Computer Graphics (fall 2009)

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Computer Graphics (fall 2009). School of Computer Science University of Seoul. Chap 5: Viewing. Classical and Computer Viewing Viewing with a Computer Positioning of the Camera Simple Projections Projections in OpenGL Hidden-Surface Removal Interactive Mesh Displays

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### Computer Graphics(fall 2009)

School of Computer Science

University of Seoul

Chap 5: Viewing
• Classical and Computer Viewing
• Viewing with a Computer
• Positioning of the Camera
• Simple Projections
• Projections in OpenGL
• Hidden-Surface Removal
• Interactive Mesh Displays
• Parallel-Projection Matrices
• Perspective-Projection Matrices

### 5.1 Classical and Computer Viewing

Basic Elements
• Object
• Projector
• Projection Plane
• COP (Center of Projection) or DOP (Direction of Projection)
• Principal faces/directions
Classical Viewing
• Orthographic projections
• Axonometric projections
• Isometric, dimetric, trimetric
• Oblique projections
• Perspective projections
• One-, two-, three-point perspectives
1. Orthographic Projections
• Projectors perpendicular to the projection plane
• Multivieworthographic projection

(image courtesy of http://img19.imageshack.us/i/importedpsk.jpg)

2. Axonometric Projections
• Projectors orthogonal to the projection plane
• Projection plane can have any orientation with respect to the object
• Isometric, dimetric, trimetric
• How many principal faces is the projection plane is placed symmetrically with respect to
3. Oblique Projections
• Most general parallel views
• Projector can have an arbitrary angle with the projection plane
4. Perspective Viewing
• To achieve real-looking images
• One-, two-, three-point perspectives
• Number of vanishing points
• How many of three principal directions in the object are parallel to the projection plane?

### 5.2 Viewing with a Camera

OpenGL Viewing
• All the classical viewing can be implemented HOW??? (§5.9 & §5.10)
• Default camera
• Orthographic
• Projection plane: “z=0”
• At the origin pointing to –z-direction
• Up direction: +y-direction
• Viewing volume: cube with side 2  objects outside of this box cannot be rendered! (“clipping”)
• Objects behind the camera can be rendered! (only for orthogonal projections)

### 5.3 Positioning of the Camera

Positioning the Camera
• No separate “viewing” transformation  concatenated into model-view matrices
• Object coords.worldcoords.eyecoords.
• What matters is the relative position of objects and the camera
• At any given time, the state of the model-view matrix encapsulates the relationship between the camera frame and the object frame
Positioning the Camera (cont’d)
• Separate matrices For model & viewing
• Model-view matrix (OpenGL)
Viewing-Coordinate System
• Defined by
• Position (VRP: view-reference point, p)
• Viewing direction (VPN: view-plane normal , n)
• Up direction (VUP: view-up vector, vup)
• Projected on the view plane (v)
• 3rd orthogonal direction (u) obtained by cross product
• x, y, z axes  u, v, n
• gluLookAt
• Needs to be called before transforming objects
• Error in the book (p.251)
Other Viewing APIs
• Roll, pitch, yaw
• To specify the orientation
• Ex) flight simulation
• Elevation, azimuth, twist
• Direction from the viewer
• Ex) star in the sky

### 5.4 Simple Projections

Perspective Projections
• All projectors pass through the origin
• (x,y,z)(xp,yp,zp)
• Nonlinear, not affine, irreversible
• Perspective division required
• “Extended” homogeneous coordinates required  3D points = 4D lines (through the origin): (x,y,z,1)(x,y,z,z/d)=(xp,yp,zp,1)
• How does the matrix look like?
Orthogonal Projections
• No division required
• (x,y,z)(x,y,0)=(xp,yp,zp)
• How does the matrix look like?

### 5.5 Projections in OpenGL

Projections in OpenGL
• Parameters in eye coordinates
• Near/far clipping planes
Perspective Viewing in OpenGL
• glFrustum
• Need not be symmetric
• gluPerspective
• Symmetric
• Defined using fov (field of view)  error in the figure 5.28 of the textbook
• Calls glFrustum internally
• near & far parameters must be positive

(image courtesy of the redbook)

Parallel Viewing in OpenGL
• glOrtho
• gluOrtho2D
• No sign restrictions on near & far parameters

### 5.6 Hidden-Surface Removal

Hidden-Surface Removal
• How to determine which object is closer than others?
• What happens without hidden-surface removal?
• Two classes
• Object space algorithm
• BSP (Binary Space Partitioning) tree  restriction?
• Image space algorithm
• Z-buffer algorithm  used in (most) interactive graphics system including OpenGL
• Complexity proportional to the resolution
• More in Chap 7
• Culling
• Back faces of closed objects not rendered
• Number of primitives reduced early

### 5.8 Parallel-Projection Matrices

Projection Normalization
• All projections are converted into orthogonal projections by first distorting the objects such that the orthogonal projection of the distorted objects is the same as the desired projection of the original objects.
• Simplifies clipping & hidden-surface removal(Chap 7)
Orthogonal-Projection Matrices
• Normalization(by OpenGL projection matrix)
• Converts the specified viewing volume to canonical view volume ([-1,1]x[-1,1]x[-1,1] cube)
• Translation followed by scaling
• Orthographic projection
• (x,y,z)  (x,y,0)
Oblique Projections
• We can either
• Define the 4x4 matrix directly or
• Implement by shear followed by orthographic projection (In practice, normalization is required in between)

### 5.9 Perspective-Projection Matrices

Simple Perspective Projection
• Frustum defined by x=±z, y=±z, z_max, z_min
• For the perspective-normalization matrix N,what converts the planes as follows?
• x=±z  x’’=±1
• y=±z  y’’=±1
• z=z_max z’’=1
• z=z_minz’’=-1
Perspective Projection
• Depth ordering preserved by perspective-projection matrix  hidden-surface removal works in the normalized volume
• General perspective projection
• Apply shear to convert the asymmetric frustum to a symmetric one
• Scale to the “simple” frustum