Comuter graphics bresenhams circle drawing algorithm
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Comuter graphics bresenhams circle drawing algorithm, extra solved examples on midpoint circle drawing algorithm,
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Comuter graphics ellipse drawing algorithm
Comuter graphics ellipse drawing algorithm, solved example for ellipse drawing algorithm, calculate pixel positions for ellipse, region1 , region 2,
Lec05 circle ellipse
This document discusses algorithms for drawing circles and ellipses in computer graphics. It covers:
1. Circle drawing algorithms including using Cartesian coordinates, polar coordinates, and the midpoint circle algorithm which iteratively calculates the midpoint between pixels to determine if they are inside, outside, or on the circle.
2. The midpoint ellipse algorithm which is similar but uses a modified decision parameter calculation that handles two regions for the first quadrant.
3. Key aspects of the algorithms include using octants and circle symmetry to reduce calculations, and translating the drawn shape to be centered at a given point.
Bressenham’s Midpoint Circle Drawing Algorithm
This document explains Bresenham's midpoint circle drawing algorithm. It defines a circle as all points equidistant from a center point. The algorithm uses integer arithmetic to iteratively determine the next pixel on the circle circumference. It starts at an initial point and calculates a decision parameter P to determine if the next point is above or below along the y-axis. It tabulates an example of drawing a circle of radius 10 centered at the origin to demonstrate how P is used to progress around the circle octant by octant.
Midpoint circle algo
The midpoint circle algorithm is similar to Bresenham's circle algorithm and uses the midpoint between pixels to determine whether the pixel is inside or outside a circle. It defines a decision parameter pi based on the midpoint and updates pi by integer amounts at each step to determine the next pixel along the circle. The initial value of pi is set to 5/4 - r when r is an integer to determine the first pixel.
Bresenham circle
The document describes Bresenham's circle generation algorithm. It explains that circles have 8-point symmetry that can be exploited to only calculate a quarter circle. It derives the algorithm by interpolating discrete pixels along a small arc and choosing the pixel with minimum error. It determines the initial decision parameter and the recurrence relation to calculate the next decision parameter in the algorithm. The full pseudocode of the algorithm is also provided.
Mid point circle algorithm
This document describes the midpoint circle algorithm for drawing circles given a radius and center point. It works by starting at an initial point on the circumference, calculating a decision parameter, and then iteratively determining the next point by testing if the decision parameter is positive or negative and updating the parameter according to the point's coordinates. It also explains how to determine additional points in the other octants and shift the calculated pixel positions to be centered on the given center point.
Computer Graphic - Lines, Circles and Ellipse
1. The document describes algorithms for drawing lines, circles, and ellipses using a midpoint technique. It provides examples showing the steps and calculations for applying each algorithm.
2. Key steps of the line drawing algorithm include calculating slope, change in x and y, and a decision parameter to determine the next point. Circles use a decision parameter comparing radius to x and y values. Ellipses use two regions and decision parameters involving radii and x/y values.
3. Examples are provided applying each algorithm to draw specific geometric shapes given endpoint or radius values. Tables show the calculations and plotted points at each iteration.
Mid-Point Cirle Drawing Algorithm
This document discusses algorithms for drawing circles using midpoint circle algorithm. It defines key circle concepts like radius, diameter and chord. It then explains the midpoint circle algorithm which uses 8-fold symmetry and computes pixel positions for only 45° sector. The algorithm initializes the first point as (0, radius) and uses a decision parameter F to iteratively calculate the next pixel position as (x+1, y) if F < 0 or (x+1, y-1) if F >= 0 until x >= y. Pseudocode and a C program to implement the algorithm is also provided with an example.
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Comuter graphics ellipse drawing algorithm
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Lec05 circle ellipse
This document discusses algorithms for drawing circles and ellipses in computer graphics. It covers:
1. Circle drawing algorithms including using Cartesian coordinates, polar coordinates, and the midpoint circle algorithm which iteratively calculates the midpoint between pixels to determine if they are inside, outside, or on the circle.
2. The midpoint ellipse algorithm which is similar but uses a modified decision parameter calculation that handles two regions for the first quadrant.
3. Key aspects of the algorithms include using octants and circle symmetry to reduce calculations, and translating the drawn shape to be centered at a given point.
Bressenham’s Midpoint Circle Drawing Algorithm
This document explains Bresenham's midpoint circle drawing algorithm. It defines a circle as all points equidistant from a center point. The algorithm uses integer arithmetic to iteratively determine the next pixel on the circle circumference. It starts at an initial point and calculates a decision parameter P to determine if the next point is above or below along the y-axis. It tabulates an example of drawing a circle of radius 10 centered at the origin to demonstrate how P is used to progress around the circle octant by octant.
Midpoint circle algo
The midpoint circle algorithm is similar to Bresenham's circle algorithm and uses the midpoint between pixels to determine whether the pixel is inside or outside a circle. It defines a decision parameter pi based on the midpoint and updates pi by integer amounts at each step to determine the next pixel along the circle. The initial value of pi is set to 5/4 - r when r is an integer to determine the first pixel.
Bresenham circle
The document describes Bresenham's circle generation algorithm. It explains that circles have 8-point symmetry that can be exploited to only calculate a quarter circle. It derives the algorithm by interpolating discrete pixels along a small arc and choosing the pixel with minimum error. It determines the initial decision parameter and the recurrence relation to calculate the next decision parameter in the algorithm. The full pseudocode of the algorithm is also provided.
Mid point circle algorithm
This document describes the midpoint circle algorithm for drawing circles given a radius and center point. It works by starting at an initial point on the circumference, calculating a decision parameter, and then iteratively determining the next point by testing if the decision parameter is positive or negative and updating the parameter according to the point's coordinates. It also explains how to determine additional points in the other octants and shift the calculated pixel positions to be centered on the given center point.
Computer Graphic - Lines, Circles and Ellipse
1. The document describes algorithms for drawing lines, circles, and ellipses using a midpoint technique. It provides examples showing the steps and calculations for applying each algorithm.
2. Key steps of the line drawing algorithm include calculating slope, change in x and y, and a decision parameter to determine the next point. Circles use a decision parameter comparing radius to x and y values. Ellipses use two regions and decision parameters involving radii and x/y values.
3. Examples are provided applying each algorithm to draw specific geometric shapes given endpoint or radius values. Tables show the calculations and plotted points at each iteration.
Mid-Point Cirle Drawing Algorithm
This document discusses algorithms for drawing circles using midpoint circle algorithm. It defines key circle concepts like radius, diameter and chord. It then explains the midpoint circle algorithm which uses 8-fold symmetry and computes pixel positions for only 45° sector. The algorithm initializes the first point as (0, radius) and uses a decision parameter F to iteratively calculate the next pixel position as (x+1, y) if F < 0 or (x+1, y-1) if F >= 0 until x >= y. Pseudocode and a C program to implement the algorithm is also provided with an example.
Circle drawing algo.
This document describes Bresenham's circle algorithm for efficiently scan converting a circle. It begins by explaining the symmetry of a circle and prior inefficient polynomial and trigonometric methods. It then presents Bresenham's algorithm which takes advantage of the circle's symmetry and uses a decision variable to determine whether to move in the x or y direction to plot each pixel, ensuring points are always closest to the true circle. The algorithm is presented with variables initialized and steps to iterate through the first octant to plot all pixels.
Bresenhamcircle derivation
The document describes Bresenham's circle generation algorithm. It utilizes the symmetry of circles to only calculate pixels in the first 45 degrees. It calculates the error between the actual circle curve and discrete pixels, choosing the pixel with lower error to trace the circle. It derives a decision parameter formula to iteratively calculate the next pixel, tracking whether to increase x or decrease y based on the sign of the decision parameter.
Computer Graphics - Bresenham's line drawing algorithm & Mid Point Circle alg...
Computer Graphics - Bresenham's line drawing algorithm & Mid Point Circle alg...Saikrishna Tanguturu
The document discusses algorithms for drawing lines and circles on a digital display. It describes Bresenham's line drawing algorithm which plots discrete points along a line to approximate it on the pixel grid. It uses decision parameters to determine whether to increment the x or y coordinate to best fit the line. For circles, it explains using the midpoint circle algorithm which calculates decision parameters based on the distance of pixel midpoints to the circle boundary to iteratively plot points. The document provides pseudocode to implement Bresenham's circle algorithm. It also lists the initial values and decision parameter updates required.Ellipses drawing algo.
(1) An ellipse is defined by the equation (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center and a and b are the lengths of the major and minor axes.
(2) There are two methods to scan convert an ellipse - the polynomial method and trigonometric method.
(3) The midpoint ellipse algorithm uses a decision parameter p to recursively scan convert the ellipse pixel by pixel in a manner similar to the midpoint circle algorithm.
Bresenhams
The document discusses algorithms for computer graphics including Bresenham's line drawing algorithm, circle drawing algorithms, and polygon filling algorithms. It introduces Bresenham's line drawing algorithm which uses integer calculations to incrementally determine which pixel is closer to the mathematical line. It then discusses a simple circle drawing algorithm and its limitations before introducing the more efficient mid-point circle drawing algorithm.
Circle generation algorithm
The document describes the Breshenham's circle generation algorithm. It explains that the algorithm uses a decision parameter to iteratively select pixels along the circumference of a circle. It provides pseudocode for the algorithm, which initializes x and y values, calculates a decision parameter, and increments x while decrementing y at each step, plotting points based on the decision parameter. An example of applying the algorithm to generate a circle with radius 5 is also provided.
Finite elements : basis functions
1. The document discusses the basis functions used in 1D and 2D finite element analysis.
2. For 1D elements, it presents the linear, quadratic and cubic basis functions derived from imposing interpolation conditions on linear, quadratic and cubic shape functions respectively.
3. For 2D elements, it derives the linear and quadratic basis functions for triangular and rectangular elements by transforming the element coordinates and imposing interpolation conditions on linear and quadratic Ansatz shape functions.
Bresenham circles and polygons derication
The document discusses algorithms for drawing lines and circles on a discrete pixel display. It begins by describing what characteristics an "ideal line" would have on such a display. It then introduces several algorithms for drawing lines, including the simple line algorithm, digital differential analyzer (DDA) algorithm, and Bresenham's line algorithm. The Bresenham algorithm is described in detail, as it uses only integer calculations. Next, a simple potential circle drawing algorithm is presented and its shortcomings discussed. Finally, the more accurate and efficient mid-point circle algorithm is described. This algorithm exploits the eight-way symmetry of circles and uses incremental calculations to determine the next pixel point.
Bresenham's line algo.
Bresenham's line algorithm is an efficient method for drawing lines on a discrete grid such as a display. It works by calculating the next pixel to plot along the line based on the difference between the true and discrete slopes. It starts at the initial point and calculates a decision variable di to determine whether to plot the next point horizontally or diagonally. This variable is updated using the slope to continually track the closest path to the true line.
Cs580
The document discusses techniques for line drawing and generalization in computer graphics. It covers Bresenham's line drawing algorithm, which uses only integer arithmetic for efficiency. It also discusses circle drawing using the midpoint circle algorithm and extensions to draw ellipses. Anti-aliasing techniques like area sampling are introduced to reduce jagged edges when rasterizing lines.
Bresenham's line drawing algorithm
The Bresenham's line drawing algorithm is an incremental scan conversion algorithm that uses only integer calculations. It works by calculating the difference between the ideal y value and the actual plotted pixel y value at each x value, and determining whether to plot the pixel above or below the line based on whether this difference is positive or negative. The algorithm takes the line endpoints as input, calculates the change in x and y per step, sets an initial decision parameter, and then iteratively plots pixels, updating the decision parameter to determine whether to plot above or below the line at each step.
Bresenham circlesandpolygons
The document discusses algorithms for drawing lines and circles on a discrete pixel display. It begins by describing what characteristics an "ideal line" would have on such a display. It then introduces several algorithms for drawing lines, including the simple line algorithm, digital differential analyzer (DDA) algorithm, and Bresenham's line algorithm. The Bresenham algorithm is described in detail, as it uses only integer calculations. Next, a simple potential circle drawing algorithm is presented and its shortcomings discussed. Finally, the more accurate and efficient mid-point circle algorithm is introduced. This algorithm exploits the eight-way symmetry of circles and only calculates points in one octant.
Graphics6 bresenham circlesandpolygons
The document discusses several common algorithms for computer graphics rendering including Bresenham's line drawing algorithm, the midpoint circle algorithm, and scanline polygon filling. Bresenham's algorithm uses only integer calculations to efficiently draw lines. The midpoint circle algorithm incrementally chooses pixel coordinates to draw circles with eightfold symmetry and without floating point operations. Scanline polygon filling finds the edge intersections on each scanline and fills pixels between interior intersections.
Mid point line Algorithm - Computer Graphics
The document describes the midpoint line algorithm for plotting lines on a grid. It works by calculating the midpoint between each set of pixels and determining if it falls above or below the line to choose the next pixel. It only requires integer calculations, avoiding errors from division or multiplication. The algorithm is derived step-by-step and an example is provided to demonstrate how it is implemented to plot a line between two points.
bresenham circles and polygons in computer graphics(Computer graphics tutorials)
bresenham circles and polygons in computer graphics(Computer graphics tutorials)Daroko blog(www.professionalbloggertricks.com)
Do Not just learn computer graphics an close your computer tab and go away..
APPLY them in real business,
Visit Daroko blog for real IT skills applications,androind, Computer graphics,Networking,Programming,IT jobs Types, IT news and applications,blogging,Builing a website, IT companies and how you can form yours, Technology news and very many More IT related subject.
-simply google:Daroko blog(professionalbloggertricks.com)
• Daroko blog (www.professionalbloggertricks.com)
• Presentation by Daroko blog, to see More tutorials more than this one here, Daroko blog has all tutorials related with IT course, simply visit the site by simply Entering the phrase Daroko blog (www.professionalbloggertricks.com) to search engines such as Google or yahoo!, learn some Blogging, affiliate marketing ,and ways of making Money with the computer graphic Applications(it is useless to learn all these tutorials when you can apply them as a student you know),also learn where you can apply all IT skills in a real Business Environment after learning Graphics another computer realate courses.ly
• Be practically real, not just academic reader
Parabola direction , vertex ,roots, minimum and maximum
This document discusses quadratic functions of the form f(x) = ax^2 + bx + c. It provides details on:
1) The direction a parabola opens depending on whether a is positive or negative.
2) How to calculate the vertex (turning point) of a parabola using the formula (-b/2a, 4ac-b^2/4a).
3) How to calculate the roots (x-intercepts) of a quadratic function using the formula (-b ± √(b^2 - 4ac))/2a.
Several examples are provided to demonstrate finding the direction, vertex, roots, minimum/maximum values of various quadratic functions.
Local linear approximation
The document discusses local linear approximations, which provide a linear function that closely approximates a given non-linear function near a specific point. It defines the local linear approximation at a point x0 as f(x0) + f'(x0)(x - x0). Graphs and examples are provided to illustrate how the local linear approximation can be used to estimate function values close to x0. The concept of differentials is also introduced to estimate small changes in a function using its derivative. Examples demonstrate using differentials to approximate changes and estimate errors in computations involving measured values.
Computer Graphic - Transformations in 3d
1. The document describes the steps to perform a rotation of an object in 3D space about an arbitrary point or axis.
2. It provides an example of rotating a unit cube 90 degrees about an axis defined by two points and calculating the new coordinates.
3. It also gives an example of rotating the point (1,2,1) 90 degrees and showing the resulting point (1,2,3) transformed to (4,6,7).
Bresenham derivation
The document derives Bresenham's line algorithm for drawing lines on a discrete grid. It starts with the line equation and defines variables for the slope and intercept. It then calculates the distance d1 and d2 from the line to two possible pixel locations and expresses their difference in terms of the slope and intercept. By multiplying this difference by the change in x, it removes the floating point slope value, resulting in an integer comparison expression. This is defined recursively to draw each subsequent pixel, using pre-computed constants. The initial p0 value is also derived from the line endpoint coordinates.
Lab lecture 2 bresenham
Bresenham's line algorithm uses incremental integer calculations to determine which pixels to turn on when drawing a line on a pixel-based display. It works by calculating a decision parameter Pk at each step k to determine whether to plot the pixel at (Xk+1,Yk) or (Xk+1,Yk+1). For a given line from (30,20) to (40,28), the algorithm is applied by initializing P0=6 and then iteratively calculating Pk+1 at each step k by adding either 2dy or 2dy-2dx depending on whether Pk is positive or negative. This tracks the line from (30,20) to (40,28)
Comuter graphics bresenhams line drawing algorithm
bresenhams line drawing algorithm examples, solved examples, computer graphics, pixel calculation, hearn and baker text book on computer graphics, extra solved examples on bresenhams line drawing algorithm,
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Circle drawing algo.
This document describes Bresenham's circle algorithm for efficiently scan converting a circle. It begins by explaining the symmetry of a circle and prior inefficient polynomial and trigonometric methods. It then presents Bresenham's algorithm which takes advantage of the circle's symmetry and uses a decision variable to determine whether to move in the x or y direction to plot each pixel, ensuring points are always closest to the true circle. The algorithm is presented with variables initialized and steps to iterate through the first octant to plot all pixels.
Bresenhamcircle derivation
The document describes Bresenham's circle generation algorithm. It utilizes the symmetry of circles to only calculate pixels in the first 45 degrees. It calculates the error between the actual circle curve and discrete pixels, choosing the pixel with lower error to trace the circle. It derives a decision parameter formula to iteratively calculate the next pixel, tracking whether to increase x or decrease y based on the sign of the decision parameter.
Computer Graphics - Bresenham's line drawing algorithm & Mid Point Circle alg...
Computer Graphics - Bresenham's line drawing algorithm & Mid Point Circle alg...Saikrishna Tanguturu
The document discusses algorithms for drawing lines and circles on a digital display. It describes Bresenham's line drawing algorithm which plots discrete points along a line to approximate it on the pixel grid. It uses decision parameters to determine whether to increment the x or y coordinate to best fit the line. For circles, it explains using the midpoint circle algorithm which calculates decision parameters based on the distance of pixel midpoints to the circle boundary to iteratively plot points. The document provides pseudocode to implement Bresenham's circle algorithm. It also lists the initial values and decision parameter updates required.Ellipses drawing algo.
(1) An ellipse is defined by the equation (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center and a and b are the lengths of the major and minor axes.
(2) There are two methods to scan convert an ellipse - the polynomial method and trigonometric method.
(3) The midpoint ellipse algorithm uses a decision parameter p to recursively scan convert the ellipse pixel by pixel in a manner similar to the midpoint circle algorithm.
Bresenhams
The document discusses algorithms for computer graphics including Bresenham's line drawing algorithm, circle drawing algorithms, and polygon filling algorithms. It introduces Bresenham's line drawing algorithm which uses integer calculations to incrementally determine which pixel is closer to the mathematical line. It then discusses a simple circle drawing algorithm and its limitations before introducing the more efficient mid-point circle drawing algorithm.
Circle generation algorithm
The document describes the Breshenham's circle generation algorithm. It explains that the algorithm uses a decision parameter to iteratively select pixels along the circumference of a circle. It provides pseudocode for the algorithm, which initializes x and y values, calculates a decision parameter, and increments x while decrementing y at each step, plotting points based on the decision parameter. An example of applying the algorithm to generate a circle with radius 5 is also provided.
Finite elements : basis functions
1. The document discusses the basis functions used in 1D and 2D finite element analysis.
2. For 1D elements, it presents the linear, quadratic and cubic basis functions derived from imposing interpolation conditions on linear, quadratic and cubic shape functions respectively.
3. For 2D elements, it derives the linear and quadratic basis functions for triangular and rectangular elements by transforming the element coordinates and imposing interpolation conditions on linear and quadratic Ansatz shape functions.
Bresenham circles and polygons derication
The document discusses algorithms for drawing lines and circles on a discrete pixel display. It begins by describing what characteristics an "ideal line" would have on such a display. It then introduces several algorithms for drawing lines, including the simple line algorithm, digital differential analyzer (DDA) algorithm, and Bresenham's line algorithm. The Bresenham algorithm is described in detail, as it uses only integer calculations. Next, a simple potential circle drawing algorithm is presented and its shortcomings discussed. Finally, the more accurate and efficient mid-point circle algorithm is described. This algorithm exploits the eight-way symmetry of circles and uses incremental calculations to determine the next pixel point.
Bresenham's line algo.
Bresenham's line algorithm is an efficient method for drawing lines on a discrete grid such as a display. It works by calculating the next pixel to plot along the line based on the difference between the true and discrete slopes. It starts at the initial point and calculates a decision variable di to determine whether to plot the next point horizontally or diagonally. This variable is updated using the slope to continually track the closest path to the true line.
Cs580
The document discusses techniques for line drawing and generalization in computer graphics. It covers Bresenham's line drawing algorithm, which uses only integer arithmetic for efficiency. It also discusses circle drawing using the midpoint circle algorithm and extensions to draw ellipses. Anti-aliasing techniques like area sampling are introduced to reduce jagged edges when rasterizing lines.
Bresenham's line drawing algorithm
The Bresenham's line drawing algorithm is an incremental scan conversion algorithm that uses only integer calculations. It works by calculating the difference between the ideal y value and the actual plotted pixel y value at each x value, and determining whether to plot the pixel above or below the line based on whether this difference is positive or negative. The algorithm takes the line endpoints as input, calculates the change in x and y per step, sets an initial decision parameter, and then iteratively plots pixels, updating the decision parameter to determine whether to plot above or below the line at each step.
Bresenham circlesandpolygons
The document discusses algorithms for drawing lines and circles on a discrete pixel display. It begins by describing what characteristics an "ideal line" would have on such a display. It then introduces several algorithms for drawing lines, including the simple line algorithm, digital differential analyzer (DDA) algorithm, and Bresenham's line algorithm. The Bresenham algorithm is described in detail, as it uses only integer calculations. Next, a simple potential circle drawing algorithm is presented and its shortcomings discussed. Finally, the more accurate and efficient mid-point circle algorithm is introduced. This algorithm exploits the eight-way symmetry of circles and only calculates points in one octant.
Graphics6 bresenham circlesandpolygons
The document discusses several common algorithms for computer graphics rendering including Bresenham's line drawing algorithm, the midpoint circle algorithm, and scanline polygon filling. Bresenham's algorithm uses only integer calculations to efficiently draw lines. The midpoint circle algorithm incrementally chooses pixel coordinates to draw circles with eightfold symmetry and without floating point operations. Scanline polygon filling finds the edge intersections on each scanline and fills pixels between interior intersections.
Mid point line Algorithm - Computer Graphics
The document describes the midpoint line algorithm for plotting lines on a grid. It works by calculating the midpoint between each set of pixels and determining if it falls above or below the line to choose the next pixel. It only requires integer calculations, avoiding errors from division or multiplication. The algorithm is derived step-by-step and an example is provided to demonstrate how it is implemented to plot a line between two points.
bresenham circles and polygons in computer graphics(Computer graphics tutorials)
bresenham circles and polygons in computer graphics(Computer graphics tutorials)Daroko blog(www.professionalbloggertricks.com)
Do Not just learn computer graphics an close your computer tab and go away..
APPLY them in real business,
Visit Daroko blog for real IT skills applications,androind, Computer graphics,Networking,Programming,IT jobs Types, IT news and applications,blogging,Builing a website, IT companies and how you can form yours, Technology news and very many More IT related subject.
-simply google:Daroko blog(professionalbloggertricks.com)
• Daroko blog (www.professionalbloggertricks.com)
• Presentation by Daroko blog, to see More tutorials more than this one here, Daroko blog has all tutorials related with IT course, simply visit the site by simply Entering the phrase Daroko blog (www.professionalbloggertricks.com) to search engines such as Google or yahoo!, learn some Blogging, affiliate marketing ,and ways of making Money with the computer graphic Applications(it is useless to learn all these tutorials when you can apply them as a student you know),also learn where you can apply all IT skills in a real Business Environment after learning Graphics another computer realate courses.ly
• Be practically real, not just academic reader
Parabola direction , vertex ,roots, minimum and maximum
This document discusses quadratic functions of the form f(x) = ax^2 + bx + c. It provides details on:
1) The direction a parabola opens depending on whether a is positive or negative.
2) How to calculate the vertex (turning point) of a parabola using the formula (-b/2a, 4ac-b^2/4a).
3) How to calculate the roots (x-intercepts) of a quadratic function using the formula (-b ± √(b^2 - 4ac))/2a.
Several examples are provided to demonstrate finding the direction, vertex, roots, minimum/maximum values of various quadratic functions.
Local linear approximation
The document discusses local linear approximations, which provide a linear function that closely approximates a given non-linear function near a specific point. It defines the local linear approximation at a point x0 as f(x0) + f'(x0)(x - x0). Graphs and examples are provided to illustrate how the local linear approximation can be used to estimate function values close to x0. The concept of differentials is also introduced to estimate small changes in a function using its derivative. Examples demonstrate using differentials to approximate changes and estimate errors in computations involving measured values.
Computer Graphic - Transformations in 3d
1. The document describes the steps to perform a rotation of an object in 3D space about an arbitrary point or axis.
2. It provides an example of rotating a unit cube 90 degrees about an axis defined by two points and calculating the new coordinates.
3. It also gives an example of rotating the point (1,2,1) 90 degrees and showing the resulting point (1,2,3) transformed to (4,6,7).
Bresenham derivation
The document derives Bresenham's line algorithm for drawing lines on a discrete grid. It starts with the line equation and defines variables for the slope and intercept. It then calculates the distance d1 and d2 from the line to two possible pixel locations and expresses their difference in terms of the slope and intercept. By multiplying this difference by the change in x, it removes the floating point slope value, resulting in an integer comparison expression. This is defined recursively to draw each subsequent pixel, using pre-computed constants. The initial p0 value is also derived from the line endpoint coordinates.
Lab lecture 2 bresenham
Bresenham's line algorithm uses incremental integer calculations to determine which pixels to turn on when drawing a line on a pixel-based display. It works by calculating a decision parameter Pk at each step k to determine whether to plot the pixel at (Xk+1,Yk) or (Xk+1,Yk+1). For a given line from (30,20) to (40,28), the algorithm is applied by initializing P0=6 and then iteratively calculating Pk+1 at each step k by adding either 2dy or 2dy-2dx depending on whether Pk is positive or negative. This tracks the line from (30,20) to (40,28)
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Comuter graphics bresenhams line drawing algorithm
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2. Key steps in the line algorithm include calculating slope, increment values, and a decision parameter to determine the next point. Circle and ellipse algorithms similarly use initial points, decision parameters, and region-based steps to iteratively plot points.
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This document discusses integrating rational functions. It begins by defining rational functions as functions of the form P(x)/Q(x) where P and Q are polynomials. It states that integrating rational functions involves decomposing them into simpler fractions using the Rational Decomposition Theorem. Examples are then provided to demonstrate how to decompose rational functions and integrate each term using substitution.
Computer_Graphics_circle_drawing_techniq.ppt
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The document discusses algorithms for drawing lines and circles on raster displays. It describes Bresenham's line algorithm which uses only integer calculations to determine which pixels to turn on along a line. For circles, it presents the midpoint circle algorithm which uses incremental integer calculations and the implicit equation of a circle to determine the pixel positions along the circle boundary.
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The document summarizes Bresenham's line drawing algorithm. It derives the equations for calculating the next pixel position when drawing a line on a digital display. It considers cases where the slope is less than or equal to 1 and greater than 1. For each case, it calculates the distance from intersection points to pixel positions and derives the decision parameter equation to determine the next pixel.
mid point algorithm.pdf
The document describes the mid-point circle algorithm for drawing circles. It uses eight-way symmetry to only calculate points for one octant of the circle, then uses symmetry to get the remaining points. At each x-coordinate, it must choose between two possible y-coordinates by calculating the mid-point between them and determining which side of the circle it falls on based on comparing the mid-point to the circle equation. It iterates this process, incrementally updating values for efficiency, until all points along the top-right octant are drawn, then uses symmetry to plot the full circle.
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Comuter graphics bresenhams circle drawing algorithm
- 1. COMPUTER GRAPHICS Bressenham's Circle Drawing Algorithm By: Ms. Rachana Marathe
- 2. Bressenham's algorithm: Circle drawing from x=0 and x=y ex: if r=10, (x0,y0)= (0,10) p0 =1-r if Pk<0 then (xk+1,yk) and pk+1= pk+2xk+1+1 else (pk>0) then (xk+1,yk-1) and pk+1= pk+2xk+1+1-2yk+1
- 3. Example 1: radius = 5 Algorithm: ex: if r=radius, (x0,y0)= (0,r) p0 =1-r if Pk<0 then (xk+1,yk) and pk+1= pk+2xk+1+1 else (pk>0) then (xk+1,yk-1) and pk+1= pk+2xk+1+1-2yk+1 Solution: X=0, r=5 , so plot first point at ( 0,5) p0 =1-r = 1-5 = -4 X=1 , y=5 (if case) pk+1= pk+2xk+1+1 = -4+2+1 = -1 X=2, y=5 (if case) pk+1= pk+2xk+1+1 = -1 + 4+1 = 4 X= 3 y = 4 (else case) pk+1= pk+2xk+1+1-2yk+1 = 4+6+1 – 8=3 X=4, y = 3 (else case) X>y hence STOP
- 4. Therefore the pixels are (0,5) (1,5) (2,5) (3,4) (4,3)
- 5. Reference Hearn,Baker - Computer Graphics - C Version 2nd Ed http://edu.uokufa.edu.iq/staff/dr.nidhal/compressed%20comp.book/H earn,Baker%20-%20Computer%20Graphics%20- %20C%20Version%202nd%20Ed.pdf