt function is only one possible API for positioning the camera

• Others include

- View reference point, view plane normal, view up (PHIGS, GKS-3D)

- Yaw, pitch, roll

- Elevation, azimuth, twist - Direction angles

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Projections and Normalization

• The default projection in the eye (camera) frame is orthogonal

• For points within the default view volume

• Most graphics systems use view normalization

- All other views are converted to the default view by transformations that determine the projection matrix - Allows use of the same pipeline for all views

x_p = x y_p = y z_p = 0

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Homogeneous Coordinate Representation

x_p = x y_p = y z_p = 0 w_p = 1

p_p= Mp

M =

!!

"

#

$$

%

&

1 0

0 0

1 0

0 0

0 1

In practice, we can let M = I and set the z term to zero later

default orthographic projection

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Simple Perspective

• Center of projection at the origin

• Projection plane z = d, d < 0

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Perspective Equations

Consider top and side views

x_p =

d z

x /

d z

x

/ ^y^p⁼ z d

y

/ ^z^p^{= d}

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Homogeneous Coordinate Form

M =

!!

"

#

$$

%

&

0 /

1 0

0

0 1

0 0

1 0

0 0

0 1

d

consider q = Mp where

! !

"

#

$ $

%

&

1 z y x

!!

"

#

$$

%

&

d z

z y x

/

q = ⇒ p =

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Perspective Division

• However w ≠ 1, so we must divide by w to return from homogeneous coordinates

• This perspective division yields

the desired perspective equations

• We will consider the corresponding clipping volume with the OpenGL functions

x_p =

d z

x

/ ^y^p⁼ z d

y

/ ^z^p^{= d}

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OpenGL Orthogonal Viewing

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

near and far measured from camera

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OpenGL Perspective

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

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Using Field of View

• With _glFrustum it is often difficult to get the desired view

•gluPerpective(fovy, aspect, near, far)

often provides a better interface

aspect = w/h

front plane