Procedural Circles, Rings and Linked Rings in Blender Cycles

Procedural Linked Rings

Procedural Linked Rings

A step by step guide to linking rings in Blender

There are two important parts to making linked rings in Blender: Making a ring and making it repeat. The ring requires a bit of math, but it’s not hard math. Repeating requires a different bit of math, but it’s not hard either. I’ll explain the math as I go along, but you can skip it and still learn how to use ring textures.

Drawing a circle

Remember from Cartesian geometry that a circle is all of the points that are the same distance from a given point. The distance is called the radius, and the given point the center.

The Generated output from the Texture Coordinates node encodes the X, Y, Z coordinates of a point in space. Since we’re only using a flat plane, we only care about X and Y. We can feed the Generated output into the Vector input of a Separate XYZ node and obtain those coordinates, mapped so they fit in the range of 0 to 1. If we view our plane from above, the X coordinates run along the bottom of the plane from zero on the left to 1 on the right. Because blender also uses the range of 0 to 1 for the gray scale, we can visualize this using a simple node group

X to gray mapping

A simple node group that will display the X coordinate as a gray scale running from black on the left to white on the right.

Rendering this gives up the coordinates, ranging from black as 0, to 1 as white.

gray scale

A gray scale representation of the X coordinates ranging from black as 0 to white as 1

If we take the X and Y values from the Separate XYZ and combine them properly, we will know the distance of any point on the plane from the origin, the lower left hand corner. Recall from Cartesian geometry that the distance to a point is the square root of the sum of the square of X and the square of Y, SQRT(X^2+Y^2). We calculate this with a combination of 4 math nodes. Blender doesn’t have a node to calculate the square root, but recall that the square root is the same thing as raising the sum to the 1/2 power.  (generalizing: The Nth root is the same as the 1/Nth power)

calculating distance

A node group that calculates the distance from the origin at the lower left hand corner to each point on the plane

Producing

Distance from the origin as gray scale

A gray scale showing the distance from the origin to each point on the plane

That’s an interesting blob, but we want our center at the midpoint of the plane. The formula for the distance between two points is only slightly more complicated that the one for the distance from the origin: sqrt((X1 – X0)^2 + (Y1 – Y0)^2) where (X0, Y0) is the coordinate of the center and (X1, Y1) Is the coordinate of the point.  An easy adjustment:

Calculating the distance from the center

A gray scale showing the distance each point is from the center of the plane, rather than the origin.

producing

A gray scale showing the distance each point is from the center of the plane, rather than the origin.

gray scale showing distance of a point from the center of the plane

It’s a step in the right direction. Let’s turn it into a circle, using another math node, a greater than. Now the gray scale goes from black at the center to 0 at the edge, so it’s no longer a direct match for the coordinates.  Let’s place the edge of the circle half way between the center and the edge. Since the center is at .5, half way between it and the edge will be half that or .25. The greater than node will return a 1 if the point is farther away and a 0 if it’s not. We can apply that to a Mix RGB node and produce a nice red circle on a blue field.

A red circle on a blue field

Using a greater than node to divide the plane to inside the circle and outside the circle.

Renders

A red circle on a blue field.

A red circle on a blue field.

Now to make it a ring.  Let’s use a less than math node along with the greater than. We’ll make anything that’s in the range between the two red and leave everything else blue. Blender doesn’t have logic nodes, but we want to know when the distance is less than the higher value and also greater than the lower value. To do this we use a minimum math node.  If the distance is outside the range, one or the other of the compare nodes will return 0 and the minimum will select 0.  If it’s inside the range, than both will return 1 and the minimum will select 1.

We need to make another change. For the circle, we had 0 mean outside and 1 mean inside. But for the ring we have the opposite. There are several ways to fix this.  I simply switched the two colors in the mix RGB node.

A ring ring on a blue field

A ring ring on a blue field

and we have our red ring on our blue field

A ring ring on a blue field

A ring ring on a blue field

Now we can create a ring of any width. The value in the greater than provides the inner radius of the ring, while the value in the less than provides the outer radius.

Repeating patterns

It would be nice if Blender had a tile option that would resize its input and repeat it, like the checker tile, but using the whole pattern. It doesn’t, so we fake it.  Time for our last bit of math: Modulus. We’re going to repeat the pattern by repeating the coordinates, going from 0 to 1 many times. Recall that if you divide one integer by another, the remainder is called the modulus of the number being divided. For instance 22 divided by 7 returns 3, 3 times 7 returns 21 and 22 minus 21 gives 1, so the modulus of 22 with respect to 7 is 1. The modulus “wraps around” every time the number is divisible by 7, so it will be a repeating pattern from 0 to 6. The function that calculates the modulus is modulo and blender has a module function.

Since Blender coordinates go from  0 to 1, the Blender  modulo function is a bit funky. Without going into the details, what we need to know is that the modulo operator takes an argument such that the pattern from 0 to 1 is repeated a certain number of times that depend on the value of the modulo’s second argument. If modulo were .5, the pattern would be reproduced twice. If it were .25, it would be reproduced 4 times. The number of patterns across is 1 divided by the second argument. Here’s an example with modulus set to .25. Notice that the value node contains the number of repetitions we want. The divide node converts it to the value expected by modulo.

Repeating coordinates 4 times

Repeating coordinates 4 times

as expected, gives

Repeating coordinates 4 times

Repeating coordinates 4 times

This is good, but it’s not quiet what we want. (Why it’s not is left as an exercise for the curious.) It repeats the pattern 0..1 4 times, but what we really want the pattern 0..1..0 Let’s add a color ramp with white at both ends and black in the middle

Trying to make the repetition symmetric

Trying to make the repetition symmetric

Unfortunately, we get symmetry but it doesn’t go 0..1..0. (The math for this is left as an exercise for the curious.) Without going into details, I will replace the modulus operator with a group of math nodes that give us useful values to work with.

A symmetric modulus

A symmetric modulus

giving what we want

A symmetric modulus

A symmetric modulus

Putting it all together

There are only two more things to do: have the repeating pattern be the rings, and then do it again, offset, to make the links. Putting it together is easy:

Repeating RIngs

Node group that puts the ring drawing pattern together with the pattern replication node to produce repeating rings.

renders as

Repeated Rings

Render that puts the ring drawing pattern together with the pattern replication node to produce repeating rings.

The next step requires duplicating all those nodes and adding two more.  This is a good time to turn some of the repetition into node groups. Calculating the ring become two node groups, distance and in range

calculate the distance from the origin

calculate the distance from the origin

is in Range

return 1 if x lies between the lower and upper bound, 0 otherwise.

and symmetric modulus becomes another

Generate a symmetric modulus

Generate a symmetric modulus

Put it all together and duplicate it we get our final material

Linked Rings

Blender Cycles Material that will generate linked rings on a plane.

We’ve added a mapping node to the second copy, using its parameters to move the X and Y coordinates slightly, resulting in the image we started with.

Extra credit: One last node group.

Suppose we wanted to reuse the repeated ring pattern in other contexts. We can easily create a node group that does that, if we have our own version of the mapping node. I won’t go into the custom mapping node, but having it handy, we create a new node group that produced the repeating rings pattern

Repeating Rings Node Group

Repeating Rings Node Group

In our example, we want the rings to be the same size, so we use a few value parameters to insure this and our final material is the very simple

Repeating rings material

Having created node groups, our material is simplified down to 7 nodes, although we add 3 value nodes to make it clear that we meant the rings to be the same size

Appendix: The custom mapping node group

This is not a precise duplicate of the mapping node, but it often serves when we would like to control the mapping node. It duplicates the group behavior of applying the translation first, the rotation second, and the scaling last. It could be made simpler by using vector math for the translation and scaling. This version was written for clarity.

mapping parameterized

Vector mapping is translation, rotation, and scaling, in that order

vector translate

Vector translate is simply adding constants to each of X, Y and Z

Vector Rotate XYZ

When applying rotation, the order matters. The mapping node rotates around X, then around Y, finally around Z.

vector scale

Vector scale is similar to vector translate. It multiplies X, Y and Z by a constant value.

Vector Rotate 2D

Rotating around an axis works the same way for each axis, so we use a single node group. It makes Vector Rotate XYZ a little hard to follow, but it works.

degrees to radians

Blender does trig using radians, so we apply the conversion factor.

A procedural layered wall in Blender Cycles – Part 1: creating the layers

Introduction

A section of a wall

This texture was a contest entry in the Facebook page “Blender Procedural Textures”

The Facebook “Blender Procedural Textures” page runs a weekly contest to use procedural textures to match a specific texture, usually from a photograph. This image is similar to my entry for one of the contests. Most of the texture will be familiar to you if you create your own bump pattern and cracks, that are made with common techniques for doing so. The method for creating the layers may not be: it is not the typical approach.

This is a step by step explanation of the approach taken. If you understand what each of the individual nodes do, you will have no trouble following it. The examples were created using Blender 2.79b. Most of the approach should apply to any release with cycles. The displacement will not, but a bump map may be substituted.

Analyzing the texture

This texture can be described as having these parts

  1. The displacement of the underlying wall
  2. Obviously, the paint layers
  3. soft wear, in the forms of speckles and dots
  4. scratches

Normally this could be accomplished using the techniques described in BlenderInsight, Joakim Tornhill’s excellent manual Procedural Wear from A to Z in Blender. The manual can be found in the files section of the Facebook procedural textures page.

A close inspection of the original photograph, (which I do not have permission to post) shows that there is different kinds of wear in each of the segments, possibly because a different pigment was used for each and wore differently.

How to create regions and use them to select between textures

The simplest approach is to create a color ramp and use it to select the colors. Use a gradient texture for the color ramp fac and get something like

Gradient bands with linear color ramp

There are two obvious problems: the bands are vertical instead of horizontal, and they blend. The first is easy to solve: rotate the plane. The second is also easy: switch the color ramp from linear to constant

Rotating the plane is left as an exercise for the reader. The edges are too straight, so add some noise.

Use the multiply to control the strength of the noise texture. Use the color ramp to set the color, number, order and width of bands. This will do for most purposes, but it is very difficult to use it to put a different texture in the blue region than in the green. Is there another way?

Yes.  One way is with math.  It’s easy math, so take a calming breath and follow along. Start by thinking about how to mix two colors, so that one or the other is chosen, but not both. We can do this simply using a greater than node. We can easily split the plane in half this way

This image is produced by this simple node group.

The gradient texture produces a number between 0 and 1. Greater than compares that value to its other value, in this case .5. If the upper value, from the gradient, is greater than the lower value, greater than returns 1, otherwise 0. When the mix input factor is 0, only the top color is used, otherwise the bottom color. You change the width of the region by changing the value being compared to.

Add another region by adding another greater than and mix.

This image is produced by this node group

We can continue to add as many bands as we want.

There is one more feature of the banding to model: the bands slope downward from left to right. One way to do this is through the use of texture coordinates. Replace the gradient texture with a texture coordinates node’s generated coordinates and get

This is not precisely what we want.

This node group will give us exactly the same results as the gradient texture, but for a different reason. Any point on the plane can be described by two numbers, the distance along the bottom, X and the distance along the side, Y. X varies from 0, on the left, to 1, on the right. Y from 0 on the bottom to 1 on the top.

Add the slant by combining X and Y.  We could simply add them, but we only want X to have a small effect, so multiply it by a small number.

This looks good, three bands, slanted in the right direction. Add a little noise and you have what you want.  (Notice that I also rotated the bands).

The texture coordinate provides the position of a point.  The y value is used to select the band edges. As x gets larger going from left to right, it is added to y, giving the slant.

Let’s add a texture that applies only to the blue band, just a simple Voronoi to demonstrate.

This is straightforward. Replace the color in the color mix with the mixed texture. Now let’s add a texture that crosses all the bands.  This is ugly, but it shows the effect.

Add the noise on the borders back in and the result looks like this

and is produced by this

The overlay texture is mixed with the output of the band mixture and provides the color to the diffuse node.

The evolution of a character model

As I follow the steps in Blender Production, (see this blog entry for details), one of the things I’ve discovered that the author didn’t discuss is how evolving skills with blender impact the animation work flow. The character I’m developing for my little animation has evolved from a simple stick man to a full human model with a complete rigging, but it has been a several week diversion and required redoing my master reel.

My first idea was to animate a literal stick man.  It would have a body, arms, and legs that were simple cylinders. There would be no arms or feet. The head would be a featureless sphere.  Easy to model, trivial to animate.  I soon realized that I wanted arms and feet, and that the body should have a different cross section than the limbs.  I came up with a design that was still trivial to model.

Rough stick man figure

My first attempt at modeling a character. The idea was to have a very abstract character in a very realistic environment

The awful colors were not meant to be used in the final animation. They were simply to make it easy to distinguish the bits. It took a whole ten minutes to model.  Most of the time was spent getting the proportions I wanted.  It worked fine for the story reel, but there was no way it would work in the finished animation.  I thought I could get away with simply modifying it a bit to fix the proportions.  This result was also disappointing.

Stick man evolves

Some size and shape tweeking and horrible cone feet and hands are not enough.

The tweeked version looked good enough to use for rough animation.  I redid the storyboards and story reel using it. I realized I had no idea how to build the rig of controls that would allow me to animate it.  This led to my first diversion, Blender’s Armatures: A Crash Course, which is a brief but dense introduction to blender’s armature, the main tool for controlling characters in animation.

Having completed the tutorial, and armed with a little knowledge, I made a simple rig of 14 bones. This took me nearly two days, as I learned by doing the things I hadn’t really understood from the tutorial. In the end I had a simple, difficult to use rig for my character. I spent a couple of weeks using it to make a master reel, with the character posed for each of the key frames.

While I was doing all of this, I was also trying to learn character animation from tradigital Blender. I got as far as the example of creating a walk and was overwhelmed by the amount of control a believable character would need. It became clear that my simple rig would not serve.  I found another excellent tutorial, DVD training 8: Humane Rigging. Unlike previous tutorials, it consists of more than one video.  There are, in fact, 35 videos, a DVD’s worth, which explains the “DVD training” in the title.

If you want to learn how to rig characters in blender, I highly recommend this tutorial.  It goes through five different characters and rigs them. Each character is briefly introduced and its animation requirements are discussed. A series of videos introduce a particular rig designed to solve a specific problem. The problem is described, alternatives are presented and discussed, and the rig is put together step by step to show how it is done. One or more videos are then spent on putting the  various rigs together into a single rig.

I learned a great deal from the video and feel it was time well spent. The downside was that I spent two weeks watching the tutorials and playing along at home, building each rig in Blender.

Between reading tradigital Blender, watching the two tutorials, and setting up the first attempt at a master reel, it became clear that rigging was going to be difficult and time consuming and that I needed a more realistic character. An abstract character simply wasn’t going to work. I started another digression, starting to follow a tutorial on how to model characters for animation.

Then I was reminded of makehuman, an open source tool designed to, well, make humans.  With makehuman, you can create a usable human mesh in very little time. Makehuman has a very shallow learning curve, at least to reach the point where you can export a usable character for blender to import.  I also encountered rigify, a blender add on that makes character rigging easy. I spent an hour stumbling around with makehuman and blender, finally reaching my current character. It was generated in makehuman and imported into blender.  The import allows selecting between the rigging that makehuman generated, or a rigify rig. I don’t see anything wrong with the makehuman rig, but am trying the rigify rig to see how well it will work.

Current stick man figure

A makehuman character, slightly tweeked from the default, with “clothing” being nothing more than colors

It took another hour of fumbling around to replace my abomination with this character and generate a new master template file. In the new master file, the character is posed in each of the keyframe positions, but the poses are not very good.  But it’s good enough to use as a starting point.

In total, I was diverted from my animation project for six weeks.  I will need some of the things I learned, so it wasn’t time wasted. I may never rig anything from scratch, but the tutorial taught me a good deal I didn’t know about how the controls of a good rig are implemented and how they work.  That will help a lot when I use rigify to animate the makehuman character.

Had I remembered makehuman, even without being aware of rigify, I could have generated a usable humanoid character, even with fumbling around learning how to export an import, in less than an hour. This would have saved me significant time, but I wouldn’t understand rigging as well as I do now.  I consider the knowledge a fair trade for the time.