Download presentation

Presentation is loading. Please wait.

Published byBrooks Sandal Modified over 2 years ago

1
CP411 polygon Lecture 6 1. Scan conversion algorithm for circle 2. Draw polygons 3. Antialiasing

2
CP411 Computer Graphics 2 Draw a circle_ straight algorithm #include const float DEG2RAD = 3.14159/180; void drawCircle(float radius, float x, float y) { glBegin(GL_LINE_LOOP); for (int i=0; i < 360; i++) { float degInRad = i*DEG2RAD; glVertex2f(x+cos(degInRad)*radius, y+sin(degInRad)*radius); } glEnd }

3
Midpoint point algorithm ● See class note CP411 polygon

4
Polygon ● What is a polygon? ■ A geometric shape bounded by a closed polyline formed by a sequence of vertices. ● Why choose polygon as a primitive ? ■ In interactive graphics, polygons rule the CG world ■ Two main reasons: ○ Can represent any surface with arbitrary accuracy Splines, mathematical functions, etc. ○ Mathematical simplicity lends itself to simple, regular rendering algorithms Such algorithms embed well in hardware

5
CP411 polygon Convex / Concave Polygons ● A two-dimensional shape is convex if and only if every line segment connecting two points on the boundary is entirely contained ● OpenGL can only draw convex polygon ■ How to recognize a convex polygon ?

6
CP411 polygon OpenGL polygon command ● glBegin (GL_POLYGON) specify vertices of the polygon glend ● glPolygonMode( GLenum face, GLenum mode ) controls the interpretation of polygons for rasterization ■ face Specifies the polygons that mode applies to. Must be GL_FRONT for front-facing polygons, GL_BACK for back-facing polygons, or GL_FRONT_AND_BACK for front- and back-facing polygons. ■ mode Specifies the way polygons will be rasterized. Accepted values are GL_POINT, GL_LINE, and GL_FILL. The default is GL_FILL for both front- and back-facing polygons.

7
CP411 polygon Rasterizing Polygons ● Triangle is the minimal unit of a polygon ■ All polygons can be broken up into triangles ○ Convex, concave, complex ■ Triangles are guaranteed to be: ○ Planar ○ Convex ● We often draw polygons by breaking them into triangles

8
CP411 polygon Triangles ● Two basic methods for drawing triangles: ■ Edge walking ○ Use a line drawing algorithm to compute edges (edge setup) ○ Fill in pixels between edges ○ Finicky, lots of special cases ○ Touches only pixels involved in triangle ■ Edge equations

9
CP411 polygon Edge Walking ● Basic idea: ■ Draw edges vertically ■ Fill in horizontal spans for each scanline ■ Interpolate colors down edges ■ At each scanline, interpolate edge colors across span

10
CP411 polygon Edge Walking: Notes ● Order vertices in x and y ■ 3 cases: break left, break right, no break ● Walk down left and right edges ■ Fill each span ■ Until breakpoint or bottom vertex is reached ● Advantage: can be made very fast ● Disadvantages: ■ Lots of finicky special cases ■ Need to pay attention to fractional offsets

11
CP411 polygon Edge Walking: Notes ● Fractional offsets: ● Also: beware gaps between adjacent edges

12
CP411 polygon Edge Equations ● An edge equation is simply the equation of the line containing that edge ■ What is the equation of a 2D line? ■ Ax + By + C = 0 ■ Given a point (x,y), what does plugging x & y into this equation tell us? ■ Whether the point is: ○ On the line: Ax + By + C = 0 ○ “Above” the line: Ax + By + C > 0 ○ “Below” the line: Ax + By + C < 0

13
Edge Equations ● Edge equations thus define two half-spaces: CP411 polygon

14
Edge Equations ● And a triangle can be defined as the intersection of three positive half-spaces: A 1 x + B 1 y + C 1 < 0 A 2 x + B 2 y + C 2 < 0 A 3 x + B 3 y + C 3 < 0 A 1 x + B 1 y + C 1 > 0 A 3 x + B 3 y + C 3 > 0 A 2 x + B 2 y + C 2 > 0 CP411 polygon

15
Edge Equations ● So…simply turn on those pixels for which all edge equations evaluate to > 0: + + + - - - CP411 polygon

16
Using Edge Equations ● Which pixels: compute min,max bounding box ● Edge equations: compute from vertices ● Orientation: ensure area is positive (why?) CP411 polygon

17
Computing Edge Equations ● Want to calculate A, B, C for each edge from (x i, y i ) and (x j, y j ) ● Treat it as a linear system: Ax 1 + By 1 + C = 0 Ax 2 + By 2 + C = 0 ● Notice: two equations, three unknowns ● Does this make sense? What can we solve? ● Goal: solve for A & B in terms of C CP411 polygon

18
Computing Edge Equations ● Set up the linear system: ● Multiply both sides by matrix inverse: ● Let C = x 0 y 1 - x 1 y 0 for convenience ■ Then A = y 0 - y 1 and B = x 1 - x 0 CP411 polygon

19
Computing Edge Equations: Numerical Issues ● We can avoid the problem in this case by using our definitions of A and B: A = y 0 - y 1 B = x 1 - x 0 C = x 0 y 1 - x 1 y 0 Thus: C = -Ax 0 - By 0 or C = -Ax 1 - By 1 CP411 polygon

20
Edge Equations ● So…we can find edge equation from two verts. ● Given three corners C 0, C 1, C 0 of a triangle, what are our three edges? ● How do we make sure the half-spaces defined by the edge equations all share the same sign on the interior of the triangle? ● Be consistent (Ex: [C 0 C 1 ], [C 1 C 2 ], [C 2 C 0 ]) ● How do we make sure that sign is positive? ● A: Test, and flip if needed (A= -A, B= -B, C= -C) CP411 polygon

21
Edge Equations: Code ● Basic structure of code: ■ Setup: compute edge equations, bounding box ■ (Outer loop) For each scanline in bounding box... ■ (Inner loop) …check each pixel on scanline, evaluating edge equations and drawing the pixel if all three are positive CP411 polygon

22
findBoundingBox(&xmin, &xmax, &ymin, &ymax); setupEdges (&a0,&b0,&c0,&a1,&b1,&c1,&a2,&b2,&c2); /* Optimize this: */ for (int y = yMin; y <= yMax; y++) { for (int x = xMin; x <= xMax; x++) { float e0 = a0*x + b0*y + c0; float e1 = a1*x + b1*y + c1; float e2 = a2*x + b2*y + c2; if (e0 > 0 && e1 > 0 && e2 > 0) setPixel(x,y); } CP411 polygon

23
General Polygon Rasterization ● Consider the following polygon: ● How do we know whether a given pixel on the scanline is inside or outside the polygon? A B C D E F CP411 polygon

24
General Polygon Rasterization ● Does it still work? A B C D E F G I H CP411 polygon

25
General Polygon Rasterization ● Basic idea: use a parity test for each scanline edgeCnt = 0; for each pixel on scanline (l to r) if (oldpixel->newpixel crosses edge) edgeCnt ++; // draw the pixel if edgeCnt odd if (edgeCnt % 2) setPixel(pixel); CP411 polygon

26
Faster Polygon Rasterization ● How can we optimize the code? for each scanline edgeCnt = 0; for each pixel on scanline (l to r) if (oldpixel->newpixel crosses edge) edgeCnt ++; // draw the pixel if edgeCnt odd if (edgeCnt % 2) setPixel(pixel); ● Big cost: testing pixels against each edge ● Solution: active edge table (AET) CP411 polygon

27
Active Edge Table ● Idea: ■ Edges intersecting a given scanline are likely to intersect the next scanline ■ Within a scanline, the order of edge intersections doesn’t change much from scanline to scanline CP411 polygon

28
Active Edge Table ● Algorithm: ■ Sort all edges by their minimum y coord ■ Starting at bottom, add edges with Y min = 0 to AET ■ For each scanline: ○ Sort edges in AET by x intersection ○ Walk from left to right, setting pixels by parity rule ○ Increment scanline ○ Retire edges with Y max < Y ○ Add edges with Y min > Y ○ Recalculate edge intersections and resort (how?) ■ Stop when Y > Y max for last edges CP411 polygon

29
Edge Equations: Speed Hacks ● Some speed hacks for the inner loop: int xflag = 0; for (int x = xMin; x <= xMax; x++) { if (e0|e1|e2 > 0) { setPixel(x,y); xflag++; } else if (xflag != 0) break; e0 += a0; e1 += a1; e2 += a2; } ■ Incremental update of edge equation values ■ Early termination (why does this work?) ■ Faster test of equation values CP411 polygon

30
Antialiasing ● What is aliasing? The jaggies appeared in the drawings of lines are called aliasing. It is due to the approximation of a continuous line with discrete pixels. ● Aliasing effects: ■ Distant objects may disappear entirely ■ Objects can blink on and off in animations ● Antialiasing techniques involve some form of blurring to reduce contrast, smoothen image ● Three antialiasing techniques: ■ Prefiltering ■ Postfiltering ■ Supersampling CP411 polygon

31
Prefiltering ● Basic idea: ■ compute area of polygon coverage ■ use proportional intensity value ● Example: if polygon covers ¼ of the pixel ■ use ¼ polygon color ■ add it to ¾ of adjacent region color ● Cons: computing pixel coverage can be time consuming CP411 polygon

32
Prefiltering ● Conceive that a line is 1 pixel wide which covers certain pixel squares. ● A simple drawing method sets the pixels in C, G, H and L to the current color. ● Suppose that the overlapping proportion of a certain pixel is , the new pixel_color is then set to current_color + (1- ) existing_color

33
CP411 polygon glEnable(GL_BLEND); glEnable(GL_LINE_SMOOTH); glBlendFunc(GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA); glHint(GL_LINE_SMOOTH_HINT, GL_DONT_CARE); Without antialiasing With antialiasing

34
Supersampling ● Useful if we can compute color of any (x,y) value on the screen ■ Increase frequency of sampling ■ Instead of (x,y) samples in increments of 1 ■ Sample (x,y) in fractional (e.g. ½) increments ■ Find average of samples ● Example: Double sampling = increments of ½ = 9 color values averaged for each pixe CP411 polygon

35
Postfiltering ● Supersampling uses average ■ Gives all samples equal importance ● Post-filtering: use weighting (different levels of importance) ● Compute pixel value as weighted average ■ Samples close to pixel center given more weight CP411 polygon

Similar presentations

Presentation is loading. Please wait....

OK

CS 551 / 645: Introductory Computer Graphics

CS 551 / 645: Introductory Computer Graphics

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on diversity in living organisms for class 9 Ppt on rabindranath tagore poems Show ppt on samsung tv Ppt on stock market crash Download ppt on civil disobedience movement in india Ppt on natural and artificial satellites drawings Free ppt on brain machine interface invasive Chemistry ppt on solid state Ppt on power system deregulation Ppt on mammals and egg laying animals coloring