3D Rendering. Course Syllabus. Where Are We Now? Rendering. 3D Rendering Example. Overview. Rendering. I. Image processing II. Rendering III.

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1 Course Syllabus I. Image processing II. Rendering III. Modeling 3D Rendering Rendering I. Animation (Michael Bostock, CS426, Fall99) Image Processing Adam Finkelstein Princeton University COS 426, Spring 2005 (Rusty Coleman, CS426, Fall99) Modeling (Dennis Zorin, CalTech) Animation (Jon Beyer, CS426, Spring04) Where Are We ow? Rendering I. Image processing Generate an image from geometric primitives II. Rendering III. Modeling Rendering I. Animation (Michael Bostock, CS426, Fall99) Rendering Image Processing (Rusty Coleman, CS426, Fall99) Geometric Primitives Modeling (Dennis Zorin, CalTech) Animation (Jon Beyer, CS426, Spring04) 3D Rendering Example Overview 3D scene representation 3D viewer representation isible surface determination Lighting simulation What issues must be addressed by a 3D rendering system? Raster Image

2 Overview» 3D scene representation 3D viewer representation isible surface determination Lighting simulation 3D Scene Representation Scene is usually approximated by 3D primitives Point Line segment Polygon Polyhedron Curved surface Solid object etc. How is the 3D scene described in a computer? 3D Point Specifies a location 3D Point Specifies a location Represented by three coordinates Infinitely small Coordinate x; Coordinate y; Coordinate z; } Point; (x,y,z) 3D ector Specifies a direction and a magnitude 3D ector Specifies a direction and a magnitude Represented by three coordinates Magnitude = sqrt(dx dx + dy dy + dz dz) Has no location Coordinate dx; Coordinate dy; Coordinate dz; } ector; (dx,dy,dz)

3 3D ector Dot product of two 3D vectors 1 2 = 1 2 cos(") (dx 1,dy 1,dz 1 ) 3D ector Cross product of two 3D vectors 1 2 = (dy 1 dx 2 - dz 1 dy 2, dz 1 dx 2 - dx 1 dz 2, dx 1 dy 2 - dy 1 dx 2 ) 1 x 2 = vector perpendicular to both 1 and 2 1 x 2 = 1 2 sin(") (dx 1,dy 1,dz 1 ) " (dx 2,dy 2,dz 2 ) " (dx 2,dy 2,dz 2 ) 1 x 2 3D Line Segment Linear path between two points 3D Line Segment Use a linear combination of two points Parametric representation:» P = P 1 + t (P 2 - P 1 ), (0 # t # 1) Point P1; Point P2; } Segment; P 2 P 1 3D Ray Line segment with one endpoint at infinity Parametric representation:» P = P 1 + t, (0 <= t < $) Point P1; ector ; } Ray; 3D Line Line segment with both endpoints at infinity Parametric representation:» P = P 1 + t, (-$ < t < $) Point P1; ector ; } Line; P 1 P 1

4 3D Plane 3D Plane A linear combination of three points A linear combination of three points Implicit representation:» P + d = 0, or» ax + by + cz + d = 0 P2 ector ; Distance d; } Plane; P3 P1 = (a,b,c) P2 P3 P1 is the plane normal» Unit-length vector» Perpendicular to plane d 3D Polygon 3D Sphere Area inside a sequence of coplanar points Triangle Quadrilateral Convex Star-shaped Concave Self-intersecting All points at distance r from point (cx, cy, cz) Implicit representation:» (x - cx)2 + (y - cy)2 + (z - cz)2 = r 2 Parametric representation:» x = r cos(%) cos(") + cx» y = r cos(%) sin(") + cy» z = r sin(%) + cz Point *points; int npoints; } Polygon; r Point center; Distance radius; } Sphere; Points are in counter-clockwise order Holes (use > 1 polygon struct) 3D Scenes Other Geometric Primitives Comprise set of geometric primitives More detail on 3D modeling later in course (Dennis Zorin, CalTech) (Angel, Plate 1) Point Line segment Polygon Polyhedron Curved surface Solid object etc. (Michael Bostock, CS426, Fall99) H&B Figure 10.46

5 Overview Camera Models 3D scene representation» 3D viewer representation isible surface determination 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 Lighting simulation iew plane How is the viewing device described in a computer? Camera Parameters What are the parameters of a camera? Other models consider... Depth of field Motion blur Lens distortion Eye position (focal point) Camera Parameters Position Eye position (px, py, pz) Orientation iew direction (dx, dy, dz) Up direction (ux, uy, uz) Up direction Aperature Field of view (xfov, yfov) Film plane Look at Point back Look at point iew plane normal iew Frustum iew direct ion right Overview 3D scene representation Up Back 3D viewer representation» isible surface determination Towards Lighting simulation Right How can the front-most surface be found with an algorithm? iew Frustum iew Plane Eye Position

6 isible Surface Determination The color of each pixel on the view plane depends on the radiance emanating from visible surfaces Ray Casting For each sample Construct ray from eye position through view plane Find first surface intersected by ray through pixel Compute color of sample based on surface radiance Simplest method is ray casting Rays through view plane Eye position iew plane Ray Casting For each sample Construct ray from eye position through view plane Find first surface intersected by ray through pixel Compute color of sample based on surface radiance Construct Ray Up direction iew Plane back towards P 0 right Ray: P = P 0 + t P Ray Casting For each sample Construct ray from eye position through view plane Find first surface intersected by ray through pixel Compute color of sample based on surface radiance Find First Surface Intersection T 3 T 1 ' & P T 2 P 0

7 isible Surface Determination For each sample Construct ray from eye position through view plane Find first surface intersected by ray through pixel Compute color of sample based on surface radiance Rendering Algorithms Any samples can be used Rendering is a problem in sampling and reconstruction More efficient algorithms utilize spatial coherence Overview 3D scene representation 3D viewer representation isible surface determination» Lighting simulation Lighting Simulation Lighting parameters Light source emission Surface reflectance Atmospheric attenuation Camera response Surface Light Source How do we compute the radiance for each sample ray? Camera Lighting Simulation Lighting Simulation Direct illumination Ray casting Polygon shading Light Source iewer L 1 L 2 Global illumination Ray tracing Monte Carlo methods Radiosity methods Surface More on these methods later Camera

8 Summary Major issues in 3D rendering 3D scene representation 3D viewer representation isible surface determination Lighting simulation ext Lecture Ray intersections Light and reflectance models Indirect illumination Concluding note Accurate physical simulation is complex and intractable» Rendering algorithms apply many approximations to simplify representations and computations Tricycle (James Percy, CS 426, Fall99) For assignment #2, you will write a ray tracer