Lecture Objectives • Understand the basic concepts of Computer Graphics • Learn about Computer Graphics Applications • Learn about the Coordinate System of Computer Graphics • Learn about the basic and advanced transformations in Computer Graphics
Lecture Objectives • Outline: Graphics Applications What is Computer Graphics? Representations in Graphics Supporting Disciplines Coordinate Systems Basic and Advanced Transformations
Graphics Applications • Entertainment: Cinema A Bug’s Life (Pixar) Pixar: Monster’s Inc.
Graphics Applications (Cont’d) • Medical Visualization The Visible Human Project MIT: Image-Guided Surgery Project
Graphics Applications (Cont’d) • Computer Aided Design (CAD) • Scientific Visualization
Graphics Applications (Cont’d) • Training Designing Effective Step-By-Step Assembly Instructions (Maneesh Agrawala et. al)
Graphics Applications (Cont’d) • Game Modeling and Simulation GT Racer 3 Polyphony Digital: Gran Turismo 3, A Spec
Graphics Applications (Cont’d) • The major application that we will be dealing with extensively in the next coming lectures is that of developing graphical user interfaces • Windows • Menus • Buttons • Textboxes • ...
What is Computer Graphics? • Computer graphics: generating 2D images of a 3D world represented in a computer. • Main tasks: • modeling: (shape) creating and representing the geometry of objects in the 3D world • rendering: (light, perspective) generating 2D images of the objects • animation: (movement) describing how objects change in time
Representations in Graphics A) Vector Graphics: • Image is represented by continuous geometric objects: lines, curves, etc. B) Raster Graphics: • Image is represented as a rectangular grid of colored pixels • PIXEL = PIcture ELement X
Raster Graphics • Generic • Image processing techniques • Geometric Transformation: loss of information • Relatively high processing time • in terms of the number of pixels • Realistic images, textures, ... • Examples: Paint, PhotoShop, ...
Raster Graphics (Cont’d) Sample Image Processing Techniques • Edge Detection • Image Denoising
Vector Graphics Vector graphics • Graphics objects: geometry + color • Relatively low processing time • in terms of the number of graphic objects • Geometric transformation possible without loss of information (zoom, rotate, …) • Examples: PowerPoint, CorelDraw, SVG, ...
Scalable Vector Graphics (SVG) <?xml version="1.0" standalone="no"?> <!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"> <svg width="12cm" height="4cm" viewBox="0 0 1200 400" xmlns="http://www.w3.org/2000/svg" version="1.1"> <desc>Example polygon01 - star and hexagon</desc> <!-- Show outline of canvas using 'rect' element --> <rect x="1" y="1" width="1198" height="398" fill="none" stroke="blue" stroke-width="2" /> <polygon fill="red" stroke="blue" stroke-width="10" points="350,75 379,161 469,161 397,215 423,301 350,250 277,301 303,215 231,161 321,161" /> <polygon fill="lime" stroke="blue" stroke-width="10" points="850,75 958,137.5 958,262.5 850,325 742,262.6 742,137.5" /> </svg>
From Modeling to Processing Image Analysis (pattern recognition) Math. Model Image Image Synthesis(Rendering) Modeling Image processing
Supporting Disciplines • Computer science (algorithms, data structures, software engineering, …) • Mathematics (geometry, numerical, …) • Physics (Optics, mechanics, …) • Psychology (Colour, perception) • Art and design
Computer Graphics: Transformations • Types of transformations • Screen and World Coordinate Systems • Matrix representations of transformations • 2D Transformations • 3D Transformations
Graphical Transformations: Basics • There are several standard graphics transformation. • All involve plotting images on the screen in terms of points and the lines connecting those points. • Each programming language has its own particular constructs for drawing items on the screen. • "screen" here includes printer output and invisible background buffers for efficient cached graphics (see Double buffering later) • Regardless of language particulars, the various graphics transformations themselves remain logically the same. • Here, will deal with the logic of the transformations. • Consult language books/manuals re: how to do the mechanics of the actual drawing.
Graphical Transformations: Basics (Cont’d) • Translation: moving an item from one location to another, e.g., moving thru a room or landscape. • Rotation: changing the orientation of an item at a given location, e.g., spinning around. • Scaling: changing the size of an item as it appears on the screen, e.g., an item gets larger or smaller. • Clipping: knowing where to stop drawing an item because it partially extends beyond the screen.
Graphical Transformations: Advanced • More advanced operations: • Hidden surface algorithms: dealing with (removing) aspects of items that are hidden from view. • Representing 3D shapes: how to represent 3D items in a 2D medium. • Displaying depth relationships: how to achieve realistic perspective. • Shading, reflection, ambient lighting: how to achieve realistic lighting effects. • In this lecture, will deal only with basics: translation, scaling, and rotation.
Viewport into that world World and Viewport Basic ideas: World vs. Viewport The representation of the world • Representation of world stays the same. • View of world changes as you move around in it, i.e., the viewport moves.
Coordinates (0,0) (1024, 768) • Both viewport and the "world" have coordinate systems. • The entire computer screen is a set of pixels (short for picture elements). So is a window on the desktop of a screen. So is a canvas inside a window. • Pixels form a two-dimensional grid with the coordinates (0,0) being the upper-left and the max number of pixels in each dimension forming the lower-right (say, your system provides 1024,768). • So the screen coordinates are as follows:
World Coordinates • The lines to be drawn are given in world coordinates with the origin fixed at the center of the computer screen, e.g., (1024, 768 resolution means a center point of 512, 384) • All the transformations are: • applied to the world coordinates • then mapped to the real (screen) coordinates. • This allows of computation of the logical transformations to be separated from hard details of viewing surface. • Do not want to tie the model to available resolution. Mapping from world coordinates to screen coordinates allows us to keep two levels of abstraction separate: modelvs. device
World Coordinates (Cont’d) To visualize it: (512, 384, 0) y-axis x-axis z-axis (512, -384, 0) (0, 0, 0) in world coordinates(512,384,0) in real coordinates • Note that the Z-axis comes "at you" out of the computer screen, perpendicular to both the X-axis and the Y-axis.
World Coordinates (Cont’d) • For example: • world point (0,0,0) is really (512, 384) when you plot (display) it. • world point (100, -20,0) is really (612, 404) when you plot it. • Notice: real X coordinate gets larger with positive X world coordinates. • Notice: real Y coordinates gets larger with negative Y world coordinates.
Translation Basic transformations: (for simplicity, 2D) Translation: x' = x + Dx y' = y + Dy where Dx is relative distance in x dimension, Dy is relative distance in y dimension, prime indicates new point in space. Computation: [x' y'] = [x y] + [Dx Dy] P' = P + T
Translation (Cont’d) Each point gets translated [x' y'] = [x y] + [Dx Dy]
Scaling Scaling: x' = x * Sx y' = y * Sy where Sx is scale factor for x dimension, Sy is scale factor for y dimension, prime indicates new point in space. Computation: defining S as [ Sx 0 ] [ 0 Sy ] [x' y'] = [x y] * [ Sx 0 ] [ 0 Sy ] P' = P * S
Scaling (Cont’d) [x' y'] = [x y] * [ Sx 0 ] [ 0 Sy ] Question: What about stretching unequally in two dimensions?
Rotation Rotation: x' = xcos - ysin y' = xsin + ycos where is angle of rotation and prime indicates new point in space. Computation: [x' y'] = [x y] * [ cos sin ] [-sin cos ] P' = P * R Note:positive angles are counter-clockwise from x toward y; for negative angles (clockwise) use identities: cos(- ) = cos , and sin(- ) = -sin
Rotation (Cont’d) [x' y'] = [x y] * [ cos sin ] [-sin cos ]
Rotation Around a Fixed Point • Notes on Rotation: • There is a difference between: • "rotation around center point of object“ • and "rotation around origin of Cartesian world" Example: • Imagine a ball on a tether mounted to pole • Do you want the ball itself to spin around on the end of the tether? • Or do you want the ball-and-tether to rotate around the pole?To rotate an object about its own center point: • first translate object to origin, • then do rotation • then translate back