Function to create Reuleaux Polygons on an HTML 5 canvas

Question

I am working on a project to design systematic artworks using an HTML 5 canvas. To give my artworks a more organic and diverse feel I wanted to have a function that created reuleaux polygons. I think there may be a way to convert my draw_sharp_polygon(center_position, radius, number_of_sides, regular, anticlockwise) function into the function I need but I'm unsure of how to do this. Would I need to use a large amount of context.lineTo() functions or would I be able to use the context.arcTo() function in some way?

function draw_sharp_polygon(center_position, radius, number_of_sides, regular, anticlockwise)
{
    if(typeof center_position == 'undefined')
        center_position = new Position();
    if(typeof radius == 'undefined')
        radius = dice_roll(diagonal);
    if(typeof number_of_sides == 'undefined' || number_of_sides < 3)
        number_of_sides = dice_roll(10);
    if(typeof regular == 'undefined')
        regular = coin_toss();
    if(typeof anticlockwise == 'undefined')
        anticlockwise = coin_toss();
    context.moveTo(center_position.x + radius, center_position.y)
    if(regular)
    {
        var circular_angle_division = (Math.PI * 2)/number_of_sides;
        circular_angle_division = anticlockwise ? -1 * circular_angle_division : circular_angle_division;
        for(var i = 1; i < number_of_sides; i++)
        {
            context.lineTo(radius * Math.cos(circular_angle_division * i),radius * Math.sin(circular_angle_division * i));
        }
    }
    else
    {
        var amount_of_circle_taken = 0;
        var direction = anticlockwise ? -1 : 1;
        var sides_left = number_of_sides;
        for(var i = 1; i < number_of_sides; i++)
        {
            if(i < number_of_sides -1)
            {
                var circular_angle_division = get_random_value(1, (((Math.PI * 2) - amount_of_circle_taken)/number_of_sides*(sides_left / 2)));
                amount_of_circle_taken += circular_angle_division;
            }
            else
            {
                var circular_angle_division = (Math.PI * 2) - amount_of_circle_taken;
            }
            context.lineTo(radius * Math.cos(direction * circular_angle_division * i),radius * Math.sin(direction * circular_angle_division * i));
        }
    }
}

That's what I have for my flat edged polygons. I was wondering if there was a way to use context.arcTo() instead of context.lineTo() to create reuleaux polygons.

I was going to post an example of a Reuleaux Triangle but I don't have enough reputation. Wikipedia has a great example though.

P.S. I don't use jQuery in any of my personal projects yet because I feel that most of them could become part of a stand alone javascript library. So please no answers using jQuery.

Answer 1

The tangent-point needed for arcTo() is actually on the circumcircle, halfway between the arc endpoints. The radius needed could be calculated with:

reuleaux_radius = radius * sqrt(2 + 2*cos(pi / number_of_sides))

which is equivalent to

reuleaux_radius = radius * sqrt(2 + 2*cos(angle2 - angle1))

Full code:

var taget = document.getElementById('target');
var context = target.getContext('2d');

var sides = 3;
var radius = 100;
var center = new Position(150,150);

// normal polygon
context.beginPath();
draw_sharp_polygon(center, radius, sides);
context.strokeStyle = 'silver';
context.stroke();

// circle
context.beginPath();
context.arc(center.x, center.y, radius, 0, 2*Math.PI, false);
context.strokeStyle = 'silver';
context.stroke();

// reuleaux polygon
context.beginPath();
draw_reuleaux_polygon(center,radius,sides);
context.strokeStyle = 'black';
context.stroke();

function Position(x, y)
{
    this.x = x || 0;
    this.y = y || 0;
}

function draw_reuleaux_polygon(center_position, radius, number_of_sides)
{
    if(typeof center_position == 'undefined')
        throw new Error("center_position not defined");
    if(typeof radius == 'undefined')
        throw new Error("radius not defined");
    if(typeof number_of_sides == 'undefined' || number_of_sides < 3)
        throw new Error("number_of_sides not defined");

    context.moveTo(center_position.x + radius, center_position.y);
    for (var index1 = 0; index1 < number_of_sides; index1++)
    {
        // point 1 = arc start
        // point 2 = tangent intersection
        // point 3 = arc end
        var index2 = (index1 + 0.5) % number_of_sides;
        var index3 = (index1 + 1) % number_of_sides;
        var angle1 = index1*2*Math.PI/number_of_sides;
        var angle2 = index2*2*Math.PI/number_of_sides;
        var angle3 = index3*2*Math.PI/number_of_sides;
        var p1 = new Position(center_position.x + radius*Math.cos(angle1), center_position.y + radius*Math.sin(angle1));
        var p2 = new Position(center_position.x + radius*Math.cos(angle2), center_position.y + radius*Math.sin(angle2));
        var p3 = new Position(center_position.x + radius*Math.cos(angle3), center_position.y + radius*Math.sin(angle3));

        var reuleaux_radius = radius*Math.sqrt(2 + 2*Math.cos(Math.PI/number_of_sides));
        
        context.arcTo(p2.x, p2.y, p3.x, p3.y, reuleaux_radius);
    }
    
    context.closePath();
}

function draw_sharp_polygon(center_position, radius, number_of_sides)
{
    if(typeof center_position == 'undefined')
        throw new Error("center_position not defined");
    if(typeof radius == 'undefined')
        throw new Error("radius not defined");
    if(typeof number_of_sides == 'undefined' || number_of_sides < 3)
        throw new Error("number_of_sides not defined");
    
    context.moveTo(center_position.x + radius, center_position.y);
    var circular_angle_division = (Math.PI * 2)/number_of_sides;
    for(var i = 1; i < number_of_sides; i++)
    {
        var x = center_position.x + radius * Math.cos(circular_angle_division * i);
        var y = center_position.y + radius * Math.sin(circular_angle_division * i);
        console.log(i,x,y);
        context.lineTo(x,y);
    }
    
    context.closePath();
}

<canvas id="target" width="300" height="300"></canvas>

http://jsfiddle.net/hebvseju/