This algorithm is meant for basic line drawing only Initializing is not a part of Bresenham's line algorithm. By MURUGAN G. Computer Graphics By Hearn & … In this article, we will see an outline on the Line Drawing Algorithm. Bresenham's line algorithm is a line drawing algorithm that determines the points of an n-dimensional raster that should be selected in order to form a close approximation to a straight line between two points.It is commonly used to draw line primitives in a bitmap image (e.g. d1 = Yk - Y. and the distance from the actual line to the second point is . // Purpose: To implement the Bresenham's line drawing algorithm for all // slopes and line directions (using minimal routines). Bresenham 's circle algorithm is derived from the midpoint circle algorithm. Computer Graphics C Version by Donald Hearn & M Pauline Baker II Edition. Check_Pixel is used to check if the line pixel is on a valid bitmap one. Clarifying the problem. Basic algorithms rasterize lines in one color. There are many algorithms which use this equation. Step 1: If the slope is less than or equal to 1, the unit x intervals D Bresenham Line Drawing Calculator By putting x1,x2 and y1,y2 Value it Show The Result In Step By Step order,and Result Brief Calculation Which Is Calculated by Bresenham Line Drawing Algorithm. Bresenham Line Drawing Algorithm display result in tables.Starting Points is x1,y1 and Ending points is x2,y2. P0 =2dy-dx. Bresenham's line drawing algorithm for drawing a line in a computer screen by using integer arithmetic operations only.It is more efficient that the older DDA Line drawing algorithm that uses floating-point arithmetic and rounding operations although it can be optimised to perform only integer operations still pixle selection using Besenham's gives better result than DDA. We always increase x by 1, and we choose about next y, whether we need to go to y+1 or remain on y. In Computer Graphics the first basic line drawing algorithm is Digital Differential Analyzer (DDA) Algorithm. Download Our App For Source Code:- http://bit.ly/2EygXPuNote: Please turn on Subtitles then watch video.Hi Everyone..! 15CSL68 – Computer Graphics Lab Manual BMSIT & M, Bengaluru -560064 | Author: Mr. Shankar R, Asst. Draw a circle without floating point arithmetic. The vocabulary for using this draw~ ing package is: An algorithm similar to Bresenham’s line drawing algorithm, called the Midpoint Circle Algorithm, has been developed for drawing a circle e ciently. Al Hizbul Bahar says: February 10, 2015 at 6:19 PM. 01 Implement Brenham’s line drawing algorithm for all types of slope 02 Create and rotate a triangle about the origin and a fixed point 03 Draw a color cube and spin it using OpenGL transformation matrices 04 Draw a color cube and allow the user to move the camera suitably to … By Abhijeet Panda. Computer Graphics refers. Objective:-To implement Bresenham ‘s Algorithm for drawing a Line.Description: Basic Concept: Move across the x axis in unit intervals and … Draw circle in C graphics. The point is an important element of a line. A circle is defined by a centre point, and a radius. Uses only integer operations?Basic Idea ofBresenhamAlgorithm: –All lines can be placed in one of four categories: A. In the previous derivation when we checked the decision variable, we always incremented x and y by positive one. Bresenham's Line Algorithm Lecture 3 / 3rd Class / 2016-2017 2 Ms. C. Rusul Mohammed Bresenham's algorithm is generalized to lines with arbitrary slope by considering the symmetry between the various octants and quadrants of the xy plane. Bresenham Line Algorithm Using OpenGL and C++. The idea of Bresenham’s algorithm is to avoid floating point multiplication and addition to compute mx + c, and then computing round value of (mx + c) in every step. It is commonly used to draw line primitives in a bitmap image, as it uses only integer addition, subtraction and bit shifting, all of which are very cheap operations in standard computer architectures. Calculate the constants Dx, Dy, 2Dy, 2Dy – 2Dx and obtain the starting value for the decision parameter as G = 2Dy- Dx. The endpoints are obviously (X, Y) and (X + Dx, Y + Dy). Neighbors of a point on a circle using Bresenham's algorithm. Here is an explanation of my own version of Bresenham. I have 4 Years of hands on experience on helping student in completing their homework. So, in computer graphics, there are two algorithms used for drawing a line over the screen that is DDA (Digital Differential Analyser) algorithm and Bresenham algorithm. Our simple line drawing algorithm does not provide satisfactory results for line slopes greater than 1. Bresenham algorithm for x 0 < x 1 and 0 < slope < 1 6 4. Example Draw a line from (x1, y1) to (x2, y2) . By Engr Rana M Shakeel. /* WAP to draw a rectangle whose lower left corner, length and breadth are provided by the user. Properties of a Line Drawing Algorithm. The solution: symmetry. Plot the point $(x_0, y_0)$. Use Bresenham’s algorithm to draw the sides of the rectangle - Subhranath Chunder */. As it had been discussed in the previous post that when slope of line > 1, the shadow of line … lines with slopes that are not between zero and one. 1. void drawline (int x0, int y0, int x1, int y1) { int dx, dy, p, x, y; dx=x1-x0; dy=y1-y0; x=x0; y=y0; p=2*dy-dx; while (x
=0) { putpixel (x,y,7); y=y+1; p=p+2*dy-2*dx; x+=1; } else { putpixel (x,y,7); p=p+2*dy;} x+=1; } … The algorithm and derivation above assumes slopes are less than 1. for other slopes we need to adjust the algorithm slightly Otherwise, the next point to plot is (x k +1, y k +1) ... Bresenham line drawing algorithm. This completes the generalized version of the Bresenham's line drawing algorithm. Obtain the initial decision parameter as. Steep positive slope (m > 1) B. It is an incremental error algorithm. Bresenhams line draw algorithm for lines with slope < 1: Accept two endpoints from user and store the left end point in (x0, y0) as starting point. Implement DDA and Bresenham line drawing algorithm to draw: i) Simple Line ii) Dotted Line iii) Dashed Line Using mouse interface Divide the screen in four quadrants with center as (0, 0). P = 2 * dely – delx. Ellipses have vertical and horizontal symmetry; 2. #bresenham algo, #Bresenham's line algorithm, #Bresenham's line algorithm for line plotted between 45 and 90 degrees, #bresenham's line algorithm for slope < 1, #computer graphics. 01, Jun 21. The Bresenham algorithm is probably the most efficient of all line drawing algorithm. Above algorithm works, but it is slow. A simple circle drawing algorithm is presented. A novelty algorithm of spatial straight-line drawing based on a method of discriminate regions is proposed in this paper based on Bresenham’s algorithm. Uploaded by. 23 of 60 The Bresenham Line Algorithm BRESENHAM’S LINE DRAWING ALGORITHM (for |m| < 1.0) 1. // ... Write a program in C/C++ using OpenGL to implement Bresenham’s line drawing algorithm with all values of slopes. Bresenham's line algorithm is a line drawing algorithm that determines the points of an n-dimensional raster that should be selected in order to form a close approximation to a straight line between two points. on Bresenham's Line Algorithm For Slope > 1. For strains with wonderful slope increased than 1 we interchange the roles of the and directions. Bresenham’s line algorithm जो है वह केवल integer addition तथा substraction का ही प्रयोग करता है. The Otherwise plot (x+1,y-1). The Bresenham Line-Drawing Algorithm, Bresenham for negative slopes Given that a line-drawing algorithm has plotted a point at (x,y), its choice about where to plot the next point is between (x+1,y-1) and (x+1,y). A minor extension to the original algorithm also deals with drawing circles. Using Bresenham’s algorithm, generate the coordinates of the pixels that lie on a line segment having the endpoints (2, 3) and (5, 8). 05, May 17. notes bresenhams line drawing algorithm. Screens 5 through 13 contain a complete Bresenham line drawing vocabulary for all line slopes. A New Line Drawing Algorithm. Algorithm for slope |m|<1: Input two end points (x1,y1) and (x2,y2) of the line. // bresenham_line_algorithm.cpp : Defines the entry point for the console application. In Bresenham’s algorithm, we move across the x-axis in unit intervals. OpenGL program to Implement Bresenham’s line drawing algorithm for all types of slope. The basic Bresenham algorithm: Consider drawing a line on a raster grid where we restrict the allowable slopes of the line to the range 0 <= m <= 1. Calculate the constants Δx, Δy, 2Δy, and (2Δy - 2Δx) … I know how to draw lines with DDA, but I want to learn a more efficient way and Google suggests Bresenham's line drawing algorithm is better … Horizontal lines are a special case and have their own optimised routine at HLOIN, but all non-horizontal lines in Elite are drawn by the LOIN routine using Bresenham's line algorithm. In case we have to draw a line on the computer, first of all, we need to know which pixels should be on. Plot the first point (x1,y1). Line drawing on the computer means the computer screen is dividing into two parts rows and columns. 22. In Bresenham’s algorithm, we move across the x-axis in unit intervals. Drawing lines with DDA algorithm 4 3. The increment in another variable will be either 0 or 1, which is determined by analyzing the distance between the actual line and given pixel locations. The line at unit intervals is one coordinate and determine corresponding integer values nearest line for the other coordinate. Case: When slope (m) > 1 Now let’s solve the same numerical using BLA Algorithm. Bresenham Line Drawing Algorithm display result in tables.Starting Points is … This version limited to slopes in the first octant, . Otherwise the program will not work. A circle is de ned by a centre point, and a radius. For a line with positive slope greater than 1, we interchange the roles of the x and y directions. The assigning of one coordinate axis the name x and the other y was an arbitrary choice. The basic idea of the Bresenham Algorithm is shown is the previous section, but the algorithm can be easily extended to all other lines, not just the lines with slope between 0 and 1.One subset of the cases is concerned with lines with slope from -1 to 1. The idea of Bresenham’s algorithm is to avoid floating point multiplication and addition to compute mx + c, and then computing round value of (mx + c) in every step. The transformations necessary to perform correctly the digitalization of a straight line using Bresenham's algorithm on a hexagonal grid are introduced. thirumalaikumaran. Screen shot of line demo program. Step2: Declare variable x 1,x 2,y 1,y 2,d,i 1,i 2,dx,dy. Bresesnham algorithm to draw a straight line segment. For a line with positive slope greater than 1, we interchange the roles of the x and y directions. d2 = Y - (Yk - 1) Input the two line end-points, storing the left end-point in (x0, y0) 2. Prof, CSE 7 The core algorithm It can also be extended to display circles another curves. Bresenham algorithm for circles centered at the origin 16 8. Bresenham's Line Algorithm: Step1: Start Algorithm. Notice that line slopes of greater than one under one assignment result in slopes less than one when the names are reversed. Calculate. Bresenham's line algorithm is an algorithm that determines the points of an n-dimensional raster that should be selected in order to form a close approximation to a straight line between two points. Comparions between DDA and Bresenham Line Drawing algorithm. 27, Jul 17. Plot the point (x0, y0) 3. Bresenham Line Drawing Calculator By putting x1,x2 and y1,y2 Value it Show The Result In Step By Step order,and Result Brief Calculation Which Is Calculated by Bresenham Line Drawing Algorithm. At the sample position xk+1 the vertical separations from the line are labelled dupper and dlower. Having a fast line-drawing algorithm is essential in a game like Elite. implement bresenham line drawing algorithm for all type of slope View bresenham line drawing .cpp // bresenham line drawing.cpp : Defines the entry point for the console application. The algorithm can be generalized to conic sections. P 0 = 2dy - dx The program will work in Turbo C or Turbo C++ compiler as it uses graphics.h header file. It greatly simplifies line drawing by using only integer variables, and importantly removing that costly division operation for slope. 1. “The Line drawing algorithm is a graphical algorithm which is used to represent the line segment on discrete graphical media, i.e., printer and pixel-based media.” A line contains two points. Bresenham’s Line Drawing Algorithm, Rasmi M, St.Mary’s College Starting from the left endpoint (x0,y0) of a given line, we step to each successive column and plot the pixel whose scan-line y value is close to the line path. Step3: Enter value of x 1,y 1,x 2,y 2 Points on a circle are all the radius distance from the centre of the circle. Bresenham's Line Algorithm Lecture 3 / 3rd Class / 2016-2017 2 Ms. C. Rusul Mohammed Bresenham's algorithm is generalized to lines with arbitrary slope by considering the symmetry between the various octants and quadrants of the xy plane. As we examine the Bresenham algorithm for the shallow, negative slope case, we must choose between turning on two points: (Xk+1, Yk) or (Xk+1, , Yk-1) If Y is the actual value of the line, then the distance from the the first point to the actual line is . Line Drawing Algorithms- In computer graphics, Bresenham Line Drawing Algorithm is a famous line drawing algorithm. This method requires only sixteen-bit integers with addition, subtraction and multi- … To draw a line, you need two points between which you can draw a line. Often implemented in hardware? To turn it to a digital line, we want exactly one pixel per row or per column, whichever ensures a continuous line. Bresenham's Line-drawing Algorithm? The centre point has an X and Y coordinate. और हम जानते है कि कंप्यूटर integer addition तथा substraction के ऑपरेशन बहुत तेजी से करता है. To draw our line we are going to be using the standard line formula y = mx + b, where x and y are the coordinates of the pixel, m is the slope (deltay / deltax) of the line, and b is 0. Consider first a line with positive slope. As you progress away from an axis, the contour of the Those rows and columns are also known as Pixels. Bresenham's Line Drawing Algorithm in C and C++ - The Crazy Programmer Here you will get program for bresenham's line drawing algorithm in C and C++. This algorithm is used in computer graphics for drawing line. Here you will get program for bresenham's line drawing algorithm in C and C++. The line segment is the basic entity in virtually all computer graphics systems. Easy Tutor author of Program of Bresenham line drawing algorithm is from United States.Easy Tutor says . Scan conversion of Line and Line Drawing algorithms. DDA Line Algorithm The digital differential analyzer (DDA) is a scan conversion line algorithm based on calculation either Dy or Dx. Actually, x and y aren't really the coordinates of the pixel, the pixel's true coordinates are x1 + x and y1 + y. Suppose we have to draw a line PQ with coordinates P (x1, y1) and Q (x2, y2). To draw the line using Breshenam's line drawing algorithm, first of all, calculate the slope of the line from the given coordinates by using, There can be three values of slope (m) i.e. Dely = | y2 – y1 |. 6-11-1-PB.pdf. It is a basic element in graphics. Make sure to change the path of BGI folder inside initgraph() function according to your system. OR both x … A line connects two points. Uploaded by. Computer Graphics 4: Bresenham Line Drawing Algorithm, Circle Drawing - Title: Computer Graphics 4: Circle Drawing, Polygon Fill & Anti-Aliasing Algorithms Author: Brian Mac Namee Last modified by: USER Created Date: 8/22/2006 8:27:31 AM While drawing a line on computers they need to perform a set of computation, and it is not that simple as humans can do it in a simple way. Before we begin impementing the algorithm, it is advisable to revise the method for drawing line in an inefficient way. Here is a C++ implementation of the Bresenham algorithm for line segments in the first octant. The algorithm should start from the center of the 2d array, and "shoot out" a beam at angles between 0 - 2pi. Computer Graphics - Hearn & Baker. The pixel is incremented along the minor, or y-axis, only when a decision variable (based on the slope of the line) changes sign. Reply. Previously, I discussed the basic idea of Bresenham’s Line Algorithm. Lab Manual: Computer Graphics & Multimedia Department of Information Technology Experiment-2 Required Software/ Software Tool OS: Linux Operating System or Windows Operating System C/C++ TOOLS/APPARATUS: Turbo C or gcc / gprof compiler in linux. Bresenham Line Drawing Algorithm For Slope Less Than 1: Step 1: Get the line endpoints from the user. 2. Using the data storage type defined on the Bitmap page for raster graphics images, draw a line given two points with Bresenham's line algorithm. It is one of the earliest algorithms developed in the field of computer graphics. (X start, Y start) and (X end, Y end) are the endpoints of a line. DERIVATION OF THE BRESENHAM’S LINE ALGORITHM Assumptions : input: line endpoints at (X1,Y1) and (X2, Y2) X1 < X2 line slope ≤ 45 o, i.e. It is one of the earliest algorithms … Described herein are a system and method for drawing high-quality, mathematically perfect or near-perfect anti-aliased lines by using a modified integer Bresenham line-drawing algorithm that yields optimally accurate coverage values. Program for Bresenham’s Line Drawing Algorithm in C Bresenham Algorithms in C. to a straight line between two given points. The slope of the line can be greater than 1 (m>1) or less than or equal to 1 (m<=1). By Brahmi Jadhav. Now we can generalize the algorithm to handle all slopes and different orders of endpoints. Write a program to implement Bresenham’s line drawing algorithm with all values of slopes Bresenham's line drawing algorithm for drawing a line in a computer screen by using integer arithmetic operations only.It is more efficient that the older DDA Line drawing algorithm that uses floating-point arithmetic and rounding operations although it can be optimised to perform only integer operations still pixle selection using Besenham's gives better result than DDA. It is commonly used to draw lines on a computer screen, as it uses only integer addition, subtraction and bit shifting, all of which are very cheap operations in standard computer architectures. Hello Friends, I am Free Lance Tutor, who helped student in completing their homework. Implement Bresenham’s Line drawing algorithm for all types of slope. Rectangle using Bresenham's line. Bresenham’s algorithm is generaliz to strains with arbitrary slope by means of thinking about symmetry between the various octants and quadrants of the plane. a. m . Delx =| x2 – x1 |. I know that the bresenham algorithm might have to be modified to overcome slopes greater 1 and lower that 0. Note that this is a version of the standard linear equation ax + bx + c = 0. Bresenham’s line-drawing algorithm uses an iterative scheme. I add the pixel founded to a List to render point by point his element and be sure Bresenham checks the correct pixels (it does). So, our Line equation will be: Yk+1 = … For I = 0 to delx in step of 1. dx is also an integer value, so after dividing by 2, either the value was an integer, or we missed by 0.5 less than true value. Also Read: Bresenham’s Line Drawing Algorithm in C and C++. Meaning Bresenham algorithm: Efficient graphics rasterization. As we examine the Bresenham algorithm for the shallow, negative slope case, we must choose between turning on two points: (Xk+1, Yk) or (Xk+1, , Yk-1) If Yis the actual value of the line, then the distance from the the first point to the actual line is d1= Yk- Y and the distance from the actual line to the second point is d2= Y - (Yk- 1) We start with the parametric equation of a line, (X + t.Dx, Y + t.Dy), where t is a parameter in range [0, 1]. Computer Graphics C Version. bresenham's line algorithm c++. 6 2006 Wiley & Sons 11 Doubling Line-Drawing Speed Bresenham algorithm: Determines slope Chooses 1 pixel between 2 based on d Double-step algorithm: Halves the number of decisions by checking for next In Bresenham’s Line Drawing algorithm, we have to calculate the slope ( m) between the starting point and the ending point. But if the slope of line is greater then 1 (m>1). Background Theory: Bresenham’s Line Drawing Algorithm is an efficient and accurate raster line generating algorithm developed by Bresenham. "The Bresenham line algorithm is an algorithm which determines which points in an n-dimensional raster should be plotted in order to form a close approximation to a straight line between two given points. Bresenham Line Drawing Algorithm attempts to generate the points between the starting and ending coordinates. Also Read: Bresenham’s Midpoint Circle Algorithm in C and C++. c=yk-m(xk) P0 = 2dy (xk)-2dx (yk)+2dx (yk-m(xk)-dx (By using 2) = 2dy (xk)-2dx (yk)+2dxyk-2dyxk+2dy-dx. We can optimize it a bit, by noting that (eps << 1) >= dx Is equivalent to: eps >= (dx / 2) Even if it would seem that right shifting dx would lose the least significant bit, we can handle it by proper rounding: eps is an integer variable, so only has integer values. Simpler, with no doubleand no division (but be careful of integer overflow): We can take advantage of two facts to optimize this significantly: 1. glFlush(); //executes all OpenGL functions as quickly as possible } OUTPUT: ***** Bresenham’s Line Drawing Algoirhtm ***** Enter Starting vertex (x1, y1): 100 200 Enter Ending Vertex (x2, y2): 150 350 in standard computer architectures. DDA algorithm for circles centered at the origin 15 7. Midpoint circle algorithm. Step 2: Calculate dx, dy, 2dy and (2dy - 2dx) dx = X end - X start dy = Y end - Y start Step 3: Find the initial value of the decision parameter(P). draw are circles. Other Uses for the Line Algorithm A line can be represented by the equation y = mx + c, where m = deltay / deltax. I also guide them in doing their final year projects. This distance is called an error or decision parameter. In other words, each plotted point must be the closest grid point to the desired line. A better representation with multiple color gradations requires an advanced process, spatial anti-aliasing. Browse for more programs and questions. This article is about the small modification of the original Bresenham's line drawing algorithm, considering the base yk=m(xk)+c. Part 2 – Bresenham’s Line Drawing Algorithm Your second program will additionally test your GBA_drawLine() function without anti-aliasing. This algorithm is used in computer graphics for drawing line. cpp by Healthy Hawk on Apr 05 2021 Donate. Computer Graphics 4: Bresenham Line Drawing Algorithm, Circle Drawing - Title: Computer Graphics 4: Circle Drawing, Polygon Fill & Anti-Aliasing Algorithms Author: Brian Mac Namee Last modified by: USER Created Date: 8/22/2006 8:27:31 AM It involves only integer calculation so it is accurate and fast. So to draw smooth lines, you should want to look into a different algorithm. The ability to draw a straight line graphically is not included in some Forth im- plementations, nor is it to be found in many ROM support programs. Bresenham algorithm for general lines 9 5. 0 < m ≤ 1 x coordinate is incremented in steps of 1, y coordinate is computed generic line equation: y = mx + b x i ix +1 y i y +1i y = mx + b y d1 d2 Derivation These coverage values are derived from the Bresenham algorithm itself without the computational expense of an arithmetic division at each pixel. An algorithm similar to Bresenham's line drawing algorithm, called the Midpoint Circle Algorithm, has been developed for drawing a circle efficiently. Goals for line drawing algorithms 2 2. Task. Incremental (new pixel from old)? The actual coded implementation will reveal many possible efficiency considerations. To draw the line we have to compute first the slope of the line form two given points. In computer graphics, a line drawing algorithm is an algorithm for approximating a line segment on discrete graphical media, such as pixel-based displays and printers.On such media, line drawing requires an approximation (in nontrivial cases). Your program should be able to draw lines of varying slopes and directions using Bresenham’s line drawing algorithm discussed in class. 1. then our Y coordinate will always be incremented and we have to choose between xk or xk+1. Horizontal and vertical lines are treat ed as special cases for greater speed and simplicity. Points on a circle are all the radius distance from the centre of the circle. Bresenham Line Drawing Algorithm determines the points of an n-dimensional raster that should be selected in order to form a close approximation to a straight line between two points. Now, keeping in mind all the above points and calculations, here is the Bresenham algorithm for slope m < 1 − Step 1 − Input the two end-points of line, storing the left end-point in $(x_{0}, y_{0})$. Lee Chan Peter. Which are used in the calculation adder, so greatly reducing the cost of the program. Let's look at how that works. To draw the line using Breshenam's line drawing algorithm, first of all, calculate the slope of the line from the given coordinates by using, m = dy/dx Where, dy = x2 - x1 dx = y2 - y1 There can be three values of slope (m) i.e. For this, Bresenham’s Line-Drawing Algorithm always increments by one unit in either x or y depending on the slope of the line. Efficiency considerations for circle generators 14 6. In computer graphics, the midpoint circle algorithm is an algorithm used to determine the points needed for rasterizing a circle. Used in most graphics packages? Bresenham Line Drawing Algorithm contains two phases : 12 The Bresenham Line-Drawing Algorithm by Phil Koopman, Jr. I have deliberately left out my version in this document to allow an unbiased interpretation of the Bresenham derivation. Draw straight segments (MFC) is We perform Bresenham’s algorithm to the line with a positive slope and reflect back around the X-axis to get the pixels. #include. void linev6 (Screen &s, unsigned x1, unsigned y1, unsigned x2, unsigned y2, unsigned char colour ) { int dx = x2 - x1, dy = y2 - y1, y = y1, eps = 0; for ( int x = x1; x <= x2; x++ ) { s.Plot (x,y,colour); eps += dy; if ( (eps << 1) >= dx ) { y++; eps -= dx; } } } I am trying the iterate the 2d array in a circular manner, using bresenham line algorithm . 1 b. m > 1 c. m = 1 The slope of a line plays a major role in the line equation that's why Bresenham line drawing algorithm calculates the equation according to the slope of the line. There are the following properties of a good Line Drawing Algorithm. J.E. Now enough talking let's derive the equations. The common objective in creating a viable line drawing algorithm is to plot points of a grid-restricted line which most closely match an ideal continuous line joining the end points. A pixel is plotted at the starting coordinate of the line, and each iteration of the algorithm increments the pixel one unit along the major, or x-axis. As shown in the above figure let, we have initial coordinates of a line = ( xk, yk) The next coordinates of a line = (xk+1, yk+1) The intersection point between yk and yk+1 = y. The problem is that my algorithm doesn't cover all the cases but about 50% of them. 03, Jun 19.
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