CS 551 / 645: Introductory Computer Graphics Review for Midterm David Luebke 7/30/2018

Administrivia Hand out assignment 2 Hand in assignment 3 Late day policy: 1 late day = due tomorrow at noon Subsequent late days add 24 hours each Weekends and holidays count Compiling with C++ UNIX C++ compilers: g++ and /bin/CC I’ll make sure it works before next assignment David Luebke 7/30/2018

Midterm Examination Midterm is this Thursday (March 9) Study aids: This lecture Earlier lectures (available on course page) Last semester’s midterm See http://www.cs.virginia.edu/~luebke/old_cs551/ But, its not quite the same material No calculators! (you won’t need them) David Luebke 7/30/2018

Display Technologies Cathode Ray Tubes Earliest, still most common graphical display Understand the basic mechanism Vacuum tube, phosphors, electron beam Pros: bright, fairly high-res, leverages TV tech Cons: bulky, size-limited, finicky David Luebke 7/30/2018

Display Technologies Vector versus raster display Vector: traces lines like an oscilloscope Pros: bright, crisp, uniform lines Cons: wireframe only, flicker for complex scenes Raster: fixed scan pattern for electron beam, intensity controlled by scan-out from frame buffer Pros: display solid objects, image complexity limited only by framebuffer resolution Cons: discreet sampling (aliasing), memory cost David Luebke 7/30/2018

Display Technologies LCDs Understand the basic mechanism Polarized light, crystals twist 90º unless excited Basically a light valve: reflective or transmissive Pros: light-weight and thin Cons: expensive, high-power (when backlit), limited in size David Luebke 7/30/2018

Display Technologies Also know: Plasma display panels Digital Micromirror Devices David Luebke 7/30/2018

Framebuffers Memory array storing image in pixels Issues: memory speed, size, bus contention Different types in common use, motivated mainly by memory cost True-Color: 24 bits, 8 per RGB (or 32 bits with ) Hi-Color: 16 bits (R = 6, G = 6, B = 4) Pseudo-Color: 8 bits index into 256-entiry color lookup table (entries typically 24-bits) David Luebke 7/30/2018

Mathematical Foundations Geometry (2-D, 3-D) Trigonometry Vector spaces Elements: scalars and vectors Operations: Addition (identity & inverse) Scalar multiplication (distributive rule) Linear combinations, dimension, basis sets Inner (dot) product, vector (cross) product David Luebke 7/30/2018

Mathematical Foundations Affine spaces Elements: points Operations: Subtraction (point - point = vector) Addition (vector + point = point) Matrices Linear transforms, vector-matrix multiplication Matrix-matrix multiplication Composition of linear transforms = matrix concatenation David Luebke 7/30/2018

The Rendering Pipeline Transform Illuminate Clip Project Rasterize Model & Camera Parameters Rendering Pipeline Framebuffer Display David Luebke 7/30/2018

The Rendering Pipeline: 3-D Scene graph Object geometry Result: All vertices of scene in shared 3-D “world” coordinate system Vertices shaded according to lighting model Scene vertices in 3-D “view” or “camera” coordinate system Exactly those vertices & portions of polygons in view frustum 2-D screen coordinates of clipped vertices Modeling Transforms Lighting Calculations Viewing Transform Clipping Projection Transform David Luebke 7/30/2018

Geometric Transforms Modeling transforms: object coordinates  world coordinates Viewing transform: world coordinates  eye coordinates eye coordinates == camera coordinates == view coordinates Projection transform: eye coordinates  2-D screen coordinates David Luebke 7/30/2018

Geometric Transforms Understand homogeneous coordinates [x, y, z, w]T == (x/w, y/w, z/w) Allows us to capture translation and projection as matrices Know your 4x4 Euclidean transform matrices: Translation, scale, rotation about X, Y, Z Understand rotation about an arbitrary axis Understand order of composition for matrices In OpenGL: using column vectors as points  order from right to left David Luebke 7/30/2018

Perspective Projection Geometry of the perspective projection: P (x, y, z) X Z View plane d (0,0,0) x’ = ? David Luebke 7/30/2018

Perspective Projection Desired result: David Luebke 7/30/2018

Perspective Projection A matrix that accomplishes this: David Luebke 7/30/2018

Rasterizing Lines Review McMillan’s great java-enabled lecture First stab: slope-intercept + symmetry A case study in optimization Special case boundary conditions if necessary Optimize inner loops Incremental update using DDA (biggest win) Low-level tricks: integer arithmetic, compare to 0, etc. Culmination: Bresenham’s algorithm Be aware of diminishing returns and readability/portability tradeoffs David Luebke 7/30/2018

Rasterizing Triangles Triangles are nice to deal with because they are always planar and always convex Triangle rasterization techniques: REYES: recursive subdivision of primitive Warnock: recursive subdivision of screen Edge walking Edge equations David Luebke 7/30/2018

Rasterizing Triangles Edge walking: Draw edges vertically Fill in horizontal spans for each scanline Interpolate colors down edges At each scanline, interpolate edge colors across span Pros: Fast: touch only lit pixels, touch pixels only once Cons: Finicky: lots of special cases, hard to get just right David Luebke 7/30/2018

Rasterizing Triangles Edge Equations Equation of a line defines two half-spaces Triangle can be represented as intersection of three half-spaces: A1x + B1y + C1 < 0 A2x + B2y + C2 < 0 A3x + B3y + C3 < 0 A1x + B1y + C1 > 0 A3x + B3y + C3 > 0 A2x + B2y + C2 > 0 David Luebke 7/30/2018

Rasterizing Triangles Basic algorithm: Walk pixels in bounding box Evaluate three edge equations If all are greater than zero, shade pixel Issues: Computing edge equations: numerical precision Interpolating parameters (i.e., color): just like another edge equation (why?) Optimizing the algorithm Like line rasterization: DDA, early termination, etc. David Luebke 7/30/2018

Rasterizing General Polygons B C D E F G I H Parity test: Starting outside polygon, count edges crossed. Odd = inside, even = outside Big cost: testing every edge against every pixel Solution: the active edge table algorithm Sort edges by Y Keep a list of edges that intersect current scanline, sorted by their X-intersection w/ scanline David Luebke 7/30/2018

Clipping Lines Cohen-Sutherland Algorithm Clip 2-D line segments to rectangular viewport Designed for rapid trivial accept & trivial reject ops 4-bit outcodes divide screen into 9 regions Bitwise operations determine whether to accept, reject, or intersect with a viewport edge & recurse May require multiple iterations David Luebke 7/30/2018

Clipping Polygons Clipping polygons fundamentally more difficult Polygons can gain or lose edges Concave polygons can even multiply Sutherland-Hodgman Algorithm Simplify by divide-and-conquer: consider each clipping plane individually Input: polygon as ordered list of vertices Output: polygon as ordered list of vertices Lends itself to pipelined hardware implementation David Luebke 7/30/2018

Clipping Polygons Sutherland-Hodgman Algorithm Know the details: Point-plane test Line-plane intersection Rules: inside outside inside outside inside outside inside outside p p s s p s p s p output i output no output i output p output David Luebke 7/30/2018

Clipping in 3-D Problem: clipping under perspective must happen before homogeneous divide Solution 1: clip to hither plane in eye coordinates, then multiply by projection matrix, then do homogeneous divide Better: transform to canonical perspective coordinates to simplify clipping Solution 2: clip after projection (must clip all 4 homogeneous coordinates) Solution 3 (ugly but common): clip to hither & yon before projection, clip to 2-D viewport after projection and divide David Luebke 7/30/2018

Color Rods and cones Cones and color perception Gamma correction Metamers 3-D color: X, Y, and Z; CIE color space Gamma correction David Luebke 7/30/2018

Lighting Definitions: illumination, lighting, shading Illumination: Direct versus indirect Light properties: geometry, spectrum Common simplifications: ambient, directional, and point Surface material: geometry, reflectance, microstructure Common simplification: Phong lighting Diffuse (Lambertian) reflection: incoming light reflected equally in all directions, proportional to N • L Specular reflection: approximate falloff with (V • R)nshiny David Luebke 7/30/2018

Lighting Putting it all together: the Phong lighting model Note: evaluate per light, per color component Common simplification: constant V (viewer infinitely far away) David Luebke 7/30/2018

Shading Where to apply lighting calculations? Once per face: flat shading Once per vertex, interpolate resulting color: Gouraud shading Once per pixel, interpolating normal vectors from vertices: Phong shading Flat shading Phong Shading David Luebke 7/30/2018