1. CAP4730: Computational Structures in Computer Graphics Chapter 3 Hearn & Baker Portions obtained from Leonard McMillan’s COMP136 Notes: www.cs.unc.edu/~mcmillan/comp136/Lecture 6 2D Basics, Line Drawing, and Clipping

2. Definitions CG API – Computer Graphics Application Programming Interface (OpenGL, DirectX) Graphics primitives – functions in the API that describe picture components How could we describe an object? –Typically focus on object shape –Define an object’s shape with geometric primitives –Span of primitives are defined by the API –What are some types? Lines, Triangles, Quadrics, Conic sections, Curved surfaces

3. Two Dimensional Images Use Cartesian coordinates We label the two axes as –X (horizontal) –Y (vertical) Origin is in the lower left How big is the space? –So what is the image we see on a screen? –We call this space the world coordinate system X Axis Y Axis (0,0) +X +Y

4. Partition the space into pixels 1. Define a set of points (vertices) in 2D space. 2. Given a set of vertices, draw lines between consecutive vertices. 3. If you were writing OpenGL yourself, let’s talk about low level calls 4. What about 2D vs 3D? Screen Coordinates – references to frame buffer locations Q: True or Flase: Screen Coordinates == World Coordinates +X +Y (2,1) (2,7) (9,1) (9,7)

5. Pixels ?=glSetPixel(?) ?=glGetPixel(?) Scan line number – y Column number – x

6. Absolute and Relative Coordinate Specifications Absolute coordinates – location specified as a relationship to the origin Relative coordinates – location specified as a relationship to other points –Good for pen/plotters –Publishing/layout –Allows for a very object oriented approach For this class we will always use absolute coordinates +Y (2,1) (0,6) (0,-6) (7,0)

7. Specifying a World Coordinate System in OpenGL +X +Y gluOrtho2D (xmin, xmax, ymin, ymax) What should our xmin, xmax, ymin, ymax values be? Equivalent to the size of the framebuffer

8. What is a “pixel” From a geometry point of view, a pixel is a point. Q: Where is (2,1)? 2 2 1 1 0 34 3 5 Q: What is a pixel? A square or a point?

9. But when we think about images, a pixel is a rectangle. Q: Where is (2,1)? A. The center of a pixel 1 2 0 1034 2

10. Basic OpenGL Point Structure In OpenGL, to specify a point: –glVertex*(); In OpenGL, some functions require both a dimensionality and a data type –glVertex2i(80,100), glVertex2f(58.9, 90.3) –glVertex3i(20,20,-5), glVertex3f(-2.2,20.9,20) Must put within a ‘glBegin/glEnd’ pair –glBegin(GL_POINTS); –glVertex2i(50,50); –glVertex2i(60,60); –glVertex2i(60,50); –glEnd(); Let’s draw points in our assignment #1 Next up? Lines

11. Draw a line from 0,0 to 4,2 2 2 1 10 0 34 How do we choose between 1,0 and 1,1? What would be a good heuristic? (0,0) (4,2)

12. What are lines composed of? Write glBegin(GL_LINES) 2 2 1 10 0 34 (0,0) (4,2)

13. What we are working with V0: (6,2) V1: (6,8) V3: (13,2) V2: (13,8) We are still dealing with vertices Draws a line between every pair of vertices glBegin(GL_LINES); glVertex2i(6,2); glVertex2i(6,8); glEnd();

14. Let’s draw a triangle (2,0) (4,2) (0,2) 2 2 1 10 0 34

15. Consider a translation (1.8,0) (3.8,2) (-0.2,2) 2 2 1 10 0 34

16. The Ideal Line Continuous appearance Uniform thickness and brightness Pixels near the ideal line are “on” Speed (2,2) (17,8) Discretization - converting a continuous signal into discrete elements. Scan Conversion - converting vertex/edges information into pixel data for display What do we want?

17. Slope-Intercept Method From algebra: y = mx + b –m = slope b = y intercept Let’s write some code class Point { public: int x, y; int r,g,b; }; unsigned byte framebuffer[IMAGE_WIDTH*IMAGE_HEIGHT*3]; DrawLine (Point point1, Point point2) { }

18. Slope-Intercept Method From algebra: y = mx + b –m = slope b = y intercept Let’s write some code DrawLine (Point point1, Point point2){ m=(point2.y-point1.y) / (point2.x-point2.x); b=point1.y + (-point1.x) * m; for i=point1.x to point2.x SetPixel(i, round(m*i+b)), pixel1.r, pixel1.g, pixel1.b;} SetPixel(int x, int y, int r, int g, int b){ framebuffer[(y * IMAGE_WIDTH+x) * 3 + 0]=r; frame buffer[(y * IMAGE_WIDTH+x) * 3 + 1]=g; frame buffer[(y * IMAGE_WIDTH+x) * 3 + 2]=b;}

19. Example 1: Point1 V:(2,2) C:(255,102,0) Point2 V:(17,8) C:(255,102,0) What if colors were different? (17,8) (2,2) (0,0) (18,0) (0,9)

20. How do we change the framebuffer? (17,8) (2,2) (0,0) (18,0) (0,9) What’s the index into GLubyte framebuffer[]? Point is 9,5

21. Example: (7,9) (12,0) Example 2: Point1 V:(7,9 ) C:( 0,255,0) Point2 V:(12,0) C:(0,255,0) (0,0) (18,0) (0,9) What are the problems with this method? Slope>1

22. Revised Slope Intercept DrawLine (Point point1, Point point2){ m=(point2.y-point1.y) / (point2.x-point2.x); b=point1.y + (-point1.x) * m; if (m>1) { for i=point1.x to point2.x SetPixel(i, round(i*m+b));} } else { for i=point1.y to point2.y SetPixel(i, round(i-b/m));} } Which one should we use if m=1? What is the cost per pixel?

23. Optimization (DDA Algorithm) Since we increment y by the same amount, we can also inline rounding: New cost: one floating point addition, one integer addition, one cast. DrawLine (Point point1, Point point2){ m=(point2.y-point1.y) / (point2.x-point2.x); j=point1.y + (-point1.x) * m + 0.5; for i=point1.x to point2.x SetPixel(i, (int)j+=m));}

24. Bresenham’s Line Drawing In general: –Addition and Subtraction are faster than Multiplication which is faster than Division –Integer calculations are faster than Floating point Made all math just integers (Section 3.5) How?

25. What you need to know about Bresenham LDA 1)Why we use it 2)Major idea of integer-izing a decision point 3)How this reduces things to just integer form. (17,8) (2,2) (0,0) (18,0) (0,9)

26. Recap DDA/Line Intercept Algorithm –Slope Intercept y = mx + b –Easy to implement –Slow Bresenham –No floating point math –Fast Why do we spend so much time optimizing this?

27. Other Primitive Drawing Solutions What other shapes might we want to draw quickly? –Circles (and thus) Ovals –Curves Fill?