3D Viewing cgvr.korea.ac.kr

3d Rendering Pipeline 3D Primitives Model Transformation Lighting Viewing Transformation This is a pipelined sequence of operations to draw a 3D primitive into a 2D image for direct illumination Projection Transformation Clipping Viewport Transformation Scan Conversion Image cgvr.korea.ac.kr

In Pipeline Transform into3d world coordinate system 3D Primitives Model Transformation Transform into3d world coordinate system Lighting Viewing Transformation Projection Transformation Clipping Viewport Transformation Scan Conversion Image cgvr.korea.ac.kr

In Pipeline Transform into3d world coordinate system 3D Primitives Model Transformation Transform into3d world coordinate system Illustrate according to lighting and reflectance Lighting Viewing Transformation Projection Transformation Clipping Viewport Transformation Scan Conversion Image cgvr.korea.ac.kr

In Pipeline Transform into3d world coordinate system 3D Primitives Model Transformation Transform into3d world coordinate system Illustrate according to lighting and reflectance Lighting Transform into 3D viewing coordinate system Viewing Transformation Projection Transformation Clipping Viewport Transformation Scan Conversion Image cgvr.korea.ac.kr

In Pipeline Transform into3d world coordinate system 3D Primitives Model Transformation Transform into3d world coordinate system Illustrate according to lighting and reflectance Lighting Transform into 3D viewing coordinate system Viewing Transformation Transform into 2D viewing coordinate system Projection Transformation Clipping Viewport Transformation Scan Conversion Image cgvr.korea.ac.kr

In Pipeline Transform into3d world coordinate system 3D Primitives Model Transformation Transform into3d world coordinate system Illustrate according to lighting and reflectance Lighting Transform into 3D viewing coordinate system Viewing Transformation Transform into 2D viewing coordinate system Projection Transformation Clip primitives outside window’s view Clipping Viewport Transformation Scan Conversion Image cgvr.korea.ac.kr

In Pipeline Transform into3d world coordinate system 3D Primitives Model Transformation Transform into3d world coordinate system Illustrate according to lighting and reflectance Lighting Transform into 3D viewing coordinate system Viewing Transformation Transform into 2D viewing coordinate system Projection Transformation Clip primitives outside window’s view Clipping Transform into viewport Viewport Transformation Scan Conversion Image cgvr.korea.ac.kr

In Pipeline Transform into3d world coordinate system 3D Primitives Model Transformation Transform into3d world coordinate system Illustrate according to lighting and reflectance Lighting Transform into 3D viewing coordinate system Viewing Transformation Transform into 2D viewing coordinate system Projection Transformation Clip primitives outside window’s view Clipping Transform into viewport Viewport Transformation Draw pixels(includes texturing, hidden surface etc.) Scan Conversion Image cgvr.korea.ac.kr

Transformation Transform into3d world coordinate system 3D Primitives Model Transformation Transform into3d world coordinate system Illustrate according to lighting and reflectance Lighting Transform into 3D viewing coordinate system Viewing Transformation Transform into 2D viewing coordinate system Projection Transformation Clip primitives outside window’s view Clipping Transform into viewport Viewport Transformation Draw pixels(includes texturing, hidden surface etc.) Scan Conversion Image cgvr.korea.ac.kr

Transformation 3D Object Coordinate 3D World Coordinate P(x, y, z) 3D Object Coordinate 3D Viewing Coordinate Model Transformation 3D World Coordinate Viewing Transformation 3D Viewing Coordinate Projection Transformation 3D Object Coordinate 2D Projection Coordinate Viewport Transformation 3D World Coordinate 2D Device Coordinate p(x’, y’) cgvr.korea.ac.kr

Viewing Transformation P(x, y, z) 3D Object Coordinate Model Transformation 3D World Coordinate Viewing Transformation Viewing Transformation 3D Viewing Coordinate Projection Transformation 2D Projection Coordinate Viewport Transformation 2D Device Coordinate p(x’, y’) cgvr.korea.ac.kr

Viewing Transformation Mapping from world to Viewing coordinates Origin moves to eye position Up vector maps to Y axis Right vector maps to X axis Y Z Camera X cgvr.korea.ac.kr

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 General sequence of translate-rotate transformation cgvr.korea.ac.kr

Transformation from WC to VC (cont’) Translation view reference point(x0, y0, z0) 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 cgvr.korea.ac.kr

Transformation from WC to VC (cont’) Rotation by uvn system Calculate unit uvn vectors N : view-plane normal vector V : view-up vector U : perpendicular to both N and V Form the composite rotation matrix cgvr.korea.ac.kr

Camera Models The most common model is pin-hole camera All captured light rays arrive along paths toward focal point without lens distortion (everything is in focus) Sensor response proportional to radiance Other models consider… Depth of field Motion blur Lens distortion cgvr.korea.ac.kr

Viewing Parameters Position Orientation Aperture Film plane Eye position(px, py, pz) Orientation View direction(dx, dy, dz) Up direction(ux, uy, uz) Aperture Field of view(xfov, yfov) Film plane “look at” point View plane normal cgvr.korea.ac.kr

Viewing Coordinate Canonical coordinate system Convention is right-handed (looking down – z axis) Convention for projection, clipping, etc. Viewing up vector maps to Y axis Y Viewing back vector maps to Z axis (potting out of page) Viewing right vector maps to X axis X cgvr.korea.ac.kr

Viewing Transformation Transformation matrix maps camera basis vectors to canonical vectors in viewing coordinate system (0, 1, 0) Back Up Matrix (1, 0, 0) Eye Right (0, 0, 1) cgvr.korea.ac.kr

Viewing Transformation P(x, y, z) 3D Object Coordinate Model Transformation 3D World Coordinate Viewing Transformation 3D Viewing Coordinate Projection Transformation Projection Transformation 2D Projection Coordinate Viewport Transformation 2D Device Coordinate p(x’, y’) cgvr.korea.ac.kr

Projection General definition In computer graphics Transform points in n-space to m-space(m<n) In computer graphics Map viewing coordinates to 2D screen coordinates cgvr.korea.ac.kr

Taxonomy of Projections Planar geometric projection Parallel Perspective Orthographic Oblique Top Front Side Axonometric Cabinet Cavalier Other One-point Two-point Three-point cgvr.korea.ac.kr

Parallel & Perspective Parallel Projection Perspective Projection cgvr.korea.ac.kr

Taxonomy of Projections Planar geometric projection Parallel Perspective Orthographic Oblique Top Front Side Axonometric Cabinet Cavalier Other One-point Two-point Three-point cgvr.korea.ac.kr

Parallel Projection Center of projection is at infinity Direction of projection (DOP) same for all points DOP View Plane cgvr.korea.ac.kr

Taxonomy of Projections Planar geometric projection Parallel Perspective Orthographic Oblique Top Front Side Axonometric Cabinet Cavalier Other One-point Two-point Three-point cgvr.korea.ac.kr

Parallel Projection View Volume cgvr.korea.ac.kr

Orthographic & Oblique Orthographic parallel projection the projection is perpendicular to the view plane Oblique parallel projection The projectors are inclined with respect to the view plane cgvr.korea.ac.kr

Orthographic Projections DOP perpendicular to view plane cgvr.korea.ac.kr

Orthographic Projections DOP perpendicular to view plane Front Top Side cgvr.korea.ac.kr

Orthographic Coordinates cgvr.korea.ac.kr

Oblique Projections DOP not perpendicular to view plane Cavalier (DOP at 45 ) Cabinet (DOP at 63.4 ) cgvr.korea.ac.kr

Oblique Projections DOP not perpendicular to view plane Cavalier projection Cabinet projection cgvr.korea.ac.kr

Parallel Projection Matrix General parallel projection transformation Where L1 is the inverse of tan α , which is also the value of L when z=1 cgvr.korea.ac.kr

Parallel Projection Matrix General parallel projection transformation cgvr.korea.ac.kr

Parallel Projection Matrix cgvr.korea.ac.kr

Taxonomy of Projections Planar geometric projection Parallel Perspective Orthographic Oblique Top Front Side Axonometric Cabinet Cavalier Other One-point Two-point Three-point cgvr.korea.ac.kr

Perspective Projection Map points onto “view plane” along “projectors” emanating from “center of projection”(cop) Projectors Center of Projection View Plane cgvr.korea.ac.kr

Perspective Projection How many vanishing point? cgvr.korea.ac.kr

Perspective Projection How many vanishing point? Three-point perspective cgvr.korea.ac.kr

Perspective Projection How many vanishing point? Three-point perspective Two-point perspective cgvr.korea.ac.kr

Perspective Projection How many vanishing point? Three-point perspective Two-point perspective One-point perspective cgvr.korea.ac.kr

Perspective Projection View Volume cgvr.korea.ac.kr

Perspective Projection Compute 2D coordinates from 3D coordinates with similar triangles cgvr.korea.ac.kr

Perspective Projection Compute 2D coordinates from 3D coordinates with similar triangles cgvr.korea.ac.kr

Perspective Projection Matrix 4x4 matrix representation? cgvr.korea.ac.kr

Perspective Projection Matrix 4x4 matrix representation? cgvr.korea.ac.kr

Perspective Projection Matrix Orthographic projection Perspective transformation Center of Projection on the x axis Center of Projection on the y axis cgvr.korea.ac.kr

Perspective Projection Matrix 2-point perspectives 3-point perspectives cgvr.korea.ac.kr

Taxonomy of Projections Planar geometric projection Parallel Perspective Orthographic Oblique Top Front Side Axonometric Cabinet Cavalier Other One-point Two-point Three-point cgvr.korea.ac.kr

Perspective vs. Parallel Perspective projection + Size varies inversely with distance – looks realistic – Distance and angles are not(in general) preserved – Parallel line do not (in general) remain parallel Parallel projection + Good for exact measurements + Parallel lines remain parallel – Angles are not (in general) preserved – Less realistic looking cgvr.korea.ac.kr

Classical Viewing cgvr.korea.ac.kr

Thanks… cgvr.korea.ac.kr