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 Projections and Shadows
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 Multiview orthographic 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.world coords.eye coords. 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) glMatrixMode(GL_MODEL); glMultMatrix(Mo); render_object(); glMatrixMode(GL_VIEW); glMultMatrix(Mc); glMatrixMode(GL_MODELVIEW); glMultMatrix(inv(Mc)); glMultMatrix(Mo); render_object();
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 View-orientation matrix derivation? (advanced) 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 Elevation, azimuth, twist 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 Small overhead More in Chap 7 Culling Back faces of closed objects not rendered Number of primitives reduced early
5.7 Interactive Mesh Displays
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_minz’’=-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
5.10 Projections and Shadows
Shadows A point is in shadow if it is not illuminated by any light source, or equivalently if a viewer at that point cannot see any light source. Shadow polygon
Rendering Shadow Render twice: object polygon and shadow polygon How to find the shadow polygon? perspective projection (with light source as the camera) Works only for shadows on flat surface More on Chap 12