Axonometric rendering
The best way to handle axonometric (commonly called isometric) rendering is via a projection matrix.
A projection object as follows can describe all you need to do any form of axonometric projection
The object has 3 transforms for the x,y and z axis with each describing the scale and direction in the 2D projection for the x,y,z coordinates. A transform for the depth calculation and a origin that is in canvas pixels (if setTransform(1,0,0,1,0,0) or whatever the current transform for the canvas is)
To project a point call the function axoProjMat({x=10,y=10,z=10}) and it will return a 3D point with x,y being 2D coordinates of the vertex and z being the depth (with depth values positive approaching the view (opposite to 3D perspective projection));
// 3d 2d points
const P3 = (x=0, y=0, z=0) => ({x,y,z});
const P2 = (x=0, y=0) => ({x, y});
// projection object
const axoProjMat = {
xAxis : P2(1 , 0.5) ,
yAxis : P2(-1 , 0.5) ,
zAxis : P2(0 , -1) ,
depth : P3(0.5,0.5,1) , // projections have z as depth
origin : P2(), // (0,0) default 2D point
setProjection(name){
if(projTypes[name]){
Object.keys(projTypes[name]).forEach(key => {
this[key]=projTypes[name][key];
})
if(!projTypes[name].depth){
this.depth = P3(
this.xAxis.y,
this.yAxis.y,
-this.zAxis.y
);
}
}
},
project (p, retP = P3()) {
retP.x = p.x * this.xAxis.x + p.y * this.yAxis.x + p.z * this.zAxis.x + this.origin.x;
retP.y = p.x * this.xAxis.y + p.y * this.yAxis.y + p.z * this.zAxis.y + this.origin.y;
retP.z = p.x * this.depth.x + p.y * this.depth.y + p.z * this.depth.z;
return retP;
}
}
With the above object you can use the function axoProjMat.setProjection(name) to select the projection type.
Below is the associated projection types as outlined on the wiki Axonometric projections plus two modifications commonly used in pixel art and games (prefixed with Pixel). Use axoProjMat.setProjection(name) where name is one of the projTypes property names.
const D2R = (ang) => (ang-90) * (Math.PI/180 );
const Ang2Vec = (ang,len = 1) => P2(Math.cos(D2R(ang)) * len,Math.sin(D2R(ang)) * len);
const projTypes = {
PixelBimetric : {
xAxis : P2(1 , 0.5) ,
yAxis : P2(-1 , 0.5) ,
zAxis : P2(0 , -1) ,
depth : P3(0.5,0.5,1) , // projections have z as depth
},
PixelTrimetric : {
xAxis : P2(1 , 0.5) ,
yAxis : P2(-0.5 , 1) ,
zAxis : P2(0 , -1) ,
depth : P3(0.5,1,1) ,
},
Isometric : {
xAxis : Ang2Vec(120) ,
yAxis : Ang2Vec(-120) ,
zAxis : Ang2Vec(0) ,
},
Bimetric : {
xAxis : Ang2Vec(116.57) ,
yAxis : Ang2Vec(-116.57) ,
zAxis : Ang2Vec(0) ,
},
Trimetric : {
xAxis : Ang2Vec(126.87,2/3) ,
yAxis : Ang2Vec(-104.04) ,
zAxis : Ang2Vec(0) ,
},
Military : {
xAxis : Ang2Vec(135) ,
yAxis : Ang2Vec(-135) ,
zAxis : Ang2Vec(0) ,
},
Cavalier : {
xAxis : Ang2Vec(135) ,
yAxis : Ang2Vec(-90) ,
zAxis : Ang2Vec(0) ,
},
TopDown : {
xAxis : Ang2Vec(180) ,
yAxis : Ang2Vec(-90) ,
zAxis : Ang2Vec(0) ,
}
}
Example of True Isometric Projection.
The snippet is an simple example with the projection set to Isometric as detailed on the wiki link in the OP’s question and using the above functions and objects.
const ctx = canvas.getContext("2d");
// function creates a 3D point (vertex)
function vertex(x, y, z) { return { x, y, z}};
// an array of vertices
const vertices = []; // an array of vertices
// create the 8 vertices that make up a box
const boxSize = 20; // size of the box
const hs = boxSize / 2; // half size shorthand for easier typing
vertices.push(vertex(-hs, -hs, -hs)); // lower top left index 0
vertices.push(vertex(hs, -hs, -hs)); // lower top right
vertices.push(vertex(hs, hs, -hs)); // lower bottom right
vertices.push(vertex(-hs, hs, -hs)); // lower bottom left
vertices.push(vertex(-hs, -hs, hs)); // upper top left index 4
vertices.push(vertex(hs, -hs, hs)); // upper top right
vertices.push(vertex(hs, hs, hs)); // upper bottom right
vertices.push(vertex(-hs, hs, hs)); // upper bottom left index 7
const colours = {
dark: "#040",
shade: "#360",
light: "#ad0",
bright: "#ee0",
}
function createPoly(indexes, colour) {
return {
indexes,
colour
}
}
const polygons = [];
polygons.push(createPoly([1, 2, 6, 5], colours.shade)); // right face
polygons.push(createPoly([2, 3, 7, 6], colours.light)); // front face
polygons.push(createPoly([4, 5, 6, 7], colours.bright)); // top face
// From here in I use P2,P3 to create 2D and 3D points
const P3 = (x = 0, y = 0, z = 0) => ({x,y,z});
const P2 = (x = 0, y = 0) => ({ x, y});
const D2R = (ang) => (ang-90) * (Math.PI/180 );
const Ang2Vec = (ang,len = 1) => P2(Math.cos(D2R(ang)) * len,Math.sin(D2R(ang)) * len);
const projTypes = {
PixelBimetric : {
xAxis : P2(1 , 0.5) ,
yAxis : P2(-1 , 0.5) ,
zAxis : P2(0 , -1) ,
depth : P3(0.5,0.5,1) , // projections have z as depth
},
PixelTrimetric : {
xAxis : P2(1 , 0.5) ,
yAxis : P2(-0.5 , 1) ,
zAxis : P2(0 , -1) ,
depth : P3(0.5,1,1) ,
},
Isometric : {
xAxis : Ang2Vec(120) ,
yAxis : Ang2Vec(-120) ,
zAxis : Ang2Vec(0) ,
},
Bimetric : {
xAxis : Ang2Vec(116.57) ,
yAxis : Ang2Vec(-116.57) ,
zAxis : Ang2Vec(0) ,
},
Trimetric : {
xAxis : Ang2Vec(126.87,2/3) ,
yAxis : Ang2Vec(-104.04) ,
zAxis : Ang2Vec(0) ,
},
Military : {
xAxis : Ang2Vec(135) ,
yAxis : Ang2Vec(-135) ,
zAxis : Ang2Vec(0) ,
},
Cavalier : {
xAxis : Ang2Vec(135) ,
yAxis : Ang2Vec(-90) ,
zAxis : Ang2Vec(0) ,
},
TopDown : {
xAxis : Ang2Vec(180) ,
yAxis : Ang2Vec(-90) ,
zAxis : Ang2Vec(0) ,
}
}
const axoProjMat = {
xAxis : P2(1 , 0.5) ,
yAxis : P2(-1 , 0.5) ,
zAxis : P2(0 , -1) ,
depth : P3(0.5,0.5,1) , // projections have z as depth
origin : P2(150,65), // (0,0) default 2D point
setProjection(name){
if(projTypes[name]){
Object.keys(projTypes[name]).forEach(key => {
this[key]=projTypes[name][key];
})
if(!projTypes[name].depth){
this.depth = P3(
this.xAxis.y,
this.yAxis.y,
-this.zAxis.y
);
}
}
},
project (p, retP = P3()) {
retP.x = p.x * this.xAxis.x + p.y * this.yAxis.x + p.z * this.zAxis.x + this.origin.x;
retP.y = p.x * this.xAxis.y + p.y * this.yAxis.y + p.z * this.zAxis.y + this.origin.y;
retP.z = p.x * this.depth.x + p.y * this.depth.y + p.z * this.depth.z;
return retP;
}
}
axoProjMat.setProjection("Isometric");
var x,y,z;
for(z = 0; z < 4; z++){
const hz = z/2;
for(y = hz; y < 4-hz; y++){
for(x = hz; x < 4-hz; x++){
// move the box
const translated = vertices.map(vert => {
return P3(
vert.x + x * boxSize,
vert.y + y * boxSize,
vert.z + z * boxSize,
);
});
// create a new array of 2D projected verts
const projVerts = translated.map(vert => axoProjMat.project(vert));
// and render
polygons.forEach(poly => {
ctx.fillStyle = poly.colour;
ctx.strokeStyle = poly.colour;
ctx.lineWidth = 1;
ctx.beginPath();
poly.indexes.forEach(index => ctx.lineTo(projVerts[index].x , projVerts[index].y));
ctx.stroke();
ctx.fill();
});
}
}
}canvas {
border: 2px solid black;
}
body { font-family: arial; }True Isometric projection. With x at 120deg, and y at -120deg from up.<br>
<canvas id="canvas"></canvas>