Now that we have built the core of our 3D engine thanks to the previous tutorial Tutorial series- learning how to write a 3D soft engine from scratch in C#, TypeScript or JavaScript, we can work on enhancing the rendering. The next step is then to connect the dots to draw some lines in order to render what you probably know as a **“wireframe” rendering**.

1 – Writing the core logic for camera, mesh & device object

2 – Drawing lines and triangles to obtain a wireframe rendering (this article)

3 – Loading meshes exported from Blender in a JSON format

4 – Filling the triangle with rasterization and using a Z-Buffer

4b – Bonus: using tips & parallelism to boost the performance

5 – Handling light with Flat Shading & Gouraud Shading

6 – Applying textures, back-face culling and WebGL

In this tutorial, you will learn how to draw lines, what a face is and how cool is the Bresenham algorithm to draw some triangles.

Thanks to that, at the end, you will know how to code something as cool as that:

Yes! Our 3D rotating cube starts to really live on our screens!

## First basic algorithm to draw a line between two points

Let’s start by coding a simple algorithm. To **draw a line between 2 vertices**, we’re going to use the following logic:

– if the distance between the 2 points (point0 & point1) is less than 2 pixels, there’s nothing to do

– otherwise, we’re finding the **middle point** between both points (point0 coordinates + (point1 coordinates – point0 coordinates) / 2)

– we’re drawing that point on screen

– we’re launching this **algorithm recursively** between point0 & middle point and between middle point & point1

Here is the code to do that:

public void DrawLine(Vector2 point0, Vector2 point1) { var dist = (point1 - point0).Length(); // If the distance between the 2 points is less than 2 pixels // We're exiting if (dist < 2) return; // Find the middle point between first & second point Vector2 middlePoint = point0 + (point1 - point0)/2; // We draw this point on screen DrawPoint(middlePoint); // Recursive algorithm launched between first & middle point // and between middle & second point DrawLine(point0, middlePoint); DrawLine(middlePoint, point1); }

public drawLine(point0: BABYLON.Vector2, point1: BABYLON.Vector2): void { var dist = point1.subtract(point0).length(); // If the distance between the 2 points is less than 2 pixels // We're exiting if (dist < 2) return; // Find the middle point between first & second point var middlePoint = point0.add((point1.subtract(point0)).scale(0.5)); // We draw this point on screen this.drawPoint(middlePoint); // Recursive algorithm launched between first & middle point // and between middle & second point this.drawLine(point0, middlePoint); this.drawLine(middlePoint, point1); }

Device.prototype.drawLine = function (point0, point1) { var dist = point1.subtract(point0).length(); // If the distance between the 2 points is less than 2 pixels // We're exiting if(dist < 2) { return; } // Find the middle point between first & second point var middlePoint = point0.add((point1.subtract(point0)).scale(0.5)); // We draw this point on screen this.drawPoint(middlePoint); // Recursive algorithm launched between first & middle point // and between middle & second point this.drawLine(point0, middlePoint); this.drawLine(middlePoint, point1); };

You need to update the rendering loop to use this new piece of code:

for (var i = 0; i < mesh.Vertices.Length - 1; i++) { var point0 = Project(mesh.Vertices[i], transformMatrix); var point1 = Project(mesh.Vertices[i + 1], transformMatrix); DrawLine(point0, point1); }

for (var i = 0; i < cMesh.Vertices.length -1; i++){ var point0 = this.project(cMesh.Vertices[i], transformMatrix); var point1 = this.project(cMesh.Vertices[i + 1], transformMatrix); this.drawLine(point0, point1); }

for (var i = 0; i < cMesh.Vertices.length -1; i++){ var point0 = this.project(cMesh.Vertices[i], transformMatrix); var point1 = this.project(cMesh.Vertices[i + 1], transformMatrix); this.drawLine(point0, point1); }

And you should now obtain something like that:

I know this looks weird but this was the expected behavior. It should help you starting to understand what you need to do to display a 3D mesh. But to have a better rendering, we need to discover a new concept.

## Displaying faces with triangles

Now that we know how to draw lines, we need a better way to render the mesh with them. The **simplest geometric 2D shape is a triangle**. The idea in 3D is then to draw all our meshes by using those triangles. We then need to split each side of our cube into 2 triangles. We’re going to do this “manually” but we’ll see in the next tutorial that 3D modelers are doing this step automatically for us now.

To draw triangles, you need to have 3 points/vertices. A face is then simply a structure containing 3 values which are indexes pointing to the proper vertices array of the mesh to be rendered.

To be understand this concept, let’s take our previous figure with a Cube displayed by Blender:

We have 4 vertices displayed on this figure with the following indices: 0, 1, 2, 3. To draw the upper side of the cube, we need to draw 2 triangles. The first one, Face 0, will be drawn with 3 lines from vertex **0** (-1, 1, 1) to vertex **1** (1, 1, 1), from vertex **1** (1, 1, 1) to vertex **2 **(-1, –1, 1) and finally from vertex** 2** (-1, –1, 1) to vertex** 0** (-1, 1, 1). The second triangle, Face 1, will be drawn with the lines from vertex **1 **to vertex **2**, vertex **2** to vertex **3** and vertex **3 **to vertex **1**.

The equivalent code would be something like that:

var mesh = new SoftEngine.Mesh("Square", 4, 2); meshes.Add(mesh); mesh.Vertices[0] = new Vector3(-1, 1, 1); mesh.Vertices[1] = new Vector3(1, 1, 1); mesh.Vertices[2] = new Vector3(-1, -1, 1); mesh.Vertices[3] = new Vector3(1, -1, 1); mesh.Faces[0] = new Face { A = 0, B = 1, C = 2 }; mesh.Faces[1] = new Face { A = 1, B = 2, C = 3 };

If you want to draw to whole cube, you need to find the 10 remaining faces as we’ve got **12 faces** for the 6 sides of our cube to draw.

Let’s now define the code for a **Face** object. It’s a very simple object as this is just **a set of 3 indexes**. Here’s the code of Face and the new Mesh definition which also now use it:

namespace SoftEngine { public struct Face { public int A; public int B; public int C; } public class Mesh { public string Name { get; set; } public Vector3[] Vertices { get; private set; } public Face[] Faces { get; set; } public Vector3 Position { get; set; } public Vector3 Rotation { get; set; } public Mesh(string name, int verticesCount, int facesCount) { Vertices = new Vector3[verticesCount]; Faces = new Face[facesCount]; Name = name; } } }

///<reference path="babylon.math.ts"/> module SoftEngine { export interface Face { A: number; B: number; C: number; } export class Mesh { Position: BABYLON.Vector3; Rotation: BABYLON.Vector3; Vertices: BABYLON.Vector3[]; Faces: Face[]; constructor(public name: string, verticesCount: number, facesCount: number) { this.Vertices = new Array(verticesCount); this.Faces = new Array(facesCount); this.Rotation = new BABYLON.Vector3(0, 0, 0); this.Position = new BABYLON.Vector3(0, 0, 0); } } }

var SoftEngine; (function (SoftEngine) { var Mesh = (function () { function Mesh(name, verticesCount, facesCount) { this.name = name; this.Vertices = new Array(verticesCount); this.Faces = new Array(facesCount); this.Rotation = new BABYLONTS.Vector3(0, 0, 0); this.Position = new BABYLONTS.Vector3(0, 0, 0); } return Mesh; })(); SoftEngine.Mesh = Mesh; })(SoftEngine || (SoftEngine = {}));

We now need to update our **Render()** function/method of our **Device **object to iterate through all the faces defined and to draw the triangles associated.

foreach (var face in mesh.Faces) { var vertexA = mesh.Vertices[face.A]; var vertexB = mesh.Vertices[face.B]; var vertexC = mesh.Vertices[face.C]; var pixelA = Project(vertexA, transformMatrix); var pixelB = Project(vertexB, transformMatrix); var pixelC = Project(vertexC, transformMatrix); DrawLine(pixelA, pixelB); DrawLine(pixelB, pixelC); DrawLine(pixelC, pixelA); }

for (var indexFaces = 0; indexFaces < cMesh.Faces.length; indexFaces++) { var currentFace = cMesh.Faces[indexFaces]; var vertexA = cMesh.Vertices[currentFace.A]; var vertexB = cMesh.Vertices[currentFace.B]; var vertexC = cMesh.Vertices[currentFace.C]; var pixelA = this.project(vertexA, transformMatrix); var pixelB = this.project(vertexB, transformMatrix); var pixelC = this.project(vertexC, transformMatrix); this.drawLine(pixelA, pixelB); this.drawLine(pixelB, pixelC); this.drawLine(pixelC, pixelA); }

for (var indexFaces = 0; indexFaces < cMesh.Faces.length; indexFaces++) { var currentFace = cMesh.Faces[indexFaces]; var vertexA = cMesh.Vertices[currentFace.A]; var vertexB = cMesh.Vertices[currentFace.B]; var vertexC = cMesh.Vertices[currentFace.C]; var pixelA = this.project(vertexA, transformMatrix); var pixelB = this.project(vertexB, transformMatrix); var pixelC = this.project(vertexC, transformMatrix); this.drawLine(pixelA, pixelB); this.drawLine(pixelB, pixelC); this.drawLine(pixelC, pixelA); }

We finally need to declare the mesh associated with our **Cube** properly with its **12 faces** to make this new code working as expected.

Here is the new declaration:

var mesh = new SoftEngine.Mesh("Cube", 8, 12); meshes.Add(mesh); mesh.Vertices[0] = new Vector3(-1, 1, 1); mesh.Vertices[1] = new Vector3(1, 1, 1); mesh.Vertices[2] = new Vector3(-1, -1, 1); mesh.Vertices[3] = new Vector3(1, -1, 1); mesh.Vertices[4] = new Vector3(-1, 1, -1); mesh.Vertices[5] = new Vector3(1, 1, -1); mesh.Vertices[6] = new Vector3(1, -1, -1); mesh.Vertices[7] = new Vector3(-1, -1, -1); mesh.Faces[0] = new Face { A = 0, B = 1, C = 2 }; mesh.Faces[1] = new Face { A = 1, B = 2, C = 3 }; mesh.Faces[2] = new Face { A = 1, B = 3, C = 6 }; mesh.Faces[3] = new Face { A = 1, B = 5, C = 6 }; mesh.Faces[4] = new Face { A = 0, B = 1, C = 4 }; mesh.Faces[5] = new Face { A = 1, B = 4, C = 5 }; mesh.Faces[6] = new Face { A = 2, B = 3, C = 7 }; mesh.Faces[7] = new Face { A = 3, B = 6, C = 7 }; mesh.Faces[8] = new Face { A = 0, B = 2, C = 7 }; mesh.Faces[9] = new Face { A = 0, B = 4, C = 7 }; mesh.Faces[10] = new Face { A = 4, B = 5, C = 6 }; mesh.Faces[11] = new Face { A = 4, B = 6, C = 7 };

var mesh = new SoftEngine.Mesh("Cube", 8, 12); meshes.push(mesh); mesh.Vertices[0] = new BABYLON.Vector3(-1, 1, 1); mesh.Vertices[1] = new BABYLON.Vector3(1, 1, 1); mesh.Vertices[2] = new BABYLON.Vector3(-1, -1, 1); mesh.Vertices[3] = new BABYLON.Vector3(1, -1, 1); mesh.Vertices[4] = new BABYLON.Vector3(-1, 1, -1); mesh.Vertices[5] = new BABYLON.Vector3(1, 1, -1); mesh.Vertices[6] = new BABYLON.Vector3(1, -1, -1); mesh.Vertices[7] = new BABYLON.Vector3(-1, -1, -1); mesh.Faces[0] = { A:0, B:1, C:2 }; mesh.Faces[1] = { A:1, B:2, C:3 }; mesh.Faces[2] = { A:1, B:3, C:6 }; mesh.Faces[3] = { A:1, B:5, C:6 }; mesh.Faces[4] = { A:0, B:1, C:4 }; mesh.Faces[5] = { A:1, B:4, C:5 }; mesh.Faces[6] = { A:2, B:3, C:7 }; mesh.Faces[7] = { A:3, B:6, C:7 }; mesh.Faces[8] = { A:0, B:2, C:7 }; mesh.Faces[9] = { A:0, B:4, C:7 }; mesh.Faces[10] = { A:4, B:5, C:6 }; mesh.Faces[11] = { A:4, B:6, C:7 };

var mesh = new SoftEngine.Mesh("Cube", 8, 12); meshes.push(mesh); mesh.Vertices[0] = new BABYLON.Vector3(-1, 1, 1); mesh.Vertices[1] = new BABYLON.Vector3(1, 1, 1); mesh.Vertices[2] = new BABYLON.Vector3(-1, -1, 1); mesh.Vertices[3] = new BABYLON.Vector3(1, -1, 1); mesh.Vertices[4] = new BABYLON.Vector3(-1, 1, -1); mesh.Vertices[5] = new BABYLON.Vector3(1, 1, -1); mesh.Vertices[6] = new BABYLON.Vector3(1, -1, -1); mesh.Vertices[7] = new BABYLON.Vector3(-1, -1, -1); mesh.Faces[0] = { A:0, B:1, C:2 }; mesh.Faces[1] = { A:1, B:2, C:3 }; mesh.Faces[2] = { A:1, B:3, C:6 }; mesh.Faces[3] = { A:1, B:5, C:6 }; mesh.Faces[4] = { A:0, B:1, C:4 }; mesh.Faces[5] = { A:1, B:4, C:5 }; mesh.Faces[6] = { A:2, B:3, C:7 }; mesh.Faces[7] = { A:3, B:6, C:7 }; mesh.Faces[8] = { A:0, B:2, C:7 }; mesh.Faces[9] = { A:0, B:4, C:7 }; mesh.Faces[10] = { A:4, B:5, C:6 }; mesh.Faces[11] = { A:4, B:6, C:7 };

You should now have this beautiful rotating cube:

Congrats! 🙂

## Enhancing the line drawing algorithm with Bresenham

There is an optimized way to draw our lines using the Bresenham’s line algorithm. It’s faster & sharper than our current simple recursive version. The story of this algorithm is fascinating. Please read the Wikipedia definition of this algorithm to discover how Bresenham build it and for which reasons.

Here are the versions of this algorithm in C#, TypeScript and JavaScript:

public void DrawBline(Vector2 point0, Vector2 point1) { int x0 = (int)point0.X; int y0 = (int)point0.Y; int x1 = (int)point1.X; int y1 = (int)point1.Y; var dx = Math.Abs(x1 - x0); var dy = Math.Abs(y1 - y0); var sx = (x0 < x1) ? 1 : -1; var sy = (y0 < y1) ? 1 : -1; var err = dx - dy; while (true) { DrawPoint(new Vector2(x0, y0)); if ((x0 == x1) && (y0 == y1)) break; var e2 = 2 * err; if (e2 > -dy) { err -= dy; x0 += sx; } if (e2 < dx) { err += dx; y0 += sy; } } }

public drawBline(point0: BABYLON.Vector2, point1: BABYLON.Vector2): void { var x0 = point0.x >> 0; var y0 = point0.y >> 0; var x1 = point1.x >> 0; var y1 = point1.y >> 0; var dx = Math.abs(x1 - x0); var dy = Math.abs(y1 - y0); var sx = (x0 < x1) ? 1 : -1; var sy = (y0 < y1) ? 1 : -1; var err = dx - dy; while (true) { this.drawPoint(new BABYLON.Vector2(x0, y0)); if ((x0 == x1) && (y0 == y1)) break; var e2 = 2 * err; if (e2 > -dy) { err -= dy; x0 += sx; } if (e2 < dx) { err += dx; y0 += sy; } } }

Device.prototype.drawBline = function (point0, point1) { var x0 = point0.x >> 0; var y0 = point0.y >> 0; var x1 = point1.x >> 0; var y1 = point1.y >> 0; var dx = Math.abs(x1 - x0); var dy = Math.abs(y1 - y0); var sx = (x0 < x1) ? 1 : -1; var sy = (y0 < y1) ? 1 : -1; var err = dx - dy; while(true) { this.drawPoint(new BABYLON.Vector2(x0, y0)); if((x0 == x1) && (y0 == y1)) break; var e2 = 2 * err; if(e2 > -dy) { err -= dy; x0 += sx; } if(e2 < dx) { err += dx; y0 += sy; } } };

In the render function, replace the call do DrawLine by DrawBline and you should notice that this is a bit more fluid & a bit more sharp:

If you’re observing it with attention, you should see that this version using Bresenham is less choppy than our first algorithm.

Again, you can **download the solutions** containing the source code:

– **C#** : SoftEngineCSharpPart2.zip

– **TypeScript **: SoftEngineTSPart2.zip

– **JavaScript** : SoftEngineJSPart2.zip or simply right-click –> view source on the embedded iframe

In next tutorial, you will learn how to **export some Meshes from Blender**, a free 3D modeler tool, into a JSON format. We will then **load this JSON file to display it with our wireframe engine**. Indeed, we already have everything setup to display much more complex meshes like these one:

See you in the third part: learning how to write a 3D soft engine in C#, TS or JS – loading meshes exported from Blender

Awesome tutorial! Looking forward to part 3

These are amazing! Keep up the great work!

wow

Why do you prefer to draw triangles instead of quads? (I've searched for it on the internet but there's much controversy about it.)

Triangles are the most simple 2D shape with area and you can build whatever polygon from them, but you will have more triangles than quads.

Isn't a better solution to use arrays with a variable size which fit their size to the number of vertexes of each face?

Why do you prefer to draw triangles instead of quads? (I've searched for it on the internet but there's much controversy about it.)

Triangles are the most simple 2D shape with area and you can build whatever polygon from them, but you will have more triangles than quads.

Isn't a better solution to use arrays with a variable size which fit their size to the number of vertexes of each face?

Because triangles can only be on one plane, quads might mess up stuff, if not all 4 points are on 1 plane.

Quads are easier to implement, but if you want to do it clean, then use triangles.

Or in other words, if a quad isn’t on a plane, the computer by itself doesnt know how to behave.

So the 3d model would look weird.

Yes I’m necroing this, but just to clear it up.

David,

Thanks for your wonderful works! To calculate the middlePoint better to use

C#: Vector2 middlePoint = (point0 + point1)/2;

TypeScript: var middlePoint = point0.add(point1).scale(0.5);

Javar: var middlePoint = point0.add(point1).scale(0.5);

Looking forward to see more from you!

Very informative tutorial. Looking forward to part 3.

Regards,

Shalin

http://creately.com

Hello David!

Thanks for the great tutorial series, I’ve really enjoyed it so far (I’ve just implemented .obj loading). I noticed that there’s one simple, yet crucial optimization you seem to have overlooked, and I wanted to bring it to your attention.

The Render() method quickly becomes a major bottleneck when rendering a model that has more than just a couple of vertices. The problem is the calling of Project() once for every vertex of every face. The result is that every vertex in the model is projected _multiple times_. And Project() – mostly because of the TransformCoordinate() it uses – is a costly method, so minimizing the amount of calls to it helps the performance.

Here, I made some calculations with the suzanne.obj model:

Vertices in the model: 507

Faces in the model: 968, each consisting of 3 vertices

Projections per frame:

Number of faces * 3

= 968 * 3

= 2904

Projections per vertex per frame:

Projections per frame / vertices in model

= 2904 / 507

= ~5.7

So, on average, a single vertex is projected 5.7 times per frame. So, the program is working as if it was rendering a model 5.7 times as complex (vertex-wise) as suzanne.obj.

Now, how many projections per second would be needed for a nice performance of 60 frames per second?

Unoptimized:

Number of projections per frame * 60

= 2904 * 60

= 174240

Optimized:

Number of vertices * 60

= 507 * 60

= 30420

The optimization? Simple, just add an array of Vector2 to mesh and make it hold as many elements as the vertex array. Then, before rendering a mesh, reset the array. Before projecting a vertex, check if the matching index in the Vector2 array is set and if so, use that value and skip the projection. Otherwise do the projection and save the result in the array with an index matching that of the vertex in question. Now every vertex is projected only once per frame.

For my implementation of the engine this optimization alone doubled the performance for the suzanne.obj model. From 20 fps to 40. Of course, the optimization comes at the prize of memory, and it’s up to the developer to decide if it’s worth it or not.

———-

I’ve been implementing the engine on this rather old game creation software called Game Editor. It’s a 2D engine without support for using external libraries. So I had to implement all the matrix operations by myself (found great help from .NET’s reference source, though). I still don’t quite understand the way the 3D math works, but I’ve become a little more familiar with matrices and 3D transformations. The project can be found here: https://github.com/lclMetal/software3D

Again, thanks for the tutorial!

Lassi

Thanks for this great feedback Lassi!