1. Linear Interpolation, Brief Introduction to OpenGL Glenn G. Chappell CHAPPELLG@member.ams.org U. of Alaska Fairbanks CS 381 Lecture Notes Monday, September 8, 2003

2. 8 Sep 2003CS 3812 Review: Course Overview — Goals Students will: Gain an overall understanding of 3-D graphics, based on the synthetic-camera model, implemented with a rendering pipeline. Learn to use a professional-quality graphics API (OpenGL) to do 3-D graphics. Learn simple event-driven programming. Learn to deal with issues and tools involved in 3-D graphics: transformations, viewing, hidden-surface removal, lighting. Understand and be able to use facilities for rendering complex scenes using simple primitives (hierarchical objects, texturing, etc.). Understand basic issues/algorithms involved in rasterization, clipping. Demonstrate proficiency by writing 10–11 short graphics programs in C/C++.

3. 8 Sep 2003CS 3813 Review: Introduction to CG [1/4] In 2-D CG, we often think in terms of screen positions. Put this object in that spot in the image. So the scene “lives” on the screen. In 3-D CG, the user is (usually) inside the scene. The screen (or other image) is a (movable?) window through which the user looks at the scene. Therefore, we base our graphics on the synthetic camera model.

4. 8 Sep 2003CS 3814 Review: Introduction to CG [2/4] We base our 3-D viewing on a model similar to a camera. A point is chosen (the center of projection). Given an object in the scene, draw a line from it, through the center of projection, to the image. The image lies in a plane, like film in a film camera, or the sensor array in a digital camera. Where this line hits the image is where the object appears in the image. This model is similar to the way the human visual system works. How is it different? What pitfalls might this model have?

5. 8 Sep 2003CS 3815 Review: Introduction to CG [3/4] Images of 3-D scenes are generated in two steps: Modeling Rendering Modeling means producing a precise description of a scene, generally in terms of graphics primitives. Primitives may be points, lines, polygons, bitmapped images, various types of curves, etc. Rendering means producing an image based on the model. Images are produced in a frame buffer. A modern frame buffer is a raster: a 2-D array of pixels. This class focuses on rendering.

6. 8 Sep 2003CS 3816 Review: Introduction to CG [4/4] In modern graphics architectures, rendering is accomplished via a pipeline. Why are pipeline-style designs good (in general)? Vertices enter. A vertex might be a corner of a polygon. Fragments leave. A fragment is a pixel-before-it-becomes-a-pixel. At the end of the pipeline, values are stored in the frame buffer. The above picture differs from that in the book. Both are over- simplifications; but they are over-simplified in different ways. Later in the class, we will be adding more detail to this picture. Vertices (window coordinates) Vertex Operations Rasterization Fragment Operations Vertices (object coordinates) Fragments Vertex enters here To framebuffer

7. 8 Sep 2003CS 3817 Linear Interpolation: The Problem In CG we often need to determine the value of some quantity in between two places where its value is known. For example, We are rendering a line. One endpoint is white; the other is blue. What color should the rest of the line be? We are animating a spinning wheel. We know its angular position at two key frames; what is the angle between the two frames? We are rendering a partially transparent polygon. We know the transparency at two of its vertices; what is the transparency between these vertices? We are drawing a smooth curve. We know its slope at two different points; what is the slope between the two points? We are texturing a polygon (painting an image on it). What color is the polygon between two pixels in the image (“texels”)? We are doing lighting calculations for a surface. We know the effect of lighting at two points; what is the effect between the two points?

8. 8 Sep 2003CS 3818 Linear Interpolation: Lirping Approximating a quantity between places where the quantity is known, is called interpolation. Approximating the quantity in other places is called extrapolation. The simplest interpolation method is linear interpolation. Linear interpolation assumes that the graph of the quantity between the two known values is a straight line. Of course, this assumption is often false, but remember that we are only trying to approximate. We abbreviate “linearly interpolate” as lirp [some people spell it “lerp”].

9. 8 Sep 2003CS 3819 Linear Interpolation: Simple Case When we lirp, we are given two values of a quantity; we determine its value somewhere between these. To do this, we assume that the graph of the quantity is a line segment (between the two given values). Here is a simple case: When t = 0, the quantity is equal to a. When t = 1, the quantity is equal to b. Given a value of t between 0 and 1, the interpolated value of the quantity is Check that this satisfies all the requirements. Note: We only do this for t between 0 and 1, inclusive.

10. 8 Sep 2003CS 38110 Linear Interpolation: Example 1 An object lies at position y = 2 at time t = 0 and position y = 3.1 at time t = 1. Using linear interpolation, approximate the position of the object at time t = 0.6. Answer is on the last slide.

11. 8 Sep 2003CS 38111 Linear Interpolation: General Case What if we are not given values of the quantity at 0 and 1? Say the value at s 1 is a and the value at s 2 is b, and we want to find the interpolated value at s. We set and proceed as before:

12. 8 Sep 2003CS 38112 Linear Interpolation: Example 2 The brightness of a moving light source is 1.0 when it lies at position x = 1.1 and 0.5 when it lies at position x = 1.5. Using linear interpolation, approximate the brightness when it lies at position x = 1.4. Answer is on the last slide.

13. 8 Sep 2003CS 38113 Linear Interpolation: Lirping Colors & Points Much of the interpolation done in CG involves several numbers at once. An important example is colors, which are typically specified as three numbers: R, G, and B. To lirp between two colors, we actually lirp three times: between the two R values, between the two G values, and between the two B values. The three lirping calculations are carried out independently of each other. This method actually ignores some important characteristics of human vision; however, when the two colors are very similar, it is a fine method to use. Lirping between 2-D and 3-D points is done similarly.

14. 8 Sep 2003CS 38114 Linear Interpolation: Example 3 The position of an object is (0, 1) at time t = 0 and (2, 10) at time t = 1. Using linear interpolation, approximate the position of the object at time t = 0.1. Answer is on the last slide.

15. 8 Sep 2003CS 38115 Introduction to OpenGL: API API = Application Programmer’s Interface. An API specifies how a program uses a library. What calls/variables/types are available. What these do. A well-specified API allows implementation details of a library to be changed without affecting programs that use the library. The graphics API we will be using is called OpenGL.

16. 8 Sep 2003CS 38116 Introduction to OpenGL: What is OpenGL? Professional-quality 2-D & 3-D graphics API Developed by Silicon Graphics Inc. in 1992. Based on Iris GL, the SGI graphics library. Available in a number of languages. We will use the C-language API. There is no C++-specific OpenGL API. System-Independent API Same API under Windows, MacOS, various Unix flavors. Programmer does not need to know hardware details. Can get good performance from varying hardware. An Open Standard A consortium of companies sets the standard. Anyone can implement the OpenGL API. Review board handles certification of implementations.

17. 8 Sep 2003CS 38117 Introduction to OpenGL: What does OpenGL Do? OpenGL provides a system-independent API for 2-D and 3-D graphics. It is primarily aimed at: Graphics involving free-form 3-D objects made up (at the lowest level) of polygons, lines, and points. Simple hidden-surface removal and transparency- handling algorithms are built-in. Scenes may be lit with various types of lights. Polygons may have images painted on them (texturing). OpenGL does not excel at: Text generation (although it does include support for text). Page description or high-precision specification of images.

18. 8 Sep 2003CS 38118 Introduction to OpenGL: Parts of OpenGL OpenGL Itself The interface with the graphics hardware. Designed for efficient implementation in hardware. Particular OpenGL implementations may be partially or totally software. C/C++ header:. The OpenGL Utilities (GLU) Additional functions & types for various graphics operations. Designed to be implemented in software; calls GL. C/C++ header:. OpenGL Extensions Functionality that anyone can add to OpenGL. OpenGL specifies rules that extensions are to follow. May be system-dependent. We will not use any extensions.

19. 8 Sep 2003CS 38119 Answers Example 1 a = 2, b = 3.1, and the relevant value of t is 0.6. Approximation: (1 – 0.6)  2 + 0.6  3.1 = 2.66. Example 2 We start at 1.1 and end at 1.5. We want to approximate at 1.4. So t = (1.4 – 1.1)/(1.5 – 1.1) = 0.75. Now, a = 1.0, b = 0.5, and t = 0.75. Approximation: (1 – 0.75)  1.0 + 0.75  0.5 = 0.625. Example 3 We lirp the two coordinates separately. 1 st coord.: a = 0, b = 2, t = 0.1. (1 – 0.1)  0 + 0.1  2 = 0.2. 2 nd coord.: a = 1, b = 10, t = 0.1. (1 – 0.1)  1 + 0.1  10 = 1.9. We finish by putting the two coordinates together: (0.2, 1.9).