There is a family of pseudocylindrical equal-area projections that starts with the cylindrical equal area and ends with the sinusoidal, parameterised by t ∈ [0,1].
The formulas are:
x = λ cos^t φ
y = ∫_0^φ cos^(1-t) φ
where the integral has no closed form in general.
Update: It turns out this family was described, briefly, by Waldo Tolber in 1973: he calls it “a weighted geometrical mean of the meridians from the Lambert and sinusoidal projections”.
| <!DOCTYPE html> | |
| <meta charset="utf-8"> | |
| <title>A family of equal-area projections</title> | |
| <script charset="utf-8" src="http://d3js.org/d3.v3.min.js"></script> | |
| <script charset="utf-8" src="http://d3js.org/topojson.v1.min.js"></script> | |
| <script charset="utf-8" src="integrate.js"></script> | |
| <style> | |
| .graticule { fill: none; stroke: #777; stroke-width: .5px; stroke-opacity: .5; } | |
| .outline { fill: none; stroke: #777; stroke-width: 2px; stroke-opacity: 1; } | |
| .land { fill: #222; } | |
| .boundary { fill: none; stroke: #fff; stroke-width: .5px; } | |
| #controls { width: 100%; position: relative; height: 48px; } | |
| #slider { position: absolute; left: 80px; width: 800px; } | |
| </style> | |
| <div id="map"></div> | |
| <div id="controls"> | |
| <input id="slider" type="range" min="0" max="1" step="0.01" value="0.5"> | |
| </div> | |
| <script> | |
| var width = 960, | |
| height = 500; | |
| var t, scale; | |
| function set_t(new_t) { | |
| t = new_t; | |
| // Set the scale so the map has 2-to-1 proportions | |
| var max_y = integrate(dy, 0, Math.PI/2, [t]); | |
| scale = Math.sqrt(2*max_y/Math.PI); | |
| } | |
| function dy(φ, args) { | |
| var t = args[0]; | |
| return Math.pow(Math.cos(φ), 1-t); | |
| } | |
| set_t(0.5); | |
| var projection = d3.geo.projection(function(λ, φ) { | |
| return [ | |
| λ * Math.pow(Math.cos(φ), t) * scale, | |
| integrate(dy, 0, Math.abs(φ), [t]) * (φ < 0 ? -1 : 1) / scale | |
| ]; | |
| }); | |
| var path = d3.geo.path() | |
| .projection(projection); | |
| var graticule = d3.geo.graticule(); | |
| var svg = d3.select("#map").append("svg") | |
| .attr("width", width) | |
| .attr("height", height); | |
| svg.append("path") | |
| .datum(graticule) | |
| .attr("class", "graticule") | |
| .attr("d", path); | |
| svg.append("path") | |
| .datum(graticule.outline()) | |
| .attr("class", "outline") | |
| .attr("d", path); | |
| d3.json("/mbostock/raw/4090846/world-50m.json", function(error, world) { | |
| svg.insert("path", ".graticule") | |
| .datum(topojson.feature(world, world.objects.land)) | |
| .attr("class", "land") | |
| .attr("d", path); | |
| svg.insert("path", ".graticule") | |
| .datum(topojson.mesh(world, world.objects.countries, function(a, b) { return a !== b; })) | |
| .attr("class", "boundary") | |
| .attr("d", path); | |
| }); | |
| d3.select(self.frameElement).style("height", (height + 48) + "px"); | |
| d3.select("#slider").on("change", function() { | |
| set_t(parseFloat(this.value)); | |
| svg.selectAll("path").transition().attr("d", path); | |
| }); | |
| </script> |
| (function() { | |
| // Simple Simpson’s rule integration | |
| var DELTA = 0.1; | |
| function integrate_one_slice(f, a, b, args) { | |
| return (b-a) * ( | |
| f(a, args) | |
| + 4*f((a+b)/2, args) | |
| + f(b, args) | |
| ) / 6; | |
| } | |
| function integrate(f, a, b, args) { | |
| var x = a, | |
| integral = 0; | |
| while (x < b) { | |
| integral += integrate_one_slice(f, x, Math.min(x+DELTA, b), args); | |
| x += DELTA; | |
| } | |
| return integral; | |
| } | |
| window.integrate = integrate; | |
| })(); |