2/28/2016 CS 551 / 645: Introductory Computer Graphics Framebuffer Mathematical Foundations The Rendering Pipeline

2/28/2016 Organization l Class List l ITC accounts l Class web page is up l Assignment 1 goes out on Thursday l Read OGLPG Chapter 1 for Thursday

2/28/2016 Basic Definitions Raster: A rectangular array of points or dots. Pixel (Pel): One dot or picture element of the raster Scan line: A row of pixels Video raster devices display an image by sequentially drawing out the pixels of the scan lines that form the raster.

2/28/2016 Frame Buffers l A frame buffer may be thought of as computer memory organized as a two-dimensional array with each (x,y) addressable location corresponding to one pixel. l Bit Planes or Bit Depth is the number of bits corresponding to each pixel. l A typical frame buffer resolution might be 640 x 480 x x 1024 x x 1024 x 24

2/28/ Bit Memory, Monochrome Display (Bitmap Display)

2/28/ Bit Color Display

2/28/2016 True Color Display Red Blue bitplanes, 8 bits per color gun = 16,777,216

2/28/2016 Color Map Look-Up Tables Extends the number of colors that can be displayed by a given number of bit-planes RG B RED GREEN BLUE Pixel displayed at x', y' Pixel in bit map at x', y' 0 x 0 y x max y Frame bufferLook-up tableDisplay

2/28/2016 Pseudo color 2 8 x 24 Color Map LUT

2/28/2016 Display processor l Specialized hardware to assist in scan converting output primitives into the frame buffer. l Fundamental difference among display systems is how much the display processor does versus how much must be done by the graphics subroutine package executing on the general-purpose CPU. l Now often called graphics accelerator

2/28/2016 Mathematical Foundations l FvD appendix gives good review l I’ll give a brief, informal review of some of the mathematical tools we’ll employ –Geometry (2D, 3D) –Trigonometry –Vector spaces n Points, vectors, and coordinates –Dot and cross products –Linear transforms and matrices

2/28/2016 2D Geometry l Know your high school geometry: –Total angle around a circle is 360° or 2π radians –When two lines cross: n Opposite angles are equivalent n Angles along line sum to 180° –Similar triangles: n All corresponding angles are equivalent

2/28/2016 Trigonometry l Sine: “opposite over hypotenuse” l Cosine: “adjacent over hypotenuse” l Tangent: “opposite over adjacent” l Unit circle definitions: –sin (  ) = x –cos (  ) = y –tan (  ) = x/y –Etc… (x, y)

2/28/2016 Slope-intercept Line Equation Slope = m = rise / run Slope = (y - y1) / (x - x1) = (y2 - y1) / (x2 - x1) Solve for y: y = [(y2 - y1)/(x2 - x1)]x + [-(y2-y1)/(x2 - x1)]x1 + y1 or: y = mx + b x y P2 = (x2, y2) P1 = (x1, y1) P = (x, y)

2/28/2016 Parametric Line Equation l Given points P1 = (x1, y1) and P2 = (x2, y2) x = x1 + t(x2 - x1) y = y1 + t(y2 - y1) l When: –t=0, we get (x1, y1) –t=1, we get (x2, y2) –(0<t<1), we get points on the segment between (x1, y1) and (x2, y2) x y P2 = (x2, y2) P1 = (x1, y1)

2/28/2016 Other helpful formulas l Length = sqrt (x2 - x1) 2 + (y2 - y1) 2 l Midpoint, p2, between p1 and p3 –p2 = ((x1 + x3) / 2, (y1 + y3) / 2)) l Two lines are perpendicular if: –M1 = -1/M2 –cosine of the angle between them is 0

2/28/2016 3D Geometry l To model, animate, and render 3D scenes, we must specify: –Location –Displacement from arbitrary locations –Orientation

2/28/2016 Vector Spaces l Two types of elements: –Scalars (real numbers):  … –Vectors (n-tuples): u, v, w, … l Supports two operations: –Addition operation u + v, with: Identity 0 v + 0 = v Inverse - v + (- v ) = 0 –Scalar multiplication: Distributive rule:  ( u + v ) =  ( u ) +  ( v ) (  +  ) u =  u +  u

2/28/2016 Vector Spaces l A linear combination of vectors results in a new vector: v =  1 v 1 +  2 v 2 + … +  n v n l If the only set of scalars such that  1 v 1 +  2 v 2 + … +  n v n = 0 is  1 =  2 = … =  3 = 0 then we say the vectors are linearly independent l The dimension of a space is the greatest number of linearly independent vectors possible in a vector set l For a vector space of dimension n, any set of n linearly independent vectors form a basis

2/28/2016 Vector Spaces: A Familiar Example l Our common notion of vectors in a 2D plane is (you guessed it) a vector space: –Vectors are “arrows” rooted at the origin –Scalar multiplication “streches” the arrow, changing its length (magnitude) but not its direction –Addition uses the “trapezoid rule”: u+v y x u v

2/28/2016 Vector Spaces: Basis Vectors l Given a basis for a vector space: –Each vector in the space is a unique linear combination of the basis vectors –The coordinates of a vector are the scalars from this linear combination –Best-known example: Cartesian coordinates –Note that a given vector v will have different coordinates for different bases

2/28/2016 Vectors And Points l We commonly use vectors to represent: –Points in space (i.e., location) –Displacements from point to point –Direction (i.e., orientation)

2/28/2016 Points l Points support these operations –Point-point subtraction: Q - P = v n Result is a vector pointing from P to Q –Vector-point addition: P + v = Q n Result is a new point –Note that the addition of two points is not defined P Q v

2/28/2016 Dot Product l The dot product or, more generally, inner product of two vectors is a scalar: v 1 v 2 = x 1 x 2 + y 1 y 2 + z 1 z 2 (in 3D) l Useful for many purposes –Computing the length of a vector: length( v ) = sqrt( v v) –Normalizing a vector, making it unit-length –Computing the angle between two vectors: u v = |u| |v| cos(θ) –Checking two vectors for orthogonality –Projecting one vector onto another θ u v

2/28/2016 Cross Product l The cross product or vector product of two vectors is a vector: l The cross product of two vectors is orthogonal to both l Right-hand rule dictates direction of cross product

2/28/2016 Linear Transformations l A linear transformation: –Maps one vector to another –Preserves linear combinations l Thus behavior of linear transformation is completely determined by what it does to a basis l Turns out any linear transform can be represented by a matrix

2/28/2016 Matrices By convention, matrix element M rc is located at row r and column c: l By (OpenGL) convention, vectors are columns:

2/28/2016 Matrices l Matrix-vector multiplication applies a linear transformation to a vector: l Recall how to do matrix multiplication

2/28/2016 Matrix Transformations l A sequence or composition of linear transformations corresponds to the product of the corresponding matrices –Note: the matrices to the right affect vector first –Note: order of matrices matters! The identity matrix I has no effect in multiplication l Some (not all) matrices have an inverse:

2/28/2016 The Rendering Pipeline: A Whirlwind Tour Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

2/28/2016 The Display You Know Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

2/28/2016 The Framebuffer You Know Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

2/28/2016 The Rendering Pipeline Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

2/28/ D Rendering: Rasterization (Coming Soon) Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

2/28/2016 The Rendering Pipeline: 3-D Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

2/28/2016 The Rendering Pipeline: 3-D Modeling Transforms Scene graph Object geometry Lighting Calculations Viewing Transform Clipping Projection Transform Result: All vertices of scene in shared 3-D “world” coordinate system All vertices of scene in shared 3-D “world” coordinate system Vertices shaded according to lighting model Vertices shaded according to lighting model Scene vertices in 3-D “view” or “camera” coordinate system Scene vertices in 3-D “view” or “camera” coordinate system Exactly those vertices & portions of polygons in view frustum Exactly those vertices & portions of polygons in view frustum 2-D screen coordinates of clipped vertices 2-D screen coordinates of clipped vertices

2/28/2016 The Rendering Pipeline: 3-D Scene graph Object geometry Lighting Calculations Clipping Result: All vertices of scene in shared 3-D “world” coordinate system All vertices of scene in shared 3-D “world” coordinate system Vertices shaded according to lighting model Vertices shaded according to lighting model Scene vertices in 3-D “view” or “camera” coordinate system Scene vertices in 3-D “view” or “camera” coordinate system Exactly those vertices & portions of polygons in view frustum Exactly those vertices & portions of polygons in view frustum 2-D screen coordinates of clipped vertices 2-D screen coordinates of clipped vertices Modeling Transforms Viewing Transform Projection Transform

2/28/2016 Rendering: Transformations l So far, discussion has been in screen space l But model is stored in model space (a.k.a. object space or world space) l Three sets of geometric transformations: –Modeling transforms –Viewing transforms –Projection transforms

2/28/2016 Rendering: Transformations l Modeling transforms –Size, place, scale, and rotate objects and parts of the model w.r.t. each other –Object coordinates  world coordinates Z X Y X Z Y

2/28/2016 Rendering: Transformations l Viewing transform –Rotate & translate the world to lie directly in front of the camera n Typically place camera at origin n Typically looking down -Z axis –World coordinates  view coordinates

2/28/2016 Rendering: Transformations l Projection transform –Apply perspective foreshortening n Distant = small: the pinhole camera model –View coordinates  screen coordinates

2/28/2016 Rendering: Transformations l All these transformations involve shifting coordinate systems (i.e., basis sets) l That’s what matrices do… l Represent coordinates as vectors, transforms as matrices l Multiply matrices = concatenate transforms!                     Y X Y X   cossin cos

2/28/2016 The Rendering Pipeline: 3-D Modeling Transforms Scene graph Object geometry Lighting Calculations Viewing Transform Clipping Projection Transform Result: All vertices of scene in shared 3-D “world” coordinate system All vertices of scene in shared 3-D “world” coordinate system Vertices shaded according to lighting model Vertices shaded according to lighting model Scene vertices in 3-D “view” or “camera” coordinate system Scene vertices in 3-D “view” or “camera” coordinate system Exactly those vertices & portions of polygons in view frustum Exactly those vertices & portions of polygons in view frustum 2-D screen coordinates of clipped vertices 2-D screen coordinates of clipped vertices

2/28/2016 Rendering: Lighting l Illuminating a scene: coloring pixels according to some approximation of lighting –Global illumination: solves for lighting of the whole scene at once –Local illumination: local approximation, typically lighting each polygon separately l Interactive graphics (e.g., hardware) does only local illumination at run time

2/28/2016 The Rendering Pipeline: 3-D Modeling Transforms Scene graph Object geometry Lighting Calculations Viewing Transform Clipping Projection Transform Result: All vertices of scene in shared 3-D “world” coordinate system All vertices of scene in shared 3-D “world” coordinate system Vertices shaded according to lighting model Vertices shaded according to lighting model Scene vertices in 3-D “view” or “camera” coordinate system Scene vertices in 3-D “view” or “camera” coordinate system Exactly those vertices & portions of polygons in view frustum Exactly those vertices & portions of polygons in view frustum 2-D screen coordinates of clipped vertices 2-D screen coordinates of clipped vertices

2/28/2016 Rendering: Clipping l Clipping a 3-D primitive returns its intersection with the view frustum: l See Foley & van Dam section 19.1

2/28/2016 Rendering: Clipping l Clipping is tricky! In: 3 vertices Out: 6 vertices Clip In: 1 polygon Out: 2 polygons

2/28/2016 The End l Next: Basic OpenGL