This guide is heavy on theory and light on practice. Don't let it discourage you! This only means that constructing roundness between points is practically easy and straightforward in Crocotile 3D! As soon as you familiarize yourself with the principles and techniques presented in this guide, you will be able to construct any roundness you desire between any set of points without even thinking about what are you doing!

Let's say we have two points somewhere in space.


What sort of roundness can connect them? In other words, where could we place these points on a circle?


Well, we could place points like this:


Or like this:


Or like that:


Or even like this:


To answer the question, roundness of any curvature can connect any set of points. It does not matter how far the points are or their relative position. We can always position those points on a circle to achieve the desired roundness between them.

Roundness of any curvature can connect any set of points.

Don't be confused by the pictures: points do not move; circles do.

A circle can be defined as a set of all points in a plane that are at a given distance from the centre.

This definition implies that if you place points on circle's circumference, these points will be at equal distance from the central point.

Points are at equal distance from the central point

There are 360 degrees in a circle. The picture shows that a circle can be broken down into 4 quarters, each of which measures 90 degrees, contributing to the total of 360 degrees in a circle. The more degrees between points, the longer the roundness between them.



There are 90° of roundness between these points:


There are 180° of roundness between these points:


There are about +-60° of roundness between these points:

There are 360 degrees in a circle. The more degrees between points, the longer the roundness between them.

How do find the central point of a circle that contains the desired roundness?



Let's take a look at certain properties of a circle to get an idea:


Let's connect two points that lie on a circle's circumference with a line:


Let's mark the line's middlepoint:


Now, let's connect the middle point with the central point of the circle:


The line that connects the middle and central points is perpendicular to the line that connects the given two points! Perpendicular means that lines form a 90-degree angle.

The line that connects the middle and central points is perpendicular to the line that connects the given two points.

Keeping circle's perpendicular lines in mind, let's plot a circle that contains the desired roundness!

Let's place two points:


Connect them with a line:


Mark the line's middle point:


Draw a perpendicular line (a line that forms a 90-degree angle with another line) from the middle point:


Where should we place the central point? The answer will scare you: anywhere you wish.

You can place the circle's central point anywhere on the perpendicular line drawn from the midpoint of the line connecting the given points.

Placement of the central point on the line determines the roundness between the points:

Placement of the central point on the perpendicular line determines the roundness between the points.

Don't worry, you won't have to play the game of luck each time you have to place a central point. While the central point's position is crucial, it's not the primary focus of our search. The most crucial factor is the angle of roundness.

Position of the central point depends on the desired angle of roundness.

The answer lies in understanding angles of a triangle.

Why triangle? Because when we connect points with a central, triangles are formed!


By controlling angles of triangles, we can set the desired roundness angle! Don't worry, it is simple. All you will have to do is to divide the desired number by 2 and minus in from 90. Why? Consider the following:

There are 180 degrees in a triangle.


The perpendicular line always forms 90° angle with the line that connects given points.


So, if we draw a line at angle X to the perpendicular line, the other angle Y of the formed triangle will be equal to 180 - 90 - X.



180 - the total of degrees in a triangle.
90 - the angle formed by the perpendicular line
X - angle that we CONTROL.
Y - part of the angle that forms the DESIRED roundness.

Since 180 - 90 = 90 is constant, we might as well simplify:

90 - X = Y

However, since we want to get a certain Y value and X angle is used to get this value, it makes sense to rearange variables:

90 - Y = X

Consider the following examples:

Let's draw a line at a 45-degree angle to the perpendicular line. The other angle will then be equal to 45 degrees; 90 - 45 = 45.



Let's draw a line at a 60-degree angle to the perpendicular line. The other angle will then be equal to 30 degrees; 90 - 60 = 30.



When choosing the angle for the line drawn to the perpendicular, you need to remember that the resulting angle will be doubled (multiplied by 2). Why?

Because there are two equal triangles, from which two equal angles form the degrees of roundness.




Here's a formula to calculate at what angle do you have to draw a line to the perpendicular in order to get the desired roundness:

90 - (Y/2) = X,

where X is the angle by which you have to draw the line to the perpendicular, and Y is the desired degree of roundness.

Examples:
If I wanted roundness of 90°, I would have to draw the line at 45° to the perpendicular; 90 - (90/2) = 45.


If I wanted roundness of 75°, I would have to draw the line at 52.5° to the perpendicular; 90 - (75/2) = 52.5.



However, there are special cases. For instance, if we wanted roundness of 180°, the equation would produce 0. Values above 180 produce minus values. Don't worry, these cases will be covered in the practical section!


You are armed with all the theory required, it's time to put it into practice!

Two tiles:


Roundness is desired between the purple points:


First of all, connect the points with a tile. This is the construction tile:


Subdivide the construction tile into two equal segments (Alt+D). The resulting vertices are the middle point:






Set the crosshair tilt (yellow axes) along an edge of the construction tile:




Place the crosshair on the middle point vertex:


Tilt the crosshair (T):


Rotate half of the construction tile by 90° (direction depends on where you wish the roundness to curve). This is the perpendicular tile.


For the next step, I recommend you to extend the perpendicular tile along the tilted invisible planes:






Rotate the other half of the construction tile by the angle that will produce the desired roundness. This is the angle tile. IT IS ESSENTIAL THAT THE CROSSHAIR IS TILTED ALONG EDGES OF THE CONSTRUCTION TILE.

In this example, we construct a 90° roundness. Using the equation: 90-(Y/2) = X, where Y is the value of desired roundness, and X is the angle that will produce the desired roundness, we get the following calculations: 90 - (90/2) = 45.

So, the angle that will produce the desired roundness, in this case, equals to 45°.




Set the crosshair tilt (yellow axes) along an edge of the angle tile:




Tilt the crosshair:


Select the vertices of the angle tile that are nearest to the perpendicular tile:


Use the align function "Against", while the camera faces the perpendicular tile. Note: the align functions can be found in the Transform tab.




Vertices of the angle tile that touch the perpendicular tile are the central point.


Delete the perpendicular tile if you wish:




Place the crosshair on the central point vertex:


This is the part where you decide how many tiles will be used to create the desired roundness. Simply divide the value of the desired roundness by the number of tiles you wish to use.

In this example, we want to use 9 tiles to create the desired roundness. Since the value of the desired roundness is 90, we divide 90 by 9. So, in order to capture the desired roundness in 9 tiles, we have to rotate the angle tile by 10° (90/9 = 10).

Copy the angle tile, paste it and rotate by the value you've decided upon:




Fill the resulting gap between the angle tile and its copy. This is the roundness tile.


Copy, paste and rotate the roundness tile. Repeat until it touches the other set of points:








Delete the angle tiles:


90° roundness between two pairs of points:



That's it! Although it may appear cumbersome on the first look, the whole process takes no more than a minute when you know what you are doing!

However, there are nuances. For instance, 90° roundness shown in this example is not the only way 90° roundness can manifest itself. Check the next section to see what I mean!

Consider the following examples of 90° roundness:






Even though roundness has the same value of 90°, it looks very different from example to example. Why is that? Consider the following image:


The colored lines correspond to the squares in the lower right corner of the demonstration pictures. Notice how varying line slopes capture different aspects of the circle's roundness, creating distinct appearances.



Take a look at the same examples put next to their construction tiles:





As you may observe, the slope of the construction tile determines what part of roundness will be taken from a circle.

The slope of the construction tile determines what part of roundness will be taken from a circle.

Before continuing, let's quickly classify the various looks of roundness:

Smooth. Roundness flows from one tile to the other. There is a feeling of positive tension and elegance in such look. On the other hand, it may look "too perfect" and boring.


Relaxed. Roundness hangs between the tiles. There is a feeling of relaxation and lack of tension. On the other hand, it may look lifeless, careless and lazy.


Tensed. Roundness stretches between the tiles. There is a feeling of high tension and energy in it. On the other hand, it may look crude and mechanical.



Does the slope of the construction tile limit you to a certain look? NOT AT ALL! Here's an example:

A set of points with the construction tile slope of 20°:


Smooth roundness of 40°:


Relaxed roundness of 90°:


Tensed roundness of 10°:

There is a simple relationship between the construction tile angle value and the visual appearance of roundness. Consider the following diagram:



Assuming the construction tile slope value equals to Z, the following holds true:

• The closer the value of roundness is to 2z (double the value of Z), the smoother the roundness will appear. If the value of roundness is equal to 2z, it will appear as smooth as possible.

• If the value of roundness is less than 2z, it will appear tense (the bigger the difference, the tenser the roundness will appear).

• If the value of roundness is more than 2z, it will appear relaxed (the bigger the difference, the more relaxed the roundness will appear).


For example:

This slope is 20° (Z = 20):


Maximum smoothness is achieved on 40° roundness (20*2 = 40. 40 = 2z). Here's how it looks:


Less than 40° will produce tensed roundness. Here's roundness of 10°:


More than 40° will produce relaxed roundness. Here's roundness of 90°:



It was mentioned that following this rule you may achieve the roundness of "as smooth as possible". What does it mean? Take a look at the following smooth curves:

40°:


90°:


135°:



All of these examples of roundness are as smooth as possible. However, you may notice that 40° looks a bit tense compared to 90° and 135° looks relaxed compared to 90°. Consider the following diagram:


• The closer the roundness to 90°, the more potential for smooth roundness it has. 90° degrees have the maximum potential for smoothness.

• Roundness less than 90° has more potential for tensed appearance.

• Roundness more than 90° has more potential for relaxed appearance.

You can refer to a specialized guide to learn how to measure the construction tile slope:
https://gtm.you1.cn/sharedfiles/filedetails/?id=2879781450

Here's a quick method!

Let's assume we wish to measure this slope:


Place the invisible planes on the lowest point of the slope:


Place a tile under the slope:


Select the tile and rotate it towards the slope until it touches the slope. Add degrees from your rotations together. Crude? Yes. Functional? Absolutely.


In this example I will start rotating the tile by 10°. As I rotate, I count:

10°...


20°...


30°...


40°...


The tile is very close to the slope. I will stat rotating the tile by 1°. As I rotate, I count:

41°...


42°...


43°...


44°...


45°!


The tile touches the slope and lays on it perfectly. The slope is 45°.


But what if the tile overshoots? For instance, what if I were to rotate the tile by 1°:


And it went right through the construction tile:


In such situations, you can:

a) Decrease the value of rotation below 1°, for example, 0.1°
b) Ignore the overshoot and count it as 1°

Unless there is a good reason to get the most precise measurement, I recommend going with the b option. Inaccuracy of less than 1° is negligible. It most likely won't matter that you assumed that the slope is 45° although it is 44.7°.


All said, I recommend this method in situations where you don't expect to measure more than one angle. If you anticipate measuring multiple angles, check the guide link at the beginning of this section for a more suitable method.

There are two tiles:


The natural slope between them would not produce the desired roundness look. It is 20°, but 45° is desired in order to produce 90° smooth roundness.


Place a tile in front of the given set of points:


Rotate the tile by the desired angle. In this example, the tile is rotated by 45°.


Set the crosshair tilt along the tile's edge and extend the tile beyond the other set of points:




Now, you have at least two options:

Option 1:
Disable the crosshair tilt and use the align functions to connect the other set of points to the tile.


If you have a trouble with this step, check this guide: https://gtm.you1.cn/sharedfiles/filedetails/?id=2883486583
Connect the tile's vertices with the other set of points. Now the tile became the construction tile and you can construct the desired roundness.





Option 2:
Copy and paste the tile that contains the other set of points. Place it in front of the original tile.


Disable the crosshair tilt and use the align functions to connect the other set of points to the tile.


If you have a trouble with this step, check this guide: https://gtm.you1.cn/sharedfiles/filedetails/?id=2883486583

Connect the tile's vertices with the other set of points. Now the tile became the construction tile and you can construct the desired roundness.




Delete the copied tile:




Place Gizmo Crosshair on the set of points where roundness begins. Toggle "Scale" mode. Move the roundness along the axis as close to the other set of points as possible:




Connect vertices of the roundness with the other set of points:

There are two tiles. Pairs of points have different distances between them:


Connect the points with the construction tile:


Subdivide the construction tile:


Tilt the crosshair along the construction tile and extrude the construction tile (preferably in the direction of where the roundness will curve).



Delete every extruded tile, leaving only the ones on sides:








If you, like me, made a mistake of extruding tiles in the opposite direction to roundness, extend the side tiles along the tilted planes:




Hide the side tiles. We will need them later.




Disable the crosshair tilt and place the crosshair on one of the points:


Select every vertex of the construction tile on the corresponding edge to the selected point:


Use the align function "Flatten" to align the vertices along the invisible plane:


Repeat on the other side:


Now you should have a normal construction tile which has all of its vertices along the same invisible plane. If not, align vertices until you do.


Construct roundness per usual method:






Activate (Show) the side tiles again:




Select every vertex on sides of the roundness tiles and use the align function "Against" to align them to the corresponding side tile:








Do the same on the other side:




Delete the side tiles:


Roundness complete!

Let's construct a 180° roundness!

First of all, let's see the roundness angle we need. Let's use the formula 90 - (Y/2) = X

90 - (180/2) = 0.

Oh. Zero. What now?

Do not fear! This zero means that we don't have to do any extra steps with tilting crosshairs and connecting vertices to tiles. Divide the construction tile into two and use one side to create angle tiles!

Divide the construction tile into two:


Decide how many tiles you wish to dedicate to the roundness. In this case, I wish to dedicate 10 tiles. 180/10 = 18. So, I will rotate the construction tile by 18°:


Fill the space:


Copy, paste and rotate the roundness tiles:






Delete unnecessary tiles:

Personally, I don't see much use for roundness beyond 180° degree in the context of construction between given points. It is possible to construct such roundness, though.

If the desired roundness exceeds 180°, then:

1) Minus it from 360. This will be your Y value. Let's say, you want to create a roundness of 270 degree. 360 - 270 = 90.
2) Use the resulting value for the formula 90 - (Y/2) = X. In this case, 90-(90/2) = 45.
3) Rotate the construction tile in the opposite direction you would otherwise with a value less than 180°. In this example, if you were to construct a 90 degree roundness you would rotate the tile by 90. However, since you are constructing a 270 degree roundness, rotate the tile by -90°.
4) Rotate the angle tile by the value of X. 45° in this case.
5) Decide how many tiles you wish to dedicate to the roundness. In this case, there are 270° of roundness. For this example, let's use 15 tiles to represent it. 270/15 = 18. So, the angle tile must be rotated by 18° to form the desired roundness tile.
6) Rotate the angle tile IN THE OPPOSITE DIRECTION you would otherwise with a value less than 180°
7) Copy, paste and rotate the roundness tile until the desired roundness is constructed.

Here's how the roundness from the example would look like:


I hope this image will excuse me from not including a more detailed instruction and a variety of images to support it.