Sign in to follow this  
tonemgub

2D Triangle rasterization algorithm

Recommended Posts

Hello,

I’m writing research paper on software rasterization algorithms and at one point I gave example of triangle rasterization algorithm.

The algorithm is really basic. If the triangle is flat top or flat bottom it’s possible to determine the minimum and the maximum x values for each scan line using the equation of line for the edges. Then for each scan line fill the pixels between minimum x and maximum x values.
If the triangle is of other kind it’s possible to split it to flat top and flat bottom triangles (finding the fourth vertex) and draw it using the previous algorithm.

I need to cite a reference for this algorithm. I saw it in some book in the ‘90s and I can’t just write it without a reference. The problem is that I can’t remember where I saw it.
I already tried to look at “Computer Graphics: Principles and Practice” but the only similar algorithm there is the polygon rasterization algorithm, which is over engineered for this kind of problem, same with "Computer Graphics: C Version".
I also tried to look at “Black Art of 3D Game Programming”, which have similar algorithm but the algorithm that I saw was in another book and slightly different.

Anyone know a book with this kind of algorithm?
Any help is appreciated.

Thanks.

Share this post


Link to post
Share on other sites

Michael Abrash's Graphics Programming Black Book might be another reference, but polygons too IIRC.

Note that rendering polygons is much more efficient than triangles due to less edges to setup - if there is enough planar stuff around.

 

Later there was a SIMD approach which does a bounding rect per triangle and then evaluating 3 side of edge tests in parallel per pixel. I remember a guy named Nicolas Capens who wrote an article about this, might be another good reference.

 

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now

Sign in to follow this  

  • Announcements

  • Forum Statistics

    • Total Topics
      628326
    • Total Posts
      2982086
  • Similar Content

    • By Connor Rust
      I am currently attempting to make a navigation mesh for our 2D top down game, which is a multiplayer game using Node.js as the server communication. At the moment, I have implemented A* over an obstacle hardnessmap, which is awfully slow and laggy at times when we test our game on Heroku. I have been trying to find an algorithm to automatically generate the navmesh after map creation, instead of me having to do this manually. I am currently attempting to use Delaunay's Triangulation Divide and Conquer algorithm, but I am running into some issues. I have already asked a question on StackOverflow and am not getting many suggestions and help from it, so I figured I would come here. Is there another algorithm that might be better to use for the navmesh generation in comparison to Deluanay's Triangulation? My current implementation seems extremely buggy during the merge step and I cannot find the error. I have checked over the code countless times, comparing it to the description of the algorithm from http://www.geom.uiuc.edu/~samuelp/del_project.html. 
      My current code is this:
      class MapNode { constructor(x, y) { this.position = new Vector(x, y); this.neighbors = []; } distance(n) { return this.position.distance(n.position); } inNeighbor(n) { for (let i = 0; i < this.neighbors.length; i++) { if (this.neighbors[i] === n) return true; } return false; } addNeighbor(n) { this.neighbors = this.neighbors.filter((node) => node != n); this.neighbors.push(n); } addNeighbors(arr) { let self = this; arr.forEach((n) => self.neighbors.push(n)); } removeNeighbor(n) { this.neighbors = this.neighbors.filter((neighbor) => neighbor != n); } } class Triangle { constructor(p1, p2, p3) { this.p1 = p1; this.p2 = p2; this.p3 = p3; this.neighbors = []; } addNeighbors(n) { this.neighbors.push(n); } } function genSubMat(matrix, ignoreCol) { let r = []; for (let i = 0; i < matrix.length - 1; i++) { r.push([]); for (let j = 0; j < matrix[0].length; j++) { if (j != ignoreCol) r[i].push(matrix[i + 1][j]); } } return r; } function determinantSqMat(matrix) { if (matrix.length != matrix[0].length) return false; if (matrix.length === 2) return matrix[0][0] * matrix[1][1] - matrix[1][0] * matrix[0][1]; let det = 0; for (let i = 0; i < matrix.length; i++) { let r = genSubMat(matrix, i); let tmp = matrix[0][i] * determinantSqMat(r); if (i % 2 == 0) det += tmp; else det -= tmp; } return -det; } // if d is in the circle formed by points a, b, and c, return > 0 // d is on circle, return 0 // d is outside of circle, return < 0 function inCircle(a, b, c, d) { let arr = [a, b, c, d]; let mat = [ [], [], [], [] ]; for (let i = 0; i < arr.length; i++) { mat[i][0] = 1; mat[i][1] = arr[i].position.x; mat[i][2] = arr[i].position.y; mat[i][3] = arr[i].position.x * arr[i].position.x + arr[i].position.y * arr[i].position.y; } return determinantSqMat(mat); } function walkable(from, to, hardnessMap) { let diff = new Vector(to.x - from.x, to.y - from.y); if (Math.abs(diff.x) > Math.abs(diff.y)) diff.scale(Math.abs(1 / diff.x)); else diff.scale(Math.abs(1 / diff.y)); let current = new Vector(from.x + diff.x, from.y + diff.y); while (Math.round(current.x) != to.x || Math.round(current.y) != to.y) { if (hardnessMap[Math.floor(current.y)][Math.floor(current.x)] === 1) return false; current.x += diff.x; current.y += diff.y; } return true; } function getLowest(nodes) { let lowest = nodes[0]; for (let i = 1; i < nodes.length; i++) { if (nodes[i].position.y < lowest.position.y) lowest = nodes[i]; } return lowest; } // returns the angle between 2 vectors, if cw is true, then return clockwise angle between, // else return the ccw angle between. b is the "hinge" point function angleBetween(a, b, c, cw) { let ba = new Vector(a.position.x - b.position.x, a.position.y - b.position.y); let bc = new Vector(c.position.x - b.position.x, c.position.y - b.position.y); let v0 = new Vector(0, 1); let angleBA = v0.angleBetween(ba) * 180 / Math.PI; if (angleBA < 0) angleBA += 360; let angleBC = v0.angleBetween(bc) * 180 / Math.PI; if (angleBC < 0) angleBC += 360; let smallest = Math.min(angleBA, angleBC); let largest = Math.max(angleBA, angleBC); let angle = largest - smallest; return (cw) ? angle : 360 - angle; } function sortSmallestAngle(a, b, list, cw) { list.sort((m, n) => { let vab = new Vector(a.position.x - b.position.x, a.position.y - b.position.y); let vmb = new Vector(m.position.x - b.position.x, m.position.y - b.position.y); let vnb = new Vector(n.position.x - b.position.x, n.position.y - b.position.y); if (cw) return vab.angleBetween(vmb, cw) - vab.angleBetween(vnb, cw); else return vab.angleBetween(vnb, cw) - vab.angleBetween(vmb, cw); }); } // a is in list, b is in the other list function getPotential(a, b, list, cw) { sortSmallestAngle(b, a, list, cw); for (let i = 0; i < list.length - 1; i++) { let angle = angleBetween(b, a, list[i], cw); if (angle > 180) return false; else if (inCircle(a, b, list[i], list[i + 1]) <= 0) return list[i]; else { a.removeNeighbor(list[i]); list[i].removeNeighbor(a); } } let potential = list[list.length - 1]; if (potential) { let angle = angleBetween(a, b, potential, cw); if (angle > 180) return false; return potential; } return false; } function merge(leftNodes, rightNodes, leftBase, rightBase, hardnessMap) { leftBase.addNeighbor(rightBase); rightBase.addNeighbor(leftBase); let newLeft = leftNodes.filter((n) => n != leftBase); let newRight = rightNodes.filter((n) => n != rightBase); let potentialLeft = getPotential(leftBase, rightBase, newLeft, false); let potentialRight = getPotential(rightBase, leftBase, newRight, true); if (!potentialLeft && !potentialRight) return; else if (potentialLeft && !potentialRight) merge(newLeft, newRight, potentialLeft, rightBase, hardnessMap); else if (potentialRight && !potentialLeft) merge(newLeft, newRight, leftBase, potentialRight, hardnessMap); else { if (inCircle(leftBase, rightBase, potentialLeft, potentialRight) <= 0) merge(newLeft, newRight, potentialLeft, rightBase, hardnessMap); if (inCircle(leftBase, rightBase, potentialRight, potentialLeft) <= 0) merge(newLeft, newRight, leftBase, potentialRight, hardnessMap); } } // divide and conquer algorithm function delaunay(nodes, hardnessMap) { if (nodes.length <= 3) { for (let i = 0; i < nodes.length; i++) for (let j = 0; j < nodes.length; j++) if (i != j) nodes[i].addNeighbor(nodes[j]); return nodes; } else { nodes.sort((a, b) => { let tmp = a.position.x - b.position.x; if (tmp === 0) return b.position.y - a.position.y; return tmp; }); let l = nodes.length; let leftNodes; let rightNodes; if (l === 4) { leftNodes = delaunay(nodes.slice(0, 3), hardnessMap); rightNodes = delaunay(nodes.slice(3, 4), hardnessMap); } else { leftNodes = delaunay(nodes.slice(0, Math.floor(nodes.length / 2)), hardnessMap); rightNodes = delaunay(nodes.slice(Math.floor(nodes.length / 2), nodes.length), hardnessMap); } let leftBase = getLowest(leftNodes); let rightBase = getLowest(rightNodes); merge(leftNodes, rightNodes, leftBase, rightBase, hardnessMap); console.log("=============================MergeComplete================================"); return nodes; } }  
    • By Hilster
      Hello 2D Artists,
      I've started making a 2D Puzzle Adventure game for mobile and I'm looking for someone who would want in on creating assets for the game. The core of the programming is pretty much complete, you can walk within the grid laid out and push boxes, when there is an object on top of a pressure pad it will activate the linked objects or if there is one object with multiple linked pressure pads it requires you to activate all points for the object to become active. 

      The level iteration for the game is quick and simple, a Photoshop file that is made of individual pixels that represents objects is put into the game and it creates the level out of those pixels with the assigned objects.
      The objects that need sprites created so far is the character, box, pressure pad, door, trap door, the walls, the stairs and the tiled background.
      I intend to add more objects so the amount I'd like to add will be extended.
      My motivations for posting here is to have something that looks nice to be able to display on my portfolio, so if you're looking for a working game that you can place your art into and improve the look of your portfolio then we're in business.
      Please reply with a few past examples of your art below and I'll be in touch!
    • By suliman
      Hi!
      My game is coming along nicely and I would love some feedback.
      You play as one (or two in co-op) survivor that must travel the land and survive the infected hordes, looners and bandits. You stop in locations but are always pressed as the hordes will start pouring in. Collect resources (food, fuel, medical supplies and ammo) and weapons and head for the goal!
      Tips
      Always quickly switch to a melee weapon if running out of ammo Loot everything if you have time to loot, including cartrunks Choose locations that have the loot you need (such as gas station for fuel) Try to avoid running out of fuel or having your car break down. Walking is dangerous! Download (50 MB, works with windows only, you DON'T need dropbox to download):
      Damnation road (beta 2)




    • By Brian Paek
      Football Dash now on iOS! Over 1 million downloads on Android
      iOS:
      https://itunes.apple.com/us/app/football-dash-endless-runner/id1312590451?ls=1&mt=8
      Android:
      https://play.google.com/store/apps/details?id=com.beastattack.c1434846484727

    • By zizulot
      first and only logo , for now
  • Popular Now