8.Viewing

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Relationship among coordinate systems. The matrix underneath each stage determines the transformation applied at that stage for the perspective and parallel projections.

Workstation Transformation

(window to viewport mapping)

Projection coordinates
-1 <= x <= 1, -1 <= y <= 1, 0 <= z <= 1

Need device coordinates
x_dmin <= x <= x_dmax, y_dmin <= y <= y_dmax

Workstation Transformation

(window to viewport mapping)

Projection coordinates
-1 <= x <= 1, -1 <= y <= 1, 0 <= z <= 1

Need device coordinates
x_dmin <= x <= x_dmax, y_dmin <= y <= y_dmax

Steps:

1. Translate lower left corner to origin: T(1,1,0)

Workstation Transformation

(window to viewport mapping)

Projection coordinates
-1 <= x <= 1, -1 <= y <= 1, 0 <= z <= 1

Need device coordinates
x_dmin <= x <= x_dmax, y_dmin <= y <= y_dmax

Steps:

1. Translate lower left corner to origin: T(1,1,0)

2. Scale to correct size:

Workstation Transformation

(window to viewport mapping)

Projection coordinates
-1 <= x <= 1, -1 <= y <= 1, 0 <= z <= 1

Need device coordinates
x_dmin <= x <= x_dmax, y_dmin <= y <= y_dmax

Steps:

1. Translate lower left corner to origin: T(1,1,0)

2. Scale to correct size:

3. Translate into place:

View Volumes

View volume bounded by front and back planes, and by top, bottom, and side planes. Front and back planes are parallel to the view plane at positions z_front and z_back along the z_v axis.

Parallel Projection

Perspective Projection

View Volumes

World and View Spaces

World space
used for modeling;
right handed

World and View Spaces

World space
used for modeling;
right handed

View Space (simple)
Camera/viewer at origin
View along z_v axis
x_v and y_v aligned with display system

World and View Spaces

World space
used for modeling;
right handed

View Space (simple)
Camera/viewer at origin
View along z_v axis
x_v and y_v aligned with display system

World to View Transformation

Aligning a viewing system with the world-coordinate axes using a sequence of translate-rotate transformations.

World to View Transformation

Aligning a viewing system with the world-coordinate axes using a sequence of translate-rotate transformations.

Translate view point to origin of world coordinate space.

World to View Transformation

Aligning a viewing system with the world-coordinate axes using a sequence of translate-rotate transformations.

Rotate to align view coordinate axes (x_v,y_v,y_v) with world coordinate axes (x_w,y_w,y_w).

Two different projections of the same line

Line AB and its perspective projection A'B'.

Line AB and its parallel projection A'B'. Projectors AA' and BB' are parallel.

Simple parallel transformation (orthographic)

View plane is normal to direction of projection:
x_s = x_v, y_s = y_v, z_s = 0

Simple perspective transformation

Assume line from center of projection to center of view plane parallel to view plane normal.
Center of projection is at origin

Simple perspective transformation

Assume line from center of projection to center of view plane parallel to view plane normal.
Center of projection is at origin

Have P(x_v,y_v,z_v) want P(x_s,y_s)

Simple perspective transformation

Assume line from center of projection to center of view plane parallel to view plane normal.
Center of projection is at origin

Have P(x_v,y_v,z_v) want P(x_s,y_s)
By similar triangles:

Simple perspective transformation

Assume line from center of projection to center of view plane parallel to view plane normal.
Center of projection is at origin

Have P(x_v,y_v,z_v) want P(x_s,y_s)
By similar triangles:

In homogeneous coordinates:
x = x_v, y = y_v,z = z_v, w = z_v/d

Simple perspective transformation

Assume line from center of projection to center of view plane parallel to view plane normal.
Center of projection is at origin

Have P(x_v,y_v,z_v) want P(x_s,y_s)
By similar triangles:

In homogeneous coordinates:
x = x_v, y = y_v,z = z_v, w = z_v/d

Do perspective divide to get screen coordinates:
x_s = x/w, y_s = y/w,z_s = z/w = d

Basic Viewing System

Viewing system using a camera position, a viewing direction vector N, an up vector V, and an optional vector U. The world coordinate system is right handed, the view coordinate system is left handed.

Basic Viewing System

Requirements

Camera position (C)
View direction (N)
View up (V)
View plane distance (d)

Viewing system using a camera points, a viewing direction vector N, an up vector V, and an optional vector U. The world coordinate system is right handed, the view coordinate system is left handed.

Basic Viewing System

Requirements

Camera position (C)
View direction (N)
View up (V)
View plane distance (d)

Characteristics

View direction controllable
Camera up controllable
No view volume specified
No view plane window specified
Perspective projection with viewport as center of projection

Viewing system using a camera points, a viewing direction vector N, an up vector V, and an optional vector U. The world coordinate system is right handed, the view coordinate system is left handed.

Basic Viewing System

Implementation

Translation as before T(-c_x,-c_y,-c_z)

Basic Viewing System

Implementation

Translation as before T(-c_x,-c_y,-c_z)
Rotate to align axes:

Basic Viewing System

Implementation

Translation as before T(-c_x,-c_y,-c_z)
Rotate to align axes:
Convert to left-handed coordinates

View Transformation

1. Translate origin of world coordinate system to origin of view coordinate system. (Transformation of coordinate system is inverse of that which moves points)

View Transformation

C - camera position (also called VRP)

1. Translate origin of world coordinate system to origin of view coordinate system. (Transformation of coordinate system is inverse of that which moves points)

View Transformation

2. Rotate coordinate system 90^o about x' axis. Use theta = -90.

View Transformation

2. Rotate coordinate system 90^o about x' axis.

View Transformation

3. Rotate about y' by theta so that (0,0,c_z) lies on z' axis.

View Transformation

3. Rotate about y' by theta so that (0,0,c_z) lies on z' axis.

View Transformation

4. Rotate about x' by phi so that origin of original coordinate system lies on z' axis.

View Transformation

4. Rotate about x' by phi so that origin of original coordinate system lies on z' axis.

View Transformation

5. Reflect z' axis to create left-handed coordinate system.

View Transformation

5. Reflect z' axis to create left-handed coordinate system.

View Transformation

6. Twist about z' so that y' aligns with V

View Transformation

6. Twist about z' so that y' aligns with V

Viewing Example

Camera at (6,8,7.5)
View towards (0,0,0)
VPN (-6,-8,-7.5)
View up (-3.6,-4.8,8)

Viewing Example

Camera at (6,8,7.5)
View towards (0,0,0)
VPN (-6,-8,-7.5)
View up (-3.6,-4.8,8)

1. Translate world origin to view origin

Viewing Example

2. Rotate 90⁰ about x'.

Viewing Example

3. Rotate about y' by Theta 90⁰ so that (0,0,c_z) lies on z' axis.

Viewing Example

4.Rotate about x' by phi so that origin of original coordinate system lies on z' axis.

Viewing Example

5. Reflect z' axis to create left-handed coordinate system.

Viewing Example

6. Twist about z' so that y' aligns with V

Viewing Example

Multiply it all together

Viewing Example

Cube at origin

-1

Compositions of Translations and Rotations

Resulting matrix has form

Basis Rotation Shortcut

Where u'_x,u'_y,u'_z are unit basis vectors

Basis Rotation Shortcut

Where u'_x,u'_y,u'_z are unit basis vectors

Assume we've already performed translation, so x'₀=0, y'₀=0, z'₀=0

Basis Rotation Shortcut

Where u'_x,u'_y,u'_z are unit basis vectors

Assume we've already performed translation, so x'₀=0, y'₀=0, z'₀=0

Can rotate to align basis vectors using

Expressed in coordinates of S

Applying the shortcut

Given view direction vector N

Given view up vector V

Shortcut Example

Camera at (6,8,7.5)
View towards (0,0,0)
VPN (-6,-8,-7.5)
View up (-3.6,-4.8,8)

Shortcut Example

Camera at (6,8,7.5)
View towards (0,0,0)
VPN (-6,-8,-7.5)
View up (-3.6,-4.8,8)

Clipping for Basic View System

Rely on 2D clipping of projected primitives (mostly wire-frame)
Only works if view point is outside entire scene

Advanced Viewing System

Advanced Viewing system: view frustum formed by six planes and the view volume. This is a left-handed coordinate system.

Advanced Viewing System

Requirements

Camera position (C)
View direction (N,Z_v)
View up (Y_v)
Distance to near (d) and far (f) plane

Advanced Viewing system: view frustum formed by six planes and the view volume. This is a left-handed coordinate system.

Advanced Viewing System

Requirements

Camera position (C)
View direction (N,Z_v)
View up (Y_v)
Distance to near (d) and far (f) plane

Characteristics

View position and direction controllable
Camera up controllable

Advanced Viewing system: view frustum formed by six planes and the view volume. This is a left-handed coordinate system.

Advanced Viewing System

Requirements

Camera position (C)
View direction (N,Z_v)
View up (Y_v)
Distance to near (d) and far (f) plane

Characteristics

View position and direction controllable
Camera up controllable
View volume specified, but view plane constrained to be coincident with near plane

Advanced Viewing system: view frustum formed by six planes and the view volume. This is a left-handed coordinate system.

Advanced Viewing System

Requirements

Camera position (C)
View direction (N,Z_v)
View up (Y_v)
Distance to near (d) and far (f) plane

Characteristics

View position and direction controllable
Camera up controllable
View volume specified, but view plane constrained to be coincident with near plane
View window square and symmetric about the center of view plane
Perspective with center of projection at view point

Advanced Viewing system: view frustum formed by six planes and the view volume. This is a left-handed coordinate system.

Advanced Viewing System

View volume specified by:

(two sides)

Advanced Viewing system: view frustum formed by six planes and the view volume. This is a left-handed coordinate system.

Advanced Viewing System

View volume specified by:

(two sides)

(top and bottom)

Advanced Viewing system: view frustum formed by six planes and the view volume. This is a left-handed coordinate system.

Advanced Viewing System

View volume specified by:

(two sides)

(top and bottom)

(near and far)

Advanced Viewing system: view frustum formed by six planes and the view volume. This is a left-handed coordinate system.

Advanced Viewing System

View volume specified by:

(two sides)

(top and bottom)

(near and far)
View plane has dimensions:
2h x 2h

Advanced Viewing system: view frustum formed by six planes and the view volume. This is a left-handed coordinate system.

Projection for Advanced View System (part 1)

Want 3D screen space because it facilitates:

3D Clipping
Visibility calculation

Chose Z_s such that:

Z_s normalized for maximum precision
Lines in view space -> lines in screen space
Planes in view space -> planes in screen space

Projection for Advanced View System (part 1)

Want 3D screen space because it facilitates:

3D Clipping
Visibility calculation

Chose Z_s such that:

Z_s normalized for maximum precision
Lines in view space -> lines in screen space
Planes in view space -> planes in screen space

Satisfied for:

where B < 0 to preserve intuitive notion of depth (if Z_v increases, so does Z_s)
and range is normalized so that maps to

Projection for Advanced View System (part 1)

Want 3D screen space because it facilitates:

3D Clipping
Visibility calculation

Chose Z_s such that:

Z_s normalized for maximum precision
Lines in view space -> lines in screen space
Planes in view space -> planes in screen space

Satisfied for:

where B < 0 to preserve intuitive notion of depth (if Z_v increases, so does Z_s)
and range is normalized so that maps to

Projection for Advanced View System (part 2)

Full perspective transformation

Where h insures that x_s and y_s fall in range [-1,1]

Projection for Advanced View System (part 2)

Full perspective transformation

Where h insures that x_s and y_s fall in range [-1,1]

Using a homogeneous coordinates:

Projection for Advanced View System (part 2)

Full perspective transformation

Where h insures that x_s and y_s fall in range [-1,1]

Using a homogeneous coordinates:

So:

Projection for Advanced View System (part 3)

Decompose T_pers:
T_pers = T_{pers 2}T_{pers 1}

Projection for Advanced View System (part 3)

Decompose T_pers:
T_pers = T_{pers 2}T_{pers 1}

Scaling in x and y to a regular pyramid (side sloping 45^o)

Projection for Advanced View System (part 3)

Decompose T_pers:
T_pers = T_{pers 2}T_{pers 1}

Projection for Advanced View System (part 3)

Map pyramid to the box:

Clipping for Advanced View System

Clipping against view frustum necessary because:

View point may be arbitrarily located
Screen space clipping may not remove all points outside the viewing frustum

Clipping for Advanced View System

Clipping against view frustum necessary because:

View point may be arbitrarily located
Screen space clipping may not remove all points outside the viewing frustum

Clipping cases:

Completely outside -> discard
Completely inside -> accept
Intersect view frustum -> clip further

Clipping for Advanced View System

Clipping against view frustum necessary because:

View point may be arbitrarily located
Screen space clipping may not remove all points outside the viewing frustum

Clipping cases:

Completely outside -> discard
Completely inside -> accept
Intersect view frustum -> clip further

Clip in homogeneous coordinates (before perspective divide) to:

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