HyperRogue uses several different types of curves as a part of its lands. Let's have a look at them:

The Great Walls do not actually look straight -- they generally look like arcs of a circle. The important property is that this arc will always be perpendicular to the horizon -- the blue circle around the game world.

In Euclidean plane, two straight lines will either cross or be parallel. In hyperbolic geometry, they can cross, as they do in Crossroads III...

...but they will be usually "ultraparallel". Two ultraparallel lines have a certain point where they are closest to each other, and their distance grows to infinity from there.

There is a third possibility: two lines can be "convergent", coming arbitrarily close to each other but never intersecting. However, HyperRogue is based on a tile system, and so it's not possible to adequately show such lines since at some point their separation would be under the width of a tile.

In PoincarĂ© projection, circles have the nice property of actually looking like circles. Any circle that lies entirely within the horizon is also a hyperbolic circle -- but it should be noted that the Euclidean and hyperbolic center of this circle are generally NOT the same.

In Euclidean plane, very large circles look almost like straight lines snce their curvature is very slight. This is not the case here. Any circle in hyperbolic plane will be visibly curved, regardless its size.

In Euclidean geometry, the ratio between circumference and diameter of a circle is a constant called "pi". In hyperbolic geometry, this is, once again, not true. The circumference of a circle grows exponentially with its radius. Big circles (with radius measuring in tens of tiles) have incredibly high and counterintuitive tile counts.

In Euclidean plane, a set of points that have the same distance from a given line is a parallel line (or two parallel lines, one on each side, depending on how exactly you define it). However, as we've seen, two lines cannot be parallel in the usual sense of word in the hyperbolic geometry. Instead, a set of points with the same distance from a given line is an equidistant curve. In PoincarĂ© projection, equidistant curves look like circle arcs that intersect the horizon in two points. In fact, straight lines can be considered a special case of equidistant curves where the distance from the guiding straight line is zero.

Segments of equidistant lines are longer than corresponding segments of their straight lines. With very distant equidistants, you can move very far, then return to their guiding line and find you are still at the same place as before.

A horocycle is a limit curve that has no Euclidean equivalent. In PoincarĂ© projection, horocycles show as circles that touch the horizon.

This gives us two ways to reach a horocycle:

One: start with a circle and move its center further and further away. When the center is in the infinity, you'll have a horocycle.

Two: start with an equidistant curve and move its guiding line further and further away. When the line disappears in the infinity, you'll have a horocycle.

Thus, horocycles lie on a border between circles and equidistant lines. Their curvature is exact: a bit more curved and it would be a very big circle, a bit less curved and it would be a very distant equidistant.

If you construct lines perpendicular to a given circle, they will all intersect, passing through a single point (the center of the circle). If you construct lines perpendicular to a given straight line or equidistant curve, they will be all ultraparallel. In case of a horocycle, all lines perpendicular to it will be convergent.

A set of all points with a given distance from a horocycle will form two other horocycles (one outside, one inside) that will be exactly the same as the original. In fact, ALL horocycles are congruent (looking the same), the same way all straight lines are congruent.

While two points can be only connected by one straight line, they are infinitely many circles and equidistant curves that pass through them. However, there are always exactly TWO horocycles that connect the points.

A horocycle has one "point at infinity". It is infinitely deep: it is always possible to move inwards in it. However, it is generally pretty difficult. If you move in a straight line, you will almost always end up outside. Only straight lines perpendicular to the horocycle will never exit it.

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