THREE-DIMENSIONAL VIEWING 12 고려대학교 컴퓨터 학과 김 창 헌

Contents Viewing Pipeline Viewing Coordinates Projections View Volumes and Projection Transformations Clipping Hardware Implementations 3D Viewing Functions Summary

Viewing Pipeline General 3D Transformation Pipeline Clipping against Modeling Coordinates Modeling Transformation World Coordinates Viewing Transformation Viewing Coordinates Clipping against Viewing volume Projection Transformation Projection Coordinates Workstation Transformation Device Coordinates Project onto Projection plane

Viewing Coordinates Specifying the View Plane Viewing-coordinate system View-plane normal vector (N) (1, 0, 0) (1, 0, 1)

Viewing Coordinates (con’t) Specifying the View Plane (con’t) View-up vector (V) uvn system

Viewing Coordinates (con’t) Specifying the View Plane (con’t) View-plane distance Series of views of a scene

Viewing Coordinates (con’t) Transformation from WC to VC Transformation sequences 1. Translate the view reference point to the origin of the WC system 2. Apply rotations to align the xv, yv, and zv axes with the world axes Translation view reference point(x0, y0, z0)

Viewing Coordinates (con’t) Transformation from WC to VC (con’t) General sequence of translate-rotate transformation Rotation rotate around the world xw axis to bring zv into the xwzw plane rotate around the world yw axis to align the zw and zv axis final rotation is about the zw axis to align the yw and yv axis

Viewing Coordinates (con’t) Transformation from WC to VC (con’t) Direct generating the rotation-transformation matrix

Projections Parallel Projection Perspective Projection

Projections Parallel Projections Orthographic parallel projection the projection is perpendicular to the view plane Oblique parallel projection

Projections (con’t) Parallel Projections (con’t) Orthographic projection Isometric projection

12.3 Projections (con’t) Parallel Projections (con’t) Orthographic projection coordinates

12.3 Projections (con’t) Parallel Projections (con’t) Oblique projection

12.3 Projections (con’t) Parallel Projections (con’t) Oblique projection

12.3 Projections (con’t) Parallel Projections (con’t) Oblique projection cavalier projection ( )

12.3 Projections (con’t) Perspective Projections P=(x, y, z) (xp,yp,zp) zvp zprp view plane

12.3 Projections (con’t) Perspective Projections (con’t)

12.3 Projections (con’t) Perspective Projections (con’t) zvp = 0 zprp = 0

12.3 Projections (con’t) Perspective Projections (con’t) Principal vanishing point

View Volumes and Projection Transformation View Window

View Volumes … (con’t) Parallel projection

View Volumes … (con’t) Perspective projection

View Volumes … (con’t) View volume boundary

View Volumes … (con’t) Changing the shape of the oblique-projection view volume

View Volumes … (con’t) Moving the projection reference point

View Volumes … (con’t) Projected object size

View Volumes … (con’t) Animation

12.4 View Volumes … (con’t) General Parallel-Projection Transformations Oblique projection view volume Regular parallelepiped view volume

12.4 View Volumes … (con’t) General Parallel-Projection Transformations (con’t) Projection vector Transformation

12.4 View Volumes … (con’t) General Parallel-Projection Transformations (con’t) General parallel-projection matrix

12.4 View Volumes … (con’t) General Parallel-Projection Transformations (con’t) Element Relationship

12.4 View Volumes … (con’t) General Perspective-Projection Transformation Perspective view volume

View Volumes … (con’t) General Perspective-Projection Transformation

View Volumes … (con’t) General Perspective-Projection Transformation (con’t)

Clipping Clipping Identify and save all surface segments within the view volum View-volume clipping boundary : planes

Clipping (con’t) Normalized View Volumes  Why? View Volume

Clipping (con’t) Normalized View Volumes (con’t) Pipeline

Clipping (con’t) Normalized View Volumes (con’t) Advantages provides a standard shape for representing any sized view volume clipping procedures are simplified and standardized with unit cube depth cueing and visible-surface determination are simplified Mapping position within a rectangular view volume to a 3D rectangular viewport

Clipping (con’t) Normalized View Volumes (con’t) Ratios of the dimensions of the viewport and view volume Translation factors

Clipping (con’t) Viewport Clipping 1. Parametric line equation 2. Calculation of parameter value 3. Intersection point

Clipping (con’t) Viewport Clipping (con’t)

Clipping (con’t) Clipping in Homogeneous Coordinates

Hardware Implementations

3D Viewing Functions Multiple Views Using Difference Camera Orientations

3D Viewing Functions (con’t) A Wide-Angle Perspective Display Same viewing position, but with slight shifts in the viewing direction

Summary Modeling Coordinates Modeling Transformation World Coordinates Viewing Transformation Viewing Coordinates Projection Transformation Projection Coordinates Workstation Transformation Device Coordinates