About Sphere Projections
This simulation flattens a sphere onto the canvas using six classic map projections — Mercator, Stereographic, Orthographic, Azimuthal Equidistant, Mollweide and Gnomonic. Each point is converted to a 3D unit vector, rotated so the chosen centre faces forward, then mapped with that projection's formula. Mercator, for example, uses y = ln(tan(π/4 + φ/2)). Because a sphere has intrinsic curvature, Gauss's Theorema Egregium guarantees no flat map can avoid distortion of areas, angles or distances.
The projection buttons switch the mapping formula, while the longitude and latitude sliders rotate the globe to recentre the view. The overlay buttons toggle the 30° graticule, Tissot indicatrices (small 8° circles that reveal local distortion) and a coarse continent outline. Such projections underpin everyday cartography: Mercator for marine navigation, equal-area maps for honest thematic data, and gnomonic charts for plotting great-circle flight paths.
Frequently Asked Questions
What is a sphere projection?
A sphere projection is a rule for mapping points on a globe, given by longitude and latitude, onto a flat plane. Since a curved surface cannot be unrolled without stretching, every projection trades off distortion in area, shape, distance or direction. This tool lets you compare six well-known choices side by side.
Why can't a sphere be flattened without distortion?
Gauss's Theorema Egregium states that Gaussian curvature is preserved under any distance-preserving deformation. A sphere has positive curvature while a flat plane has zero, so no mapping can preserve all distances between them. This is the deep reason every world map distorts something.
What do the longitude and latitude sliders do?
They set the centre of the globe, λ₀ and φ₀. The simulation converts each point to a 3D vector and rotates it so your chosen centre faces forward before projecting. This lets you recentre on any region — handy for azimuthal projections, where everything is measured relative to the centre point.
What is the difference between conformal and equal-area?
A conformal projection, such as Mercator or Stereographic, preserves local angles and shapes but distorts area. An equal-area projection, such as Mollweide, preserves the relative sizes of regions but distorts their shapes. No projection can be both conformal and equal-area at the same time.
What equation does the Mercator projection use?
Mercator is a conformal cylindrical projection. Longitude maps directly to the horizontal axis, x = λ, while latitude maps to y = ln(tan(π/4 + φ/2)). This logarithmic stretch keeps angles correct but inflates polar areas — the simulation clips it near 85° because y diverges to infinity at the poles.
What are the Tissot indicatrices in the overlay?
A Tissot indicatrix is a tiny circle drawn on the sphere — here roughly 8° in radius — that becomes an ellipse once projected. Its size and shape reveal the local distortion: circles stay circular where shape is preserved, swell where area is exaggerated, and squash into ellipses where angles are sheared.
Why does Mercator make Greenland look as big as Africa?
Mercator's area scale grows as 1/cos(φ), so distortion explodes toward the poles. Greenland sits near 70° north and is inflated enormously, appearing similar in size to Africa, which straddles the equator and is barely stretched. In reality Africa is about fourteen times larger.
What makes the Gnomonic projection special?
The Gnomonic projection maps every great circle to a straight line, because it projects from the centre of the sphere onto a tangent plane. This makes it invaluable for navigation: the shortest path between two points is simply the straight line drawn between them. It cannot, however, show a full hemisphere — points 90° or more from the centre vanish to infinity.
How is the Mollweide projection computed here?
Mollweide is an equal-area pseudocylindrical projection. It introduces an auxiliary angle θ that satisfies 2θ + sin 2θ = π sin φ, which has no closed-form solution. The simulation solves it with about ten Newton-Raphson iterations per point, then maps the whole globe onto an ellipse with elliptical meridians and straight parallels.
Are these projections physically accurate?
The projection formulae are mathematically exact, so the distortion patterns shown are genuine. The continent outlines, however, are deliberately coarse polygons for illustration rather than survey-grade coastlines. Use the tool to understand projection behaviour conceptually, not to read precise coordinates of real geography.