introducion and output primitives - final for site use 2014

8/11/2019 Introducion and Output Primitives - Final for Site Use 2014

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Computer Graphics -Introduction & Output

Primitives

Class: 5CEB

Priyank Thakkar


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Books Computer Graphics, Donald Hearn and M.

Pauline Baker, PEARSON EDUCATION.

Computer Graphics Principles and Practice,

James D. Foley, Andries van Dam, Steven K.Feiner, John F. Hughes, Addison-Wesley.

Computer Graphics: A programming approach,Steven Harrington, McGraw Hill


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Blog and Course SiteBlog Link:http://2ce321pbt.wordpress.com

Course Site:https://sites.google.com/a/nirmauni.ac.in/2ce321/


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Teaching & Evaluation SchemeTeaching Scheme:

Evaluation Methodology:

Theory Tutorial Practical Credits

3 0 2 4

LPW SEE TA

Exam Duration Continuous Evaluation + 2Hrs. End Semester Exam

3.0 Hrs. Continuous

Evaluation

Component

Weightage

0.2 0.4 0.4


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Introduction What is Computer Graphics?

Applications CAD

Product Design, Circuit Design, Building Layout Design,etc.

Presentation GraphicsSummarizing financial, statistical, mathematical, scientific

and economic data for various reports

Computer Art

Paintbrush, Mathematical Art Entertainment

Special Effects, Scenes involving animals

Education & Training


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Introduction Visualization

GUI


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Displays Raster Scan Displays

Pixels, Beam Sweeping, Frame Buffer Random Scan Displays


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Line Drawing Algorithms Line with Positive Slope

m>1, 0


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Line Drawing Algorithms Similarly, we can obtain the x interval x

corresponding to a specifiedy as

Raster System and Step Size

DDA (Digital Differential Analyzer) (1,1)->(5,3) and (1,1)->(3,5)

If 0 < m 1

Assumption for the above equations

L to R, What if R to L?


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Line Drawing Algorithms DDA (Digital Differential Analyzer)

For Positive Slope and Left to Right

DDA vs. Direct Method (Multiplication avoided)

DDAs Limitations (Incremental but Floating Point

Arithmetic, Rounding)


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Line Drawing Algorithms DDA (Digital Differential Analyzer) (Code)

What about drawing line from right to left?Beginning point needs to be (xa,ya) and end point needs tobe (xb,yb).

What about line having negative slope?Above code will work for negative slopes as well.

DDA (Digital Differential Analyzer) (Equation)

Positive Slope (L-R)Increase Both

Positive Slope (R-L)Decrease BothNegative Slope (L-R)

Increase x, Decrease y

Negative Slope (R-L)Decrease x, Increase


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Line Drawing Algorithms Sampling Concept

(1, 1) -> (5, 3) m = (3 - 1) / (5 - 1) => m = 0.5

b = y1m * x1=> 1 - (0.5 * 1) => b = 0.5

If we unit sample y then we have to find x at y = 2

x = (y - b) / m => x = (2 0.5) / 0.5 => x = 3 (1 , 1) -> (3, 2) -> (5, 3)


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Line Drawing Algorithms Sampling Concept

(1, 1) -> (5, 3) m = (3 - 1) / (5 - 1) => m = 0.5

b = ym * x => 1 - (0.5 * 1) => b = 0.5

If we unit sample x then we have to find y at x = 2,

3 and 4 x = 2 => y = (0.5 * 2) + 0.5 = 1.5 2

x = 3 => y = (0.5 * 3) + 0.5 = 2

x = 4 => y = (0.5 * 4) + 0.5 = 2.5 3

(1 , 1) -> (2, 2) -> (3, 2) -> (4, 3) -> (5, 3)


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Line Drawing Algorithms BresenhamsLine Algorithm

Incremental Integer CalculationsAssume a line with positive slope < 1

Starting Position (10, 11)

Next Position?

(11, 11) or (11, 12)?


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Assume a line with negative slope > -1Starting Position (50, 50)

Next Position?

(51, 50) or (51, 49)?

Distance between Line Path andPixel Position?

Identify an integer whose value is

proportional to the differencebetween the separations of the two pixel positionsfrom the actual line path

Sign of this integer parameter will answer the

question


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Assume positive slope less than 1 Pixels along the line path is determined by sampling

at unit x interval

Start process from the left end point (x0, y0)

Figure demonstrates kth step in this process


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Using only pkformula, all the midpoints can be calculated.

However this formula is not additive. At each stepmultiplications (however still it is only integer arithmetic) areinvolved.


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Always use absolute value of xand y to tackle negativeslopes.

For m>1 and m


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Examples


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Circle Generating Algorithms Equation of a Circle in Cartesian Coordinate System

Plotting the Circle


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Circle Generating Algorithms Parametric Equation of a Circle

Angular Step

Fixed Step Size

Step Size = 1/r Reducing the Computation by considering

Symmetry

QuadrantsOctants

Ending of Octant

Square Root Calculations

Trigonometric Calculations


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Circle Generating Algorithms Circle Function

Assume center (0, 0) and radius r


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Circle Generating Algorithms Midpoint Circle Algorithm

Assume center (0, 0) & radius r, octant from x = 0to x = y

We are at kth step and (xk, yk) is already displayed

Candidates at step k + 1 are: (xk+1, yk) & (xk+1, yk- 1)


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E l


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Examples

Ell l h


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Ellipse Generating Algorithms Properties of Ellipses

Focus

Major and Minor Axis

Ellipse in nonstandard position

Ell G l h


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Ellipse in standard position

Ell G l h


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Symmetry Concept

Direct Method and Square root or TrigonometricOperations

Elli G i l i h


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Ellipse Generating Algorithms Ellipse Function

Assume center (0, 0)

Elli G i Al i h


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Ellipse Generating Algorithms Remember:

0 |y2 y1| < |x2x1|So, if the absolute range of x is greater than

that of y then |m| [0, 1)

|m| = 1 => |y2y1| = |x2x1|

|m| > 1 => |y2y1| > |x2x1|So, if the absolute range of y is greater than

that of x then |m| > 1

Elli G i Al i h


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Ellipse Generating Algorithms Midpoint Ellipse Algorithm

Region 1 & 2 Visual Inspection of range of x

and y in regions

Mathematical Interpretation

Elli G ti Al ith


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Elli G ti Al ith


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Elli G ti Al ith


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Elli G ti Al ith


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Elli G ti Al ith


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Elli G ti Al ith


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Elli G ti Al ith


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Elli G ti Al ith


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Examples


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Examples

Examples


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Examples

Poly on Fillin Al orithms


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Polygon Filling Algorithms Scan-Line Polygon Fill Algorithm

Identify scan lines crossing a polygon For each such scan line locate the intersection

points with polygon edges.

Sort intersection points from left to right and set

corresponding frame buffer positions between eachintersection pair to the specified fill color.

Example

Polygon Filling Algorithms


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Some intersections at polygon vertices requirespecial handling

Odd vs. even number of intersections and pairing.

Topological difference between scan line y and y

and identifying vertices which require specialhandling



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Approach of Shortening Edges



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Determining edge intersections

= +

=

()

, + =+ +

, + =+ +

+ = +

+ =

+

+ = +

()



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To avoid Rounding Operation from the above equation, wecan use following:

Initialize counter to 0.

Increment the counter by x each time we move up to anew scan line

If counter >= y, then x coordinate of intersection pointbetween the current scan line and polygon edge is one more

than the x coordinate of intersection point between previousscan line and polygon edge and decrease the counter value byy.

Else x coordinate of intersection between the current scanline and polygon edge is same as the x coordinate of

intersection point between previous scan line and polygonedge.



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Polygon Filling Algorithms Sorted Edge Table (SET) and Active Edge List/Table



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Status of AET at Scanline 8



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Polygon Filling Algorithms Scan-Line Polygon Fill Algorithm (Other than standardpolygon)

Inside Outside Tests

Odd-even rule/Odd parity rule/Even-odd rule

Nonzero winding number rule



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Odd-even rule/Odd parity rule/Even-odd rule

Assume a distant point outside the coordinate extent

Assume a line from any test point P to this distantpoint not passing through any of the vertices

If this line crosses odd number of polygon edges, thenpoint P is an interior point otherwise it is an exteriorpoint.



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Assume a distant point outside the coordinate extent ofthe object

Winding number is initialized to zero for each of the testpoint P.

As we move along the line from position P to the distinctpoint, we add one to winding number every time lineintersects an edge that crosses it from right to left.(cross product is positive)

If intersecting edge crosses the line from left to right(cross product is positive), we subtract one from windingnumber.

If the final value of the winding number is nonzero, P isdefined to be an interior point otherwise it is defined asan exterior point.

Determining Directional edge crossings



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Determining Directional edge crossings

Cross product



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Polygon Filling Algorithms Boundary Fill Algorithm

Start at a point inside a region and paint the interior outward

toward the boundaryIf the boundary is specified in single color, the fill algorithm

proceeds outward pixel by pixel until the boundary color isencountered.

This method is particularly useful in interactive paintingpackages, where interior points are easily selected.

To display a solid color region (with no border), the designercan choose fill color to be the same as the boundary color.



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Which are the inputs to boundary-fill procedure?

Starting from (x, y), the procedure tests neighboringpositions to determine whether they are of the boundarycolor

If not?

This process continues until all pixels up to theboundary color for the area have been tested.

Figure 3-43



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When can boundaryfill4 fail?

Some interior pixels are already displayed in fill color

To avoid this, first change the color of any interiorpixels that are initially set to the fill color beforeapplying the boundary-fill procedure

4-connected vs. 8-connected



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olygon F ll ng lgor thms Boundary Fill Algorithm & Alternative

Since the boundaryFill4 procedure requires considerable

stacking of neighboring points, more efficient methods aregenerally employed.

These methods fill horizontal pixel spans across scan lines,instead of proceeding to 4-connected or 8-connectedneighboring points.

This requires only stacking a beginning position for eachhorizontal pixel span, instead of stacking all unprocessedneighboring positions around the current position.

Spans are defined as the contiguous horizontal string ofpositions bounded by pixels displayed in the area border

color.At each subsequent step, the next start position is unstacked

and the process is repeated.



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yg F g g m Alternative Method

Process scan lines successively from the start line to top

boundaryAfter all upper scan lines are processed, fill in the pixel spans

on the remaining scan lines in order down to the bottomboundary

The leftmost pixel position for each horizontal span is located

and stacked, in left to right order across successive scanlines.



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yg g g m Scan-Line Polygon Fill Algorithm

Boundary Fill Algorithm

Addressing stacking of neighboring points



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yg g g Flood Fill Algorithm

Sometimes we may want to recolor

Sometimes we may want to fill in an area that is not definedwithin a single color boundary

If the area we want to paint has more than one interiorcolor?

First reassign pixel values so that all interior points have

the same color.

Character Generations


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Serif fonts and Sans Serif fonts

Serif: Hello WorldSmall lines or accents at the ends of main character strokes

Sans Serif: Hello WorldDoes not have accents


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introducion and output primitives - final for site use 2014

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