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Although converting doubled coordinates to cube coordinates works, there’s also a direct formula for distances, from the Nike Cortez Ultra QS Mens Running Trainers 882493 Sneakers Shoes Black Metallic Gold Black 001 nmSwHkWHst
:

How do we draw a line from one hex to another? I use Nike 705190221 Men’s Gym Shoes Hazelnut/Baroque Brown White pGXlFuo7IT
. Evenly sample the line at N+1 points, and figure out which hexes those samples are in.

Putting these together to draw a line from A to B :

More notes:

Given a hex and a range N, which hexes are within N steps from it?

We can work backwards from the hex distance formula, . To find all hexes within N steps, we need . This means we need three to be true: and and . Removing absolute value, we get and and . In code it’s a nested loop:

This loop will work but it’s somewhat inefficient. Of all the values of we loop over, only one of them actually satisfies the constraint on cubes. Instead, let’s directly calculate the value of that satisfies the constraint:

This loop iterates over exactly the needed coordinates. In the diagram, each range is a pair of lines. Each line is an inequality (a half-plane ). We pick all the hexes that satisfy all six inequalities.

If you need to find hexes that are in more than one range, you can intersect the ranges before generating a list of hexes.

You can either think of this problem algebraically or geometrically. Algebraically, each hexagonally-shaped region is expressed as inequality constraints of the form , and we’re going to solve for the intersection of those constraints. Geometrically, each region is a cube in 3D space, and we’re going to intersect two cubes in 3D space to form a Fleet And Foster Womens Open Toe Caged Sandal with Three Adjustment Points Luxuriously Soft Leather Upper with Soft Touch Inner Fully Opening Sandal with Two Touch Fas MQGJ5cV
in 3D space, then project back to the x + y + z = 0 plane to get hexes. I’m going to solve it algebraically:

First, we rewrite constraint into a more general form, , and set and . We’ll do the same for and , and end up with this generalization of the code from the previous section:

The intersection of two ranges and is . Since a hex region is expressed as ranges over x, y, z, we can separately intersect each of the x, y, z ranges then use the nested loop to generate a list of hexes in the intersection. For one hex region we set and and likewise for and . For intersecting two hex regions we set and , and likewise for and . The same pattern works for intersecting three or more regions, and can generalize to other shapes (triangles, trapezoids, rhombuses, non-regular hexagons).

If there are obstacles, the simplest thing to do is a distance-limited flood fill (breadth first search). In this diagram, the limit is set to moves. In the code, is an array of all hexes that can be reached in steps. Each time through the main loop, we expand level into level .

Given a hex vector (difference between one hex and another), we might want to rotate it to point to a different hex. This is simple with cube coordinates if we stick with rotations of 1/6th of a circle.

A rotation 60° right shoves each coordinate one slot to the right:

A rotation 60° left shoves each coordinate one slot to the left:

As you play with diagram, notice that each 60° rotation the signs and also physically “rotates” the coordinates. Take a look at the axis legend on the bottom left to see how this works. After a 120° rotation the signs are flipped back to where they were. A 180° rotation flips the signs but the coordinates have rotated back to where they originally were.

Here’s the full recipe for rotating a position P around a center position C to result in a new position R:

It’s several conversion steps but each step is simple. You can shortcut some of these steps by defining rotation directly on axial coordinates, but hex vectors don’t work for offset coordinates and I don’t know a shortcut for offset coordinates. Also see this stackexchange discussion for other ways to calculate rotation.

Single ring #

To find out whether a given hex is on a ring of a given , calculate the distance from that hex to the center and see if it’s . To get a list of all such hexes, take steps away from the center, then follow the rotated vectors in a path around the ring.

In this code, starts out on the ring, shown by the large arrow from the center to the corner in the diagram. I chose corner 4 to start with because it lines up the way my direction numbers work but you may need a different starting corner. At each step of the inner loop, moves one hex along the ring. After steps it ends up back where it started.

The scale, add, and neighbor operations also work on axial and doubled coordinates, so the same algorithm can be used. For offset coordinates, convert to one of the other formats, generate the ring, and convert back.

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Last year researchers associated with a company called Integrated Tissue Dynamics ( INTiDYN ) in New York discovered excessive numbersof nerve fibers in the skin of the hands of a high percentage of Fibromyalgia patients. We covered that study in a blog called “ Hands up! A key to the fatigue, pain and sleep problems in Fibromyalgia found?” .

A recent update on their progress from the President and lead researcher at Clarks Cowley Faye Women’s LowTop Sneakers Pink Pink Nubuck m5pOf
, Frank Rice Ph.D, indicated that they, in collaboration with Frank Ruscetti Ph.D, Sandra Ruscetti Ph.D (formerly of the NIH), and Judy Mikovits, Ph.D (formerly of the NIH and the Whittemore-Peterson Institute) are also studying viruses in Chronic Fatigue Syndrome (ME/CFS).

We took the opportunity of the update to review a recent podcast by Dr. Rice and ask him about his Fibromyalgia and Chronic Fatigue Syndrome research.

INTiDYN , a for-profit company created in 2008, uses complex tissue analyses to understand the effects of disease, toxins, and drugs on the body. INTiDYN asserts that the integration of the nervous, cardiovascular, immune, and endocrine systems in the skin make it an easy-to-access and cost-effective portal that can help us understand many of the problems happening in the body. Their testing of small biopsies of the skin includes examining the vasculature, nerve innervation, immunohistochemistry, gene expression, and sensory function. Their integrated approach allows them to assess the effects of disease, toxins, and drugs on the morphology, biochemistry, and genomics of tissues.

They used their ability to provide detailed analyses of microvasculature and the vascular sensory and autonomic nerves innervating the skin in their Fibromyalgia study. That study grew out of work they’d done with chronic regional pain syndrome (CRPS) patients.

First we cover a podcast interview with Dr. Rice Wilson Women KAOS Comp W Tennis Shoes Green Arub CLa7NuRt6
, and then go to our interview with him. (The podcast with Dr. Rice starts about seven minutes in.)

Dr. Rice’s Podcast

A Personal Experience Flips Rice From Doing Basic Science Research To Doing Applied Research

During Dr. Rice’s Ph.D studies in neuroscience at John Hopkins University he became interested in nerve endings in the skin. New methods had been developed that helped them see nerve endings in a way they couldn’t before. He was doing basic science at that point; he was not focused on disease – he was simply determining how the nervous system and how nerve endings in the skin formed. But then after a minor accident a friend with four kids developed complex regional pain syndrome (CRPS), a “horrible condition” he said, he became interested in pain.

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