Hexagonal grids (hexagonal grids) are used in some games, but they are not as simple and common as rectangle grids. I've been collecting resources about hex grids for almost 20 years now, and I've written this guide to some of the most elegant approaches, implemented in the simplest code. This article often uses the tutorials of Charles Fu and Clark Verbrugge. I will describe the different ways to create hex meshes, how they relate, and the most common algorithms. Many parts of this article are interactive: choosing a grid type changes the corresponding schematics, code, and texts. (Approx. Lane: this applies only to the original, I advise you to study it. In the translation, all the information of the original is preserved, but without interactivity.).

The code examples in this article are written in pseudocode, so they are easier to read and understand in order to write your own implementation.

Geometry

Hexagons are hex polygons. Regular hexagons have all sides (faces) of the same length. We will only work with regular hexagons. Typically hex grids use horizontal (pointed top) and vertical (flat top) orientations.

Flat (left) and pointed (right) top hexagons

Hexagons have 6 faces. Each face is common to two hexagons. Hexagons have 6 corner points. Each corner point is shared by three hexagons. You can read more about centers, edges and corner points in my article on mesh parts (squares, hexagons and triangles).

Corners

In a regular hexagon, the interior angles are 120 °. There are six "wedges", each of which is an equilateral triangle with internal angles of 60 °. Corner point i is at a distance of (60 ° * i) + 30 °, size units from center. In the code:

Function hex_corner (center, size, i): var angle_deg = 60 * i + 30 var angle_rad = PI / 180 * angle_deg return Point (center.x + size * cos (angle_rad), center.y + size * sin (angle_rad) )
To fill the hexagon, you need to get the vertices of the polygon from hex_corner (..., 0) to hex_corner (..., 5). To draw the outline of the hexagon, use these vertices and then draw the line again in hex_corner (..., 0).

The difference between the two orientations is that x and y change places, which causes the angles to change: the angles of the flat-top hexagons are 0 °, 60 °, 120 °, 180 °, 240 °, 300 °, and the sharp-top hexagons are 30 °, 90 °, 150 °, 210 °, 270 °, 330 °.

Angles of flat and sharp top hexagons

Size and location

Now we want to position some hexagons together. In a horizontal orientation, the height of the hexagon is height = size * 2. The vertical distance between adjacent hexagons vert = height * 3/4.

The width of the hexagon is width = sqrt (3) / 2 * height. The horizontal distance between adjacent hexagons horiz = width.

Some games use pixel art for hexagons that doesn't exactly match regular hexagons. The angle and position formulas described in this section will not match the dimensions of these hexagons. The rest of the article describing the hex meshing algorithms applies even if the hexagons are slightly stretched or compressed.

Coordinate systems

Let's start assembling the hexagons into a grid. In the case of grids of squares, there is only one obvious way to assemble. There are many approaches for hexagons. I recommend using cubic coordinates as the primary representation. Axis coordinates or offset coordinates should be used to store maps and display coordinates for the user.

Offset coordinates

The most common approach is to offset each subsequent column or row. Columns are designated col or q. Rows are designated row or r. Odd or even columns / rows can be offset, so horizontal and vertical hexagons have two options.

Odd-r horizontal layout

Even-r horizontal arrangement

Odd-q vertical arrangement

Even-q vertical arrangement

Cubic coordinates

Another way to look at grids of hexagons is to see in them three major axes, not two as in grids of squares. They exhibit elegant symmetry.

Take a grid of cubes and cut out the diagonal plane at x + y + z = 0. This is an odd idea, but it will help us simplify the hex grid algorithms. In particular, we will be able to use standard operations from Cartesian coordinates: addition and subtraction of coordinates, multiplication and division by a scalar, and also distances.

Notice the three major axes on the grid of cubes and their relationship to the six. diagonal directions of the grid of hexagons. The diagonal axes of the grid correspond to the base direction of the hex grid.

Hexagons

Cuba

Since we already have algorithms for grids of squares and cubes, using cubic coordinates allows us to adapt these algorithms to grids of hexagons. I will use this system for most of the algorithms in this article. To use algorithms with a different coordinate system, I transform the cubic coordinates, run the algorithm, and then convert them back.

Explore how cubic coordinates work for a grid of hexagons. When you select hexagons, the cubic coordinates corresponding to the three axes are highlighted.

1. Each direction of the grid of cubes corresponds the lines on a grid of hexagons. Try to select a hexagon with z equal to 0, 1, 2, 3 to see the connection. The line is highlighted in blue. Try the same for x (green) and y (purple).
2. Each hex grid direction is a combination of two cube grid directions. For example, the north of the hex grid is between + y and -z, so each step north increases y by 1 and decreases z by 1.

Cubic coordinates are a smart choice for the hex grid coordinate system. The condition is x + y + z = 0, so it must be preserved in algorithms. The condition also ensures that there will always be a canonical coordinate for each hexagon.

There are many different coordinate systems for cubes and hexagons. In some of them, the condition differs from x + y + z = 0. I have shown only one of many systems. You can also create cubic coordinates with x-y, y-z, z-x, which will have their own set of interesting properties, but I will not cover them here.

But you might argue that you don't want to store 3 numbers for coordinates, because you don't know how to store the map like this.

Axial coordinates

An axial coordinate system, sometimes called a "trapezoidal" coordinate system, is constructed from two or three coordinates from a cubic coordinate system. Since we have the condition x + y + z = 0, the third coordinate is not needed. Axis coordinates are useful for storing maps and displaying coordinates to the user. As with cubic coordinates, you can use standard Cartesian addition, subtraction, multiplication, and division operations with them.

There are many cubic coordinate systems and many axial ones. I will not cover all combinations in this tutorial. I will select two variables, q (column) and r (row). In the diagrams of this article, q corresponds to x, and r corresponds to z, but this correspondence is arbitrary, because you can rotate and rotate the diagrams, getting different matches.

The advantage of this system over displacement grids is that the algorithms are more understandable. The disadvantage of the system is that storing a rectangular map is a bit odd; see the section on saving maps. Some algorithms are even clearer in cubic coordinates, but since we have the condition x + y + z = 0, we can calculate the third implied coordinate and use it in these algorithms. In my projects, I call the axes q, r, s, so the condition looks like q + r + s = 0, and I can calculate s = -q - r when needed.

Axles

Offset coordinates are the first thing most people think of because they are the same as the standard Cartesian coordinates used for square grids. Unfortunately, one of the two axes has to go against the grain, and this complicates things as a result. The cubic and axial systems are pretty straight forward and have simpler algorithms, but storing maps is a little more complex. There is another system called "alternating" or "double", but we will not consider it here; some find it easier to work with than cubic or axial.

Offset coordinates, cubic and axial

Axis is the direction in which the corresponding coordinate is increased. The perpendicular to the axis is the line on which the coordinate remains constant. The grid diagrams above show perpendicular lines.

Coordinate transformation

It is likely that you will use axial coordinates or offset coordinates in your project, but many algorithms are easier to express in cubic coordinates. Therefore, we need to be able to transform coordinates between systems.

Axial coordinates are closely related to cubic coordinates, so the transformation is simple:

# convert cubic to axial coordinates q = x r = z # convert axial to cubic coordinates x = q z = r y = -x-z
In code, these two functions can be written as follows:

Function cube_to_hex (h): # axial var q = hx var r = hz return Hex (q, r) function hex_to_cube (h): # cubic var x = hq var z = hr var y = -xz return Cube (x, y , z)
The offset coordinates are quite a bit tricky:

Adjacent hexagons

Given one hexagon, what six hexagons is it adjacent to? As you might expect, the answer is easiest in cubic coordinates, fairly simple in axial coordinates, and a little more complex in offset coordinates. You may also need to calculate six "diagonal" hexagons.

Cubic coordinates

Moving one space in the coordinates of the hexagons changes one of the three cubic coordinates by +1 and the other by -1 (the sum must remain equal to 0). Three possible coordinates can be changed by +1, and the remaining two can be changed by -1. This gives us six possible changes. Each corresponds to one of the directions of the hexagon. Simplest and fastest way- pre-compute the changes and place them in the Cube (dx, dy, dz) table at compile time:

Var directions = [Cube (+1, -1, 0), Cube (+1, 0, -1), Cube (0, +1, -1), Cube (-1, +1, 0), Cube ( -1, 0, +1), Cube (0, -1, +1)] function cube_direction (direction): return directions function cube_neighbor (hex, direction): return cube_add (hex, cube_direction (direction))

Axial coordinates

As before, we start with the cube system. Take the Cube table (dx, dy, dz) and convert it to Hex (dq, dr) table:

Var directions = [Hex (+1, 0), Hex (+1, -1), Hex (0, -1), Hex (-1, 0), Hex (-1, +1), Hex (0, +1)] function hex_direction (direction): return directions function hex_neighbor (hex, direction): var dir = hex_direction (direction) return Hex (hex.q + dir.q, hex.r + dir.r)

Offset coordinates

In axial coordinates, we make changes depending on where we are on the grid. If we are in a column / row offset, then the rule differs from the case of a column / row without offset.

As before, we create a table of numbers to add to col and row. However, this time we will have two arrays, one for the odd columns / rows and the other for the even ones. Look at (1,1) in the gridmap above and notice how col and row change as you move in each of the six directions. Now we repeat the process for (2,2). The tables and code will be different for each of the four types of displacement grids, here is the corresponding code for each type of grid.

Odd-r
var directions = [[Hex (+1, 0), Hex (0, -1), Hex (-1, -1), Hex (-1, 0), Hex (-1, +1), Hex (0 , +1)], [Hex (+1, 0), Hex (+1, -1), Hex (0, -1), Hex (-1, 0), Hex (0, +1), Hex ( +1, +1)]] function offset_neighbor (hex, direction): var parity = hex.row & 1 var dir = directions return Hex (hex.col + dir.col, hex.row + dir.row)

Even-r
var directions = [[Hex (+1, 0), Hex (+1, -1), Hex (0, -1), Hex (-1, 0), Hex (0, +1), Hex (+1 , +1)], [Hex (+1, 0), Hex (0, -1), Hex (-1, -1), Hex (-1, 0), Hex (-1, +1), Hex (0, +1)]] function offset_neighbor (hex, direction): var parity = hex.row & 1 var dir = directions return Hex (hex.col + dir.col, hex.row + dir.row)


Grid for even (EVEN) and odd (ODD) rows

Odd-q
var directions = [[Hex (+1, 0), Hex (+1, -1), Hex (0, -1), Hex (-1, -1), Hex (-1, 0), Hex (0 , +1)], [Hex (+1, +1), Hex (+1, 0), Hex (0, -1), Hex (-1, 0), Hex (-1, +1), Hex (0, +1)]] function offset_neighbor (hex, direction): var parity = hex.col & 1 var dir = directions return Hex (hex.col + dir.col, hex.row + dir.row)

Even-q
var directions = [[Hex (+1, +1), Hex (+1, 0), Hex (0, -1), Hex (-1, 0), Hex (-1, +1), Hex (0 , +1)], [Hex (+1, 0), Hex (+1, -1), Hex (0, -1), Hex (-1, -1), Hex (-1, 0), Hex (0, +1)]] function offset_neighbor (hex, direction): var parity = hex.col & 1 var dir = directions return Hex (hex.col + dir.col, hex.row + dir.row)


Grid for even (EVEN) and odd (ODD) columns

Diagonals

Moving in the "diagonal" space in the coordinates of the hexagons changes one of the three cubic coordinates by ± 2 and the other two by ∓1 (the sum must remain equal to 0).

Var diagonals = [Cube (+2, -1, -1), Cube (+1, +1, -2), Cube (-1, +2, -1), Cube (-2, +1, +1 ), Cube (-1, -1, +2), Cube (+1, -2, +1)] function cube_diagonal_neighbor (hex, direction): return cube_add (hex, diagonals)
As before, we can convert these coordinates to axial ones, discarding one of the three coordinates, or convert to offset coordinates, after calculating the results.

Distances

Cubic coordinates

In a cubic coordinate system, each hexagon is a cube in three dimensions. Adjacent hexagons are 1 space apart in the hex grid, but 2 space apart in the cube grid. This makes calculating distances easy. In a grid of squares, the Manhattan distances are abs (dx) + abs (dy). In a grid of cubes, the Manhattan distances are abs (dx) + abs (dy) + abs (dz). The distance in the grid of hexagons is equal to half of them:

Function cube_distance (a, b): return (abs (a.x - b.x) + abs (a.y - b.y) + abs (a.z - b.z)) / 2
The equivalent of this notation would be expressing that one of the three coordinates should be the sum of the other two, and then getting it as a distance. You can choose the bisection shape or the maximum value shape shown below, but they give the same result:

Function cube_distance (a, b): return max (abs (a.x - b.x), abs (a.y - b.y), abs (a.z - b.z))
In the figure, the maximum values ​​are highlighted in color. Notice also that each color represents one of six "diagonal" directions.

GIF

Axial coordinates

In the axial system, the third coordinate is implicitly expressed. Let's convert from axial to cubic to calculate the distance:

Function hex_distance (a, b): var ac = hex_to_cube (a) var bc = hex_to_cube (b) return cube_distance (ac, bc)
If the compiler in your case inline hex_to_cube and cube_distance, then it will generate the following code:

Function hex_distance (a, b): return (abs (a.q - b.q) + abs (a.q + a.r - b.q - b.r) + abs (a.r - b.r)) / 2
There are many different ways to write the distances between hexagons in axial coordinates, but regardless of the way of writing spacing between hexagons in axial system is extracted from manhattan distance in cubic system... For example, the described "difference of differences" is obtained by writing a.q + a.r - b.q - b.r as a.q - b.q + a.r - b.r and using the maximum value form instead of the bisection form cube_distance. They are all similar if you see the relationship with cubic coordinates.

Offset coordinates

As with axial coordinates, we convert the offset coordinates to cubic coordinates and then use the cubic distance.

Function offset_distance (a, b): var ac = offset_to_cube (a) var bc = offset_to_cube (b) return cube_distance (ac, bc)
We'll use the same pattern for many algorithms: convert from hexagons to cubes, run the cubic version of the algorithm, and convert the cubic results to hex coordinates (axial or offset coordinates).

Drawing lines

How do I draw a line from one hexagon to another? I am using linear interpolation to draw lines. The line is sampled evenly at N + 1 points and it is calculated in which hexagons these samples are located.

GIF

1. We first calculate N, which will be the hex distance between the endpoints.
2. Then we uniformly sample N + 1 points between points A and B. Using linear interpolation, we determine that for values ​​of i from 0 to N, including them, each point will be A + (B - A) * 1.0 / N * i. In the figure, these control points are shown in blue. The result is floating point coordinates.
3. Convert each control point (float) back to hexagons (int). The algorithm is called cube_round (see below).

Putting it all together to draw a line from A to B:

Function lerp (a, b, t): // for float return a + (b - a) * t function cube_lerp (a, b, t): // for hexagons return Cube (lerp (ax, bx, t), lerp (ay, by, t), lerp (az, bz, t)) function cube_linedraw (a, b): var N = cube_distance (a, b) var results = for each 0 ≤ i ≤ N: results.append ( cube_round (cube_lerp (a, b, 1.0 / N * i))) return results
Notes:

• There are times when cube_lerp returns a point exactly on the edge between two hexagons. Then cube_round shifts it one way or the other. Lines look better when moved in one direction. This can be done by adding an "epsilon" -hexagonal cube (1e-6, 1e-6, -2e-6) to one or both of the endpoints before starting the loop. This will "nudge" the line in one direction so that it does not hit the edge boundaries.
• The DDA line algorithm in grids of squares equates N to the maximum distance along each of the axes. We do the same in cubic space, which is analogous to the distance in a grid of hexagons.
• The cube_lerp function should return a cube with coordinates in float. If you are programming in a statically typed language, you cannot use the Cube type. Instead, you can define the FloatCube type, or inline a function in your line drawing code if you don't want to define another type.
• You can optimize your code by inline cube_lerp and then calculating B.x-A.x, B.x-A.y and 1.0 / N outside of the loop. Multiplication can be converted to repeated summation. The result is something like a DDA line algorithm.
• I use axial or cubic coordinates to draw lines, but if you want to work with offset coordinates then learn.
• There are many options for drawing lines. Sometimes overcoating is required. They sent me the code for drawing lines with overcoating in hexagons, but I haven't looked into it yet.

Travel range

Coordinate range

For a given center of a hexagon and a range of N, which hexagons are within N steps of it?

We can do the opposite work from the formula for the distance between the hexagons distance = max (abs (dx), abs (dy), abs (dz)). To find all hexagons within N, we need max (abs (dx), abs (dy), abs (dz)) ≤ N. This means that all three values ​​are needed: abs (dx) ≤ N and abs (dy) ≤ N and abs (dz) ≤ N. By removing the absolute value, we get -N ≤ dx ≤ N and -N ≤ dy ≤ N and -N ≤ dz ≤ N. In code, this will be a nested loop:

Var results = for each -N ≤ dx ≤ N: for each -N ≤ dy ≤ N: for each -N ≤ dz ≤ N: if dx + dy + dz = 0: results.append (cube_add (center, Cube (dx , dy, dz)))
This loop will work, but it will be rather ineffective. Of all the dz values ​​that we iterate over in the loop, only one really satisfies the cubes condition dx + dy + dz = 0. Instead, we will directly compute the dz value that satisfies the condition:

Var results = for each -N ≤ dx ≤ N: for each max (-N, -dx-N) ≤ dy ≤ min (N, -dx + N): var dz = -dx-dy results.append (cube_add ( center, Cube (dx, dy, dz)))
This cycle runs only along the required coordinates. In the figure, each range is a pair of lines. Each line is an inequality. We take all hexagons that satisfy six inequalities.

GIF

Overlapping ranges

If you need to find hexagons that are in multiple ranges, you can traverse the ranges before generating the list of hexagons.

You can approach this problem in terms of algebra or geometry. Algebraically, each region is expressed as inequality conditions in the form -N ≤ dx ≤ N, and we need to find the intersection of these conditions. Geometrically, each area is a cube in three-dimensional space, and we will intersect two cubes in three-dimensional space to get a rectangular parallelepiped in three-dimensional space. We then project it back onto the x + y + z = 0 plane to get the hexagons. I will solve this problem algebraically.

First, we rewrite the condition -N ≤ dx ≤ N in a more general form x min ≤ x ≤ x max, and take x min = center.x - N and x max = center.x + N. Let's do the same for y and z, resulting in a general view of the code from the previous section:

Var results = for each xmin ≤ x ≤ xmax: for each max (ymin, -x-zmax) ≤ y ≤ min (ymax, -x-zmin): var z = -xy results.append (Cube (x, y, z))
The intersection of the two ranges a ≤ x ≤ b and c ≤ x ≤ d is max (a, c) ≤ x ≤ min (b, d). Since the area of ​​the hexagons is expressed as ranges over x, y, z, we can individually intersect each of the x, y, z ranges and then use a nested loop to generate a list of hexagons at the intersection. For one area of ​​hexagons, we take x min = H.x - N and x max = H.x + N, similarly for y and z. For the intersection of two regions of hexagons, we take x min = max (H1.x - N, H2.x - N) and x max = min (H1.x + N, H2.x + N), similarly for y and z. The same pattern works for the intersection of three or more regions.

GIF

Obstacles

If there are obstacles, it is easiest to fill with a limited distance (Breadth First Search). In the picture below, we are limited to four moves. In code, fringes [k] is an array of all the hexagons that can be reached in k steps. With each pass through the main loop, we expand level k-1 to level k.

Function cube_reachable (start, movement): var visited = set () add start to visited var fringes = fringes.append () for each 1< k ≤ movement: fringes.append() for each cube in fringes: for each 0 ≤ dir < 6: var neighbor = cube_neighbor(cube, dir) if neighbor not in visited, not blocked: add neighbor to visited fringes[k].append(neighbor) return visited

Turns

For a given hexagon vector (the difference between two hexagons), we may need to rotate it so that it points to another hexagon. This is easy to do with cubic coordinates, if you stick to a rotation of 1/6 of a circle.

Rotating 60 ° to the right shifts each coordinate one position to the right:

[x, y, z] to [-z, -x, -y]
Rotating 60 ° to the left shifts each coordinate one position to the left:

[x, y, z] to [-y, -z, -x]

"Having played" [in the original article] with the scheme, you will notice that each turn by 60 ° changes signs and physically "rotates" the coordinates. After a 120 ° rotation, the signs are the same again. A 180 ° rotation changes signs, but the coordinates are rotated to their original position.

Here is the complete sequence of rotating the P position around the center C position, resulting in a new R position:

1. Convert positions P and C to cubic coordinates.
2. Calculate a vector by subtracting the center: P_from_C = P - C = Cube (P.x - C.x, P.y - C.y, P.z - C.z).
3. Rotate the P_from_C vector as described above and assign the resulting vector R_from_C.
4. Convert the vector back to position by adding center: R = R_from_C + C = Cube (R_from_C.x + C.x, R_from_C.y + C.y, R_from_C.z + C.z).
5. Convert the cubic position R back to the desired coordinate system.

There are several stages of transformation, but each of them is quite simple. It is possible to shorten some of these steps by defining the rotation directly in axial coordinates, but hex vectors do not work with offset coordinates, and I don’t know how to shorten the steps for offset coordinates. See also the stackexchange discussion of other ways to calculate pivot.

Rings

Simple ring

To find out if a given hexagon belongs to a ring of a given radius, you need to calculate the distance from this hexagon to the center, and find out if it is equal to radius. To get a list of all such hexagons, take radius steps from the center and then follow the rotated vectors along the path along the ring.

Function cube_ring (center, radius): var results = # this code doesn't work for radius == 0; do you understand why? var cube = cube_add (center, cube_scale (cube_direction (4), radius)) for each 0 ≤ i< 6: for each 0 ≤ j < radius: results.append(cube) cube = cube_neighbor(cube, i) return results
In this code, the cube starts at the ring shown by a large arrow from the center to the corner of the diagram. I chose angle 4 to start because it corresponds to the path my direction numbers are moving. You may need a different starting angle. At each step of the inner loop, the cube moves one hexagon around the ring. After 6 * radius steps, it ends where it started.

Spiral rings

By traversing the rings in a spiral pattern, we can fill in the inner parts of the rings:

Function cube_spiral (center, radius): var results = for each 1 ≤ k ≤ radius: results = results + cube_ring (center, k) return results

The area of ​​the large hexagon is the sum of all the circles plus 1 for the center. Use this formula to calculate the area.

Traversing hexagons in this manner can also be used to calculate the range of motion (see above).

Area of ​​visibility

What is visible from a given position with a given distance, and is not obstructed by obstacles? The simplest way define it - draw a line to each hexagon in a given range. If the line does not meet the walls, then you see a hexagon. Move the mouse over the hexagons [in the diagram in the original article] to see the drawing of lines to these hexagons and the walls with which the lines meet.

This algorithm can be slow over large areas, but it is easy to implement, so I recommend starting with it.

GIF

There are many different definitions of visibility. Do you want to see the center of the other hexagon from the center of the starting one? Do you want to see any part of the other hexagon from the center of the starting one? Maybe any part of another hexagon from any starting point? Are obstructions smaller than a full hexagon? Scope is a trickier and more varied concept than meets the eye. Let's start with the simplest algorithm, but expect it to correctly calculate the answer in your project. There are even cases where a simple algorithm gives illogical results.

I want to expand on this guide in the future. I have

Content:

A regular hexagon, also called a perfect hexagon, has six equal sides and six equal corners. You can draw a hexagon with a tape measure and a protractor, a rough hexagon with a round object and ruler, or an even rougher hexagon with just a pencil and a little intuition. If you want to know how to draw a hexagon in different ways, just read on.

Steps

1 Draw a perfect hexagon using a compass

1. 1 Draw a circle using a compass. Insert the pencil into the compass. Extend the compass to the desired width for the radius of your circle. The radius can be from a couple to ten centimeters wide. Next, place a compass with a pencil on the paper and draw a circle.
   • Sometimes it is easier to draw the half of the circle first and then the other half.
2. 2 Move the compass needle to the edge of the circle. Place it on top of the circle. Do not change the angle and position of the compass.
3. 3 Make a small pencil mark at the edge of the circle. Make it clear, but not too dark as you will erase it later. Remember to keep the angle you set for the compass.
4. 4 Move the compass needle to the mark you just made. Place the needle directly on the mark.
5. 5 Make another pencil mark at the edge of the circle. Thus, you will make a second mark at a certain distance from the first mark. Keep moving in one direction.
6. 6 Make four more marks in the same way. You should go back to the original mark. If not, then most likely the angle at which you held the compass and made the marks has changed. Perhaps this happened due to the fact that you squeezed it too tightly or, on the contrary, loosened it a little.
7. 7 Connect the marks with a ruler. The six places where your marks intersect with the edge of the circle are the six vertices of the hexagon. Using a ruler and pencil, draw straight lines connecting adjacent marks.
8. 8 Erase both the circle, the marks on the edges of the circle, and any other marks you made. Once you've erased all of your construction lines, your perfect hexagon should be ready.

2 Draw a rough hexagon using a round object and a ruler

1. 1 Draw a pencil around the rim of the glass. This way you will draw a circle. It is very important to draw with a pencil, as later you will need to erase all the auxiliary lines. You can also circle an inverted glass, jar, or anything else that has a round base.
2. 2 Draw horizontal lines across the center of your circle. You can use a ruler, a book, anything with a straight edge. If you do have a ruler, you can mark the middle by calculating the vertical length of the circle and dividing it in half.
3. 3 Draw an "X" over half of the circle, dividing it into six equal sections. Since you have already drawn a line through the middle of the circle, the X must be wider than it is high in order for the parts to be equal. Imagine you are cutting a pizza into six pieces.
4. 4 Make triangles out of each section. To do this, use a ruler to draw a straight line under the curved portion of each section, connecting it to the other two lines to form a triangle. Do this with the remaining five sections. Think of it like making a crust around your pizza slices.
5. 5 Erase all construction lines. Construction lines include your circle, the three lines that divided your circle into sections, and other marks that you made along the way.

3 Draw a rough hexagon with one pencil

1. 1 Draw a horizontal line. To draw a straight line without a ruler, simply draw the start and end points of your horizontal line. Then place the pencil at the starting point and extend the line towards the end. The length of this line can be as little as a couple of centimeters.
2. 2 Draw two diagonal lines from the ends of the horizontal one. The diagonal line on the left side should be facing outward in the same way as the diagonal line on the right. You can imagine these lines forming an angle of 120 degrees with respect to the horizontal line.
3. 3 Draw two more horizontal lines emanating from the first horizontal lines drawn inward. This will create a mirror image of the first two diagonal lines. The bottom left line should be a reflection of the top left line, and the bottom right line should be a reflection of the top right line. While the top horizontal lines should look outward, the bottom ones should look inward of the base.
4. 4 Draw another horizontal line connecting the bottom two diagonal lines. This will draw the base for your hexagon. Ideally, this line should be parallel to the upper horizontal line. Now you have completed your hexagon.

• The pencil and compasses should be kept sharp to minimize errors from marks that are too wide.
• If, using the compass method, you connect each mark instead of all six, you get an equilateral triangle.

Warnings

• The compass is a pretty sharp object, be very careful with it.

Principle of operation

• Each method will help you draw a hexagon formed by six equilateral triangles with a radius, equal to length all sides. The six drawn radii are the same length, and all the lines to create the hexagon are also the same length, since the width of the compass did not change. Due to the fact that six triangles are equilateral, the angles between their vertices are 60 degrees.

What do you need

• Paper
• Pencil
• Ruler
• Pair of compasses
• Something you can put under the paper to keep the compass needle from slipping.
• Eraser

Geometric patterns are very popular in Lately... In today's tutorial we will learn how to create one of these patterns. Using transition, typography and trendy colors, we will create a pattern that you can use in web and print design.

Result

Step 2
Draw another hexagon, smaller this time - select a radius in 20pt.

2. Transition between hexagons

Step 1
Select both hexagons and align them in the center (vertically and horizontally). Using the tool Blend (W), select both hexagons and give them a transition in 6 steps (Steps)... To make it easier to see, change the color of the shapes before the transition.

3. Divide into sections

Step 1
Tool Line Segment (\) draw a line intersecting the hexagons in the center from the leftmost corner to the rightmost corner. Draw two more lines, intersecting the hexagons in the center from opposite corners.

4. Paint over the sections

Step 1
Before we start painting over the sections, let's define the palette. Here's the palette from the example:

• Blue: C 65 M 23 Y 35 K 0
• Beige: C 13 M 13 Y 30 K 0
• Peach: C 0 M 32 Y 54 K 0
• Light pink: C 0 M 64 Y 42 K 0
• Dark pink: C 30 M 79 Y 36 K 4

The example immediately used CMYK mode so that the pattern could be printed without changes.

5. Finishing touches and pattern

Step 1
Group (Control-G) all sections and hexagons after you finish coloring them. Copy (Control-C) and Paste (Control-V) group of hexagons. Let's call the original group Hexagon A, and a copy of it Hexagon B... Align the groups.

Step 2
Apply Linear Gradient to the group Hexagon B. In the palette Gradient specify a fill from purple ( C60 M86 Y45 K42) to cream color ( C0 M13 Y57 K0).

We will learn how to depict a hexagonal prism in various positions.

Learn different ways to build a regular hexagon, draw hexagon drawings, check if they are correct. Draw hex prisms from the hexagons.

Consider the hex prism in fig. 3.52 and its orthogonal projections in Fig. 3.53. Regular hexagons lie at the base of a hexagonal prism (hexagon), the side faces are identical rectangles. In order to correctly depict a hexagon in perspective, you must first learn how to correctly depict its base in perspective (Fig. 3.54). In the hexagon in Fig. 3.55 tops are marked with numbers from one to six. If you connect points 1 and 3, 4 and 6 with vertical lines, you can see that these lines, together with the center point of the circle, divide the diameter 5 - 2 into four equal segments (these segments are indicated by arcs). Opposite sides of a hexagon are parallel to each other and a line passing through its center and connecting two vertices (for example, sides 6 - 1 and 4 - 3 are parallel to line 5 - 2). These observations will help you construct the hexagon in perspective, as well as verify the correctness of this construction. There are two ways to construct a regular hexagon from a representation: on the basis of the circumcircle and on the basis of the square.

Based on the circumscribed circle. Consider fig. 3.56. All vertices of a regular hexagon belong to the circumcircle, the radius of which is equal to the side of the hexagon.

Horizontal hexagon. Draw a horizontal, freehand ellipse, that is, the circumscribed circle in perspective. Now you need to find six points on it, which are the vertices of the hexagon. Draw any diameter of the given circle through its center (Fig. 3.57). The extreme points of the diameter - 5 and 2, lying on the ellipse, are the vertices of the hexagon. To find the rest of the vertices, you need to divide this diameter into four equal segments. The diameter is already divided by the center point of the circle into two radii, it remains to divide each radius in half. In a perspective drawing, all four segments are evenly reduced with distance from the viewer (Fig. 3.58). Now draw through the midpoints of the radii - points A and B - straight lines perpendicular to the straight line 5 - 2. You can find their direction using the tangents to the ellipse at points 5 and 2 (Fig. 3.59). These tangents will be perpendicular to the diameter 5 - 2, and the lines drawn through points A and B parallel to these tangents will also be perpendicular to the line 5 - 2. Designate the points obtained at the intersection of these lines with the ellipse as 1, 3, 4, 6 ( (See Figure 3.60) Connect all six vertices with straight lines (Fig. 3.61).

Check the correctness of your construction different ways... If the construction is correct, then the lines connecting the opposite vertices of the hexagon intersect at the center of the circle (Fig. 3.62), and the opposite sides of the hexagon are parallel to the corresponding diameters (Fig. 3.63). Another way to check is shown in Fig. 3.64.

Vertical hexagon. In such a hexagon, the straight lines connecting points 7 and 3, b and 4, as well as tangents to the circumscribed circle at points 5 and 2, have a vertical direction and retain it in the perspective drawing. Thus, drawing two vertical tangents to the ellipse, we find points 5 and 2 (points of tangency). Connect them with a straight line, and then divide the resulting diameter 5 - 2 into 4 equal segments, taking into account their perspective cuts (Fig. 3.65). Draw vertical lines through points A and B, and at their intersection with the ellipse find points 1,3,6L4. Then connect points 1 - 6 in series with straight lines (Fig. 3.66). Check the correctness of the construction of the hexagon in the same way as in the previous example.

The described method of constructing a hexagon allows you to get this figure based on a circle, which is easier to draw in perspective than a square of given proportions. So this way construction of a hexagon seems to be the most accurate and versatile. The construction method based on a square makes it easy to depict a hexagon in the case when there is already a cube in the drawing, in other words, when the proportions of the square and the direction of its sides are determined.

Based on a square. Consider fig. 3.67. A hexagon inscribed in a square in the horizontal direction 5 - 2 is equal to the side of the square, and in the vertical direction it is less than its length.

Vertical hexagon. Draw a vertical square in perspective. Draw a straight line through the intersection of the diagonals parallel to its horizontal sides. Divide the resulting segment 5 - 2 into four equal parts and draw vertical lines through points A and B (Fig. 3.68). The top and bottom lines of the hexagon do not line up with the sides of the square. Draw them at some distance (1114 a) from the horizontal sides of the square and parallel to them. Connecting the points 1 and 3 found in this way with point 2, and points 6 and 4 with point 5, we get a hexagon (Fig. 3.69).

The horizontal hexagon is constructed in the same sequence (Fig. 3.70 and 3.71).

This construction method is only appropriate for hexagons with sufficient opening. If the disclosure of the hexagon is insignificant, it is better to use the circumcircle method. You can use the methods you already know to test a hexagon built through a square.

In addition, there is one more - to describe a circle around the resulting hexagon (in your drawing - an ellipse). All vertices of the hexagon must belong to this ellipse.

Having mastered the skills of drawing a hexagon, you will freely move on to drawing a hexagonal prism. Take a close look at the diagram in fig. 3.72, as well as schemes for constructing hexagonal prisms based on the circumscribed circle (Fig. 3.73; 3.74 and 3.75) and on the basis of a square (Fig. 3.76; 3.77 and 3.78). Draw vertical and horizontal hexagons in different ways. In the illustration of a vertical hexagon, the long sides of the side faces will be parallel to each other vertical straight lines, and the base hexagon will be the more open the further it is from the horizon line. In the drawing of a horizontal hexagon, the long sides of the side faces will converge at the vanishing point on the horizon, and the opening of the base hexagon will be the greater the further from the viewer it is. When depicting a hexagon, also make sure that the parallel edges of both bases converge in perspective (Fig. 3.79; 3.80).

Geometric constructions are one of the main parts of the training. They shape spatial and logical thinking, and also enable understanding of primitive and natural geometric reasoning. Constructions are made on a plane using a compass and a ruler. These tools are allowed to build a large number of geometric shapes. At the same time, many figures, which seem rather difficult, are constructed using the simplest rules. For example, how to build the correct hexagon, it is allowed to describe each in a few words.

You will need

• Compass, ruler, pencil, sheet of paper.

Instructions

1. Draw a circle. Set some distance between the legs of the compass. This distance will be the radius of the circle. Select the radius in such a way that drawing the circle is quite comfortable. The circle should fit entirely on the sheet of paper. Too huge or too small distance between the legs of the compass can cause it to change during drawing. The optimal distance will be at which the angle between the legs of the compass is 15-30 degrees.

2. Plot the vertex points of the corners of the correct hexagon. Place the leg of the compass, in which the needle is fixed, at any point on the circle. The needle should pierce the drawn line. The more correctly the compass is set, the more correct the construction will be. Draw a circular arc so that it intersects the previously drawn circle. Move the compass needle to the intersection of the arc you just sketched with the circle. Draw another arc that intersects the circle. Reposition the compass needle to the intersection of the arc and circle and re-draw the arc. Repeat this action three more times, moving in one direction around the circle. Each should have six arcs and six intersection points.

3. Construct a positive hexagon. Join all six points of intersection of the arcs with the originally drawn circle in steps. Connect the points with straight lines drawn with a ruler and pencil. Later, the performed actions will receive the correct hexagon inscribed in a circle.

Hexagon a polygon with six corners and six sides is considered. Polygons are either convex or concave. For a convex hexagon, all internal angles are obtuse; for a concave one, one or more angles are acute. The hexagon is fairly easy to build. This is done in a couple of steps.

You will need

• Pencil, sheet of paper, ruler

Instructions

1. A sheet of paper is taken and 6 points are marked on it approximately as shown in Fig. one.

2. Later, after the points were marked, a ruler, a pencil are taken and, with their help, stepwise, the points are connected one after another as it looks in Fig. 2.

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Note!
The sum of all the interior angles of the hexagon is 720 degrees.

Hexagon Is a polygon, the one that has six corners. In order to draw an arbitrary hexagon, you need to do every 2 actions.

You will need

• Pencil, ruler, sheet of paper.

Instructions

1. You need to take a pencil in your hand and mark 6 arbitrary points on the sheet. In the future, these points will play the role of angles in the hexagon. (fig. 1)

2. Take a ruler and draw 6 segments according to these points, which would be connected to each other according to the previously drawn points (Fig. 2)

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Note!
A special type of hexagon is the positive hexagon. It is called such because all its sides and angles are equal to each other. It is allowed to describe or inscribe a circle around such a hexagon. It is worth noting that at the points obtained by touching the inscribed circle and the sides of the hexagon, the sides of the positive hexagon are halved.

Useful advice
In nature, positive hexagons are very popular. For example, the entire honeycomb has a positive hexagonal shape. Or crystal cell graphene (a modification of carbon) also has the shape of a positive hexagon.

How to build one or the other injection Is a big question. But for some angles the task is invisibly simplified. One of these angles is injection at 30 degrees. It is equal to? / 6, that is, the number 30 is a divisor of 180. Plus, its sine is known. This helps in building it.

You will need

• protractor, square, compasses, ruler

Instructions

1. To begin with, let's see a particularly primitive environment when you have a protractor in your hands. Then a straight line at an angle of 30 degrees to this one can easily be postponed with the support of it.

2. In addition to the protractor, there are injection nicks, one of the corners of which is equal to 30 degrees. Then another injection injection the nick will be equal to 60 degrees, that is, you need a visually smaller injection to build the required straight line.

3. Now let's move on to non-trivial methods of constructing an angle of 30 degrees. As you know, the sine of an angle of 30 degrees is 1/2. To build it, we need to erect directly injection ny tre injection nickname. Perhaps we can build two perpendicular lines. But the tangent of 30 degrees is an irrational number, therefore we can only calculate the ratio between the legs approximately (exclusively, if there is no calculator), and, therefore, build injection at about 30 degrees.

4. In this case, it is allowed to make an accurate construction. Let's erect again two perpendicular straight lines, on which the legs will be located directly injection new tre injection nick. Set aside one straight leg BC of some length with the support of a compass (B - straight injection). After that, we will increase the length between the legs of the compass by 2 times, which is elementary. Drawing a circle centered at point C with a radius of this length, we find the point of intersection of the circle with another straight line. This point will be point A straight injection new tre injection nickname ABC, and injection A will be equal to 30 degrees.

5. Erect injection at 30 degrees is allowed and with the support of a circle, applying that it is equal to? / 6. Let's erect a circle with radius OB. Consider in the theory of injection nick, where OA = OB = R is the radius of the circle, where injection OAB = 30 degrees. Let OE be the height of this isosceles tre injection nika, and, consequently, its bisector and median. Then injection AOE = 15 degrees, and, using the half angle formula, sin (15o) = (sqrt (3) -1) / (2 * sqrt (2)). Therefore, AE = R * sin (15o). From here, AB = 2AE = 2R * sin (15o). Building a circle with radius BA centered at point B, we find the intersection point A of this circle with the initial one. The AOB will be 30 degrees.

6. If we can determine the length of the arcs in some way, then, setting aside an arc of length? * R / 6, we also get injection at 30 degrees.

Note!
It must be remembered that in point 5 we can only construct an angle approximately, because irrational numbers will appear in the calculations.

Hexagon a special case of a polygon is called - a figure formed by most of the points of a plane, bounded by a closed polyline. A positive hexagon (hexagon), in turn, is also a special case - it is a polygon with six equal sides and equal angles. This figure is significant in that the length of all of its sides is equal to the radius of the circle described around the figure.

You will need

• - compasses;
• - ruler;
• - pencil;
• - paper.

Instructions

1. Select the length of the side of the hexagon. Take a compass and set the distance between the end of the needle, located on one of its legs, and the end of the stylus, located on the other leg, equal to the length of the side of the figure being drawn. To do this, it is allowed to use a ruler or to prefer a random distance, if the moment is insignificant. Fix the legs of the compass with a screw, if possible.

2. Draw a circle with a compass. The chosen distance between the legs will be the radius of the circle.

3. Divide the circle with dots into six equal parts. These points will be the vertices of the corners of the hexagon and, accordingly, the ends of the line segments representing its sides.

4. Set the leg of the compass with the needle to an arbitrary point located on the line of the outlined circle. The needle should pierce the line correctly. The accuracy of the constructions directly depends on the accuracy of the compass. Draw an arc with a compass so that it crosses at 2 points the circle drawn first.

5. Move the leg of the compass with the needle to one of the intersection points of the drawn arc with the original circle. Draw another arc, also intersecting the circle at 2 points (one of them will coincide with the point of the previous location of the compass needle).

6. In a similar way, rearrange the needle of the compass and draw arcs four more times. Move the leg of the compass with the needle in one direction around the circumference (always clockwise or counterclockwise). As a result, six points of intersection of arcs with the originally constructed circle must be identified.

7. Draw a positive hexagon. Stepwise pairwise join the six points obtained in the previous step. Draw the line segments with a pencil and ruler. As a result, the correct hexagon will be obtained. Later, the construction is allowed to erase auxiliary elements (arcs and a circle).

Note!
It is good to choose such a distance between the legs of the compass so that the angle between them is equal to 15-30 degrees, on the contrary, when constructing, this distance can easily get lost.

When building or developing home design plans, it is often required to build injection, equal to the more closely available one. Samples and school geometry skills come to the support.

Instructions

1. An angle is formed by two straight lines starting from one point. This point will be called the vertex of the corner, and the lines will be the sides of the corner.

2. Use three letters to indicate the corners: one at the top, two at the sides. Are called injection, starting with the letter that stands on one side, then they call the letter standing at the top, and after that the letter on the other side. Use other methods for marking corners if you are more comfortable with the opposite. Occasionally only one letter is called, which stands at the top. And it is allowed to designate angles with Greek letters, say, α, β, γ.

3. There are situations when you need to draw injection so that it is closer to the given angle. If there is no probability of using a protractor when constructing a drawing, it is allowed to do only with a ruler and compasses. It is possible, on the straight line marked in the drawing with the letters MN, it is necessary to erect injection at the point K, so that it is equal to the angle B. That is, from the point K you need to draw a straight line forming with the line MN injection, the one that will be equal to the angle B.

4. At the beginning, sweep along a point on the entire side of this corner, say, points A and C, then combine points C and A with a straight line. Get tre injection Nick ABC.

5. Now build on the line MN the same three injection a nick so that its vertex B is on the line at point K. Use the rule for constructing a tree injection nika on three sides. Set aside segment KL from point K. It must be equal to the BC segment. Get point L.

6. Draw a circle from point K with a radius equal to segment BA. Draw a circle from L with radius CA. Combine the obtained point (P) of intersection of 2 circles with K. injection nickname KPL, the one that will be equal to three injection Nick ABC. So you get injection K. It will be equal to the angle B. In order to make this construction more comfortable and faster, set aside equal segments from the top B, using one compass solution, without moving the legs, describe the circle with the same radius from point K.

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Note!
Avoid accidental metamorphosis of the distance between the legs of the compass. In this case, the hexagon may turn out to be wrong.

Useful advice
He is good at making constructions using a compass with a perfectly sharpened lead. This will make the constructions especially accurate.