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The most common approach is to offset every other column or row. Columns are named col ( q ). Rows are named row ( r ). You can either offset the odd or the even column/rows, so the horizontal and vertical hexagons each have two variants.

Another way to look at hexagonal grids is to see that there are primary axes, unlike the we have for square grids. There’s an elegant symmetry with these.

Let’s take a cube grid and slice out a diagonal plane at . This is a idea but it helps us make hex grid algorithms simpler. In particular, we can reuse standard operations from cartesian coordinates: adding coordinates, subtracting coordinates, multiplying or dividing by a scalar, and distances.

Notice the three six hex grid directions are halfway between two of the cube axes. We’ll see this in the neighbors section, where moving along one of the six hex grid directions involves changing of the coordinates.

Because we already have algorithms for square and cube grids, using cube coordinates allows us to adapt those algorithms to hex grids. I will be using this system for most of the algorithms on the page. To use the algorithms with another coordinate system, I’ll convert to cube coordinates , run the algorithm, and convert back.

Study how the cube coordinates work on the hex grid. Selecting the hexes will highlight the cube coordinates corresponding to the three axes.

The cube coordinates are a reasonable choice for a hex grid coordinate system. The constraint is that x + y + z = 0 so the algorithms must preserve that. The constraint also ensures that there’s a canonical coordinate for each hex.

There are many different valid cube hex coordinate systems. Some of them have constraints other than x + y + z = 0 . I’ve shown only one of the many systems. You can also construct cube coordinates with x-y, y-z, z-x , and that has its own set of interesting properties, which I don’t explore here.

“But Amit!” you say, “I don’t want to store 3 numbers for coordinates. I don’t know how to store a map that way.”

Axial coordinates #

The axial coordinate system, sometimes called “trapezoidal” or “oblique” or “skewed”, is built by taking two of the three coordinates from a cube coordinate system. Since we have a constraint , there’s some redundancy, and we don’t need to store all three coordinates. This diagram is the same as the previous one, except I don’t show :

There are many choices of cube coordinate system, and many choices of axial coordinate system. I’m not going to show all of the combinations in this guide. I’ve chosen for “column” = and as “row” = . This choice is arbitary, as you can rotate and flip the diagrams to make many different assignments of ±x,±y,±z to q,r.

The advantage of this system over offset grids is that the algorithms are cleaner when you can use add, subtract, multiply, and divide on hex coordinates. The disadvantage of this system is that storing a rectangular map is a little weird; see the map storage section for ways to handle that. In my projects, I name the axes , , so that I have the constraint , and then I can calculate when I need the third coordinate for algorithms that work better with cube coordinates.

Although I recommend axial/cube coordinates, if you are sticking to offset coordinates, consider the doubled variant. It makes many of the algorithms easier to implement. Instead of alternation, the doubled coordinates double either the horizontal or vertical step size. It has a constraint ( col + row ) % 2 == 0 . In the horizontal (pointy top hex) layout it increases the column by 2 each hex; in the vertical (flat top hex) layout it increases the row by 2 each hex. This allows the in-between values for the hexes that are halfway in between:

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Saxo's global macro strategist Kay Van-Petersen examines the big issues for the markets in the week ahead in this brief rundown.
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Squawk / 09 May 2018 at 0:38 GMT
Kay Van-Petersen
Global Macro Strategist / Saxo Bank Group - Singapore Hub
Trade View Update - Closing off the oil short at $70.19. Overall MTM on the Macro Monday Book is +6.40% YTD vs. c. +1-4% for Global Macro Funds YTD… Gooooooooood Morning Happy macro Weds to the Asia Pacific... a good evening/night to the Americas, MEA Europe... Yest KVP was speaking to some of you on simple puts ptu spreads trades... that you could close off into or post the event... The moves were big... close to -4% at one point... Which would have been great returns on the dwnside exposure... Yet would have been tough to close unless you were on the screens late Asia time... Two day range on WTI has been 70.84 / 67.63... we are now at $70.19... Where KVP closes off the short from $67.10... from few wks back... whilst it feels like closing at the highs... so far its been very tough to trade any of the dwnside moves...
09 May
* Bottom line, 6m of negotiations for a 'Better deal'... so far from immediate in rgds to Iran supply leaving the mkt... A Reuters piece suggested that Iran while condemning Trumps actions would still honour its agreement with the EU... Risk though is a harsher response from Teran.... * A client very good trader asked me, if I would be short from these lvls today for a tactical pullback the answer is yes... There is also a chance that further squeeze higher in oil leading to Gas getting over $3.00... may see Trump threaten to unleash US strategic reserves... even though I think technically the effect is crippled... Its worth re-highlighting a few things from the Macro Monday Book: *The Objective of the MM Book is to return +20-30%... under a discretionary Macro Mandate (Cross Asset Cross Products) for that we are taking risk... =>This is not a low risk, low volatility, systematic position diversified mandate
09 May
*The position sizing risk taken is view as a portfolio with individual positions either being assigned a stop loss AUM that is willing to be risked, say 100Bp to other higher conviction portfolio where a NAV % approach is taken. The latter obviously has the element of potential punchier loses on individual trades... =>So while the oil short which was 1x the $10m capital of MM was cut today at a loss of -4.42% (after initially getting ITM by +2% within 24hrs of being put on – i.e. could argue a lot of trade mismanagement went into this) =>The 1.4xshort in AUDUSD has a MtM of +8.3% (At one point 2.1x before we hit our first tgt) =>As of this morning Asia time the MM book is +6.4% for the year... after starting Feb Mar quite poorly being stuck in a -100bp to -200bp for a while...
09 May
*Obviously everything from MM the MM Book are solely views/opinions not trading advice nor investment recommendation *The underlying idea behind MM the MM Book is that research as a whole is commoditized has zero-value almost no edge, but trading views, a high level process, passion lucrative insights will always have value an edge to anything else out there. Not to mention quite a bit of humility  Thx, -KVP
09 May
Hi K! Can the MM Book be visualized anywhere besides MM? THX
15 May
Not for now - I am trying to push internally to get it put up on SaxoSelect... which would also allow folks to follow if they want, copy the trades. Happy to put my own capital behind it... Stay Tuned!
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