Convex Combination

A convex combination of three vectors in the plane allows us to represent any vector in the convex hull determined by those three vectors.

<!DOCTYPE html>
<meta charset="utf-8">
<style>

svg {
  pointer-events: all;
  cursor: none;
}

.axis path,
.axis line {
  fill: none;
  stroke: #a2a2a2;
  shape-rendering: crispEdges;
}

.axis text {
  fill: #a2a2a2;
  pointer-events: none;
}

.point {
  stroke: #000000;
  stroke-width: .5px;
}

#marker {
  fill: #5c5c5c;
  stroke: none;
  stroke-width: 2px;
}

.vector {
  stroke: #5c5c5c;
  stroke-dasharray: 5, 5;
  stroke-width: 2px;
}

.vstar {
  marker-end: url(#marker);
  stroke: #000000;
  stroke-width: 2px;
}

.support {
  stroke-linecap: round;
  stroke-width: 6px;
}

</style>
<svg width="960" height="500">
  <marker id="marker"
      viewBox="0 0 6 6"
      refY="3"
      markerWidth="4"
      markerHeight="4"
      orient="auto">
    <path d="M0,3v-3l6,3l-6,3z"></path>
  </marker>
</svg>
<script src="https://cdnjs.cloudflare.com/ajax/libs/d3/3.5.6/d3.min.js" charset="utf-8"></script>
<script src="numeric.min.js"></script>
<script>

var margin = {top: 20, right: 20, bottom: 20, left: 20},
    padding = {top: 40, right: 60, bottom: 40, left: 60},
    outerWidth = 960,
    outerHeight = 500,
    innerWidth = outerWidth - margin.left - margin.right,
    innerHeight = outerHeight - margin.top - margin.bottom,
    width = innerWidth - padding.left - padding.right,
    height = innerHeight - padding.top - padding.bottom
    radius = 5;

var svg = d3.select("svg").on("mousemove", mousemove);

var svg = d3.select("svg").append("g")
            .attr("transform", "translate(" + margin.left + "," + margin.top + ")");

var g = svg.append("g")
            .attr("transform", "translate(" + padding.left + "," + padding.top + ")");

var color = d3.scale.category10();

var x = d3.scale.linear().domain([0, width]).range([0, width]);
var y = d3.scale.linear().domain([0, height]).range([height, 0]);

var xAxis = d3.svg.axis()
    .scale(x)
    .orient("bottom")
    .tickPadding(8);

var yAxis = d3.svg.axis()
    .scale(y)
    .tickPadding(8)
    .orient("left")

g.append("g")
    .attr("class", "axis axis--x")
    .attr("transform", "translate(0," + height + ")")
    .call(xAxis);

g.append("g")
    .attr("class", "axis axis--y")
    .attr("transform", "translate(" + 0 + "," + 0 + ")")
    .call(yAxis);

var vertices = [
  {x: 300, y: 100},
  {x: 400, y: 300},
  {x: 500, y: 50}
];

var A;
var λ = d3.range(vertices.length).map(function(d) {return 0}); // weights
var p = [0,0];
update();

function update() {

A_x = [];
  A_y = [];
  A_sum = d3.range(vertices.length).map(function(d) {return 1});

for(var i = 0; i < vertices.length; i++) {
    A_x.push(vertices[i].x);
    A_y.push(vertices[i].y)
  }
  A = [];
  A.push(A_x);
  A.push(A_y);
  A.push(A_sum);

var vectors = g.selectAll(".vector")
                  .data(vertices)

vectors.enter().append("path");

vectors
        .attr("class", "vector")
        .attr("d", function(d) {return "M " + x(0) + " " + y(0) + " " + "L " + x(d.x) + " " + y(d.y) });

var points = g.selectAll("circle")
                 .data(vertices);

points.enter().append("circle");

points
        .attr("class", "point")
        .style("fill", function(d, i) {return color(i)})
        .attr("cx", function(d) {return x(d.x)})
        .attr("cy", function(d) {return y(d.y)})
        .attr("r", radius);

var supports = g.selectAll(".support")
                  .data(vertices);

supports.enter().append("path");

supports
        .attr("class", "support")
        .style("stroke", function(d, i) {return (positive(λ)) ? color(i) : "#f75454"})
        .style("stroke-opacity", function(d, i) {return Math.abs(λ[i])})
        .attr("d", function(d, i) {return "M " + x(0) + " " + y(0) + " " + "L " + (x(d.x * λ[i])) + " " + (y(d.y * λ[i])) });

var vstar = g.selectAll(".vstar")
                  .data([p].map(function(d){return {x: d[0], y: d[1]}}))

vstar.enter().append("path");

vstar
        .attr("class", "vstar")
        .attr("d", function(d) {return "M " + x(0) + " " + y(0) + " " + "L " + x(d.x) + " " + y(d.y) });

}

function mousemove() {
  var mouse = d3.mouse(g.node());
  p = [x.invert(mouse[0]), y.invert(mouse[1])]; // weights must sum to 1
  var b = p.concat([1]);

// x1, x2, ..., xm are the "weights" that must sum to 1
  λ = numeric.solve(A, b);
  update();
}

function positive(vector) {
  var l = vector.length;
  for(var i = 0; i < l; i++) {
    if(vector[i] < 0) return false;
  }
  return true;
}

</script>

https://cdnjs.cloudflare.com/ajax/libs/d3/3.5.6/d3.min.js