Topology of the body, topology of the face, topology of the hands. "rubber geometry" or topology through the eyes of a student

This tutorial is a good start for anyone who wants to learn how to model top-notch characters. Famous in his circle, Jahirul Amin will talk about the importance of correct topology, uniform mesh, the importance of quadrangular polygons and much more.

Before diving into the 3D whirlpool, I suggest having a short educational program and splashing around in the shallow water. Below we will touch on the basics of polygonal modeling, without knowledge of which it is pointless to move on.

Introduction

When geometry becomes a modeler's or animator's aid, the ideal mesh layout comes first. After this, a good topology should come into play, reducing the number of defects in character animation. In other words, a correctly (and on time) created polygon will save not only hours but days of your life.

3-gon vs 4-gon vs N-gon

So what's the difference between 3-, 4-, and N-gon polygons? The answer is obvious: the first has 3 sides, the second has 4, the third has any number of them, more than 4. If you are modeling a character for further animation, we recommend use only quadrilaterals. The process of deforming and dividing quadrangular polygons is much easier, and you will encounter less texture distortion.

It is recommended to hide triangles from your own and other people's eyes. For example, in the armpits or in the groin area of the character. In turn, an unspoken ban is imposed on polygons - they should not exist. They cause distortion and cause a lot of trouble when it comes to rigging and editing vertex groups (aka “weight-painting”).

Finally, a model that consists primarily of quad polygons will be easier to export to other modeling programs such as Mudbox.

The joys of four and trigon polygons and the horror of the N-gon

The contours of the face, which by definition resemble an N-gon, should be brought as close as possible to a quadrangular format. Little of - the location of the polygons should be as uniform as possible in principle. This is what the geometry of the same name calls for. Following these rules will make it easier to go through the rigging stage and will help when deforming the character during the animation process. In addition, the scale of distortion associated with the use of textures will be reduced, although here we should not forget about the importance of the UV scan itself.

To perform the described task, Maya provides the Sculpt Geometry tool.

The Sculpt Geometry tool in Maya will help you “smooth out” your model’s mesh

Responsible for the smooth transition of each individual edge (aka Edge Flow). It may sound simple, but in practice it is a very insidious thing.

If you set out to create a realistic character, it is recommended to study the basics of anatomy before starting work. By following the structure of the human body and the natural movement of muscles, the animator ultimately obtains a copy that is close to the original. This is especially clearly seen during the deformation process. We recommend starting with the process of wrinkle formation and skin stretching.

For stylized and cartoon characters, Edge Flow is much less important. But still, I highly recommend getting at least a basic understanding of human anatomy.

To make the shape realistic, create a good topology and be sure to take into account the smooth direction of the mesh (edges, polygons).

It is also non-manifold. Means that a three-dimensional object cannot be cut and made flat.

Example: Create a cube, select any edge (edge) and extrude it Edit Mesh > Extrude. In front of you is a somewhat shaped object. (Example below on the left) If the cube were made of paper, then when unfolded you would get a cross-shaped figure with broken proportions. Using such an object in Boolean operations is almost impossible.
To fix the situation, use the Cleanup tool.

Violation of the geometry topology can create dozens of problems. Be vigilant and periodically inspect the figure from different angles.

Each loop (edge edge) must have a target

As a rule, modeling begins with a primitive figure (for example, a cube), the structure of which is subsequently complicated by adding edge loops.

It is important that each new element is created with a specific purpose. There are situations in which “less” equals “better.” Understanding the principles of model optimization comes only with experience, so don’t get discouraged and keep working.

Don't complicate your life: detail should be appropriate

Everything we are trying to do on the screen is a reflection of the world around us in its various forms and manifestations. This is why it is so important to get up from the table from time to time. Important not only for developers, but also for animators, riggers, lighting directors, etc.

Take a closer look at the surface, its structure and shadow. How does it reflect light? How does the deformation process occur? The answer to these and other questions will help you make the right decision when modeling any object.

Topic of conversation: TOPOLOGY.

Topology (from ancient Greek τόπος - place and λόγος - word, doctrine) is a branch of mathematics that studies in its most general form the phenomenon of continuity, in particular the properties of space that remain unchanged under continuous deformations, for example, connectivity, orientability. Unlike geometry, topology does not consider the metric properties of objects (for example, the distance between a pair of points). For example, from a topological point of view, a circle and a donut (solid torus) are indistinguishable.

But this is in mathematics. How are things going with the characters? Let me put it in my own words.
Topology is the ability of a mesh to respond correctly to deformations. Be it animation, compression, stretching or other types of deformation. This is achieved by competently constructing a character’s polygonal mesh. There are some rules for this. You can familiarize yourself with some of them.

There is also a concept RE-TOPOLOGY. Changing the topological mesh while preserving the shape of the object as much as possible. The purpose of retopology is to correct the previous (incorrect) topology and/or reduce the number of polygons.

Almost all modern 3D graphics packages have tools for retopology. I personally tried:
1. Maya - both standard tools and plugins.
2. Max - standard tools (horror), plugins and scripts (I liked wrapit, but again not that much)
3. Zbrush - tight and uncomfortable..
4. Topogun - finally found something I liked... if I hadn’t met it
5. 3DCoat.... here I realized that this is so far the most convenient for retopology and UV unwrapping... although it was difficult to figure it out at first... but when I understood the principle of the program - that's it... now retopology is all about it. (don't take this as advertising.)

Well, since this booze has started, I’ll post a couple of my images on the topic of topology.
Head and face

I found an old render of this head.

topology of the face of a humanoid character. You can make both a woman and a child out of him... not to mention a man.
and here's the proof. done quickly, but clearly.
So. a man, an elf, a creature, a woman, and a girl of about 15...
I do not claim that this is the only competent topology, and that this is the ONLY way to do it.
Some studios model characters with their eyes closed. This allows you to get rid of some problems when closing the eye, and avoid deformation of the eyelid when deforming the cheek.

wrist.

I would like to point out that there are vertices here that can be approached by 6 hedgehogs... but in these places there are no problems because the deformations are minimal. Naturally, from this brush you can make the hand of a woman, a man, a child... or anyone...
Scull.

male skull. There are many differences between male and female skulls.

The differences are as follows:
Male and female skulls have a number of differences. Namely:
1. The male skull is more massive than the female and has a rather square shape. The woman's skull is slightly pointed towards the top and more rounded.
2. The upper edge of the eye socket is slightly pointed in the female skull, while in the male it has a smoother curve
3. As a result of evolution, the facial muscles have become more developed. Consequently, the place where the muscles attach to the skull is much more noticeable in men. After all, a warrior and a hunter need powerful jaws for combat and struggle.
4. A man's strong lower jaw is square in shape, while a woman's is round in shape.
5. The depth of the skull of men is greater than that of women. This provides relative safety.
6. The brow ridges on a male skull protrude noticeably more. They protect your eyes from direct sunlight.
7. Men's canines are much larger than women's. The warrior and hunter was forced to eat while on the move, and, therefore, actively chew food and do it quite quickly.
Hand and body.
If the body is female or without clearly defined muscles, then you can ignore the magnifying glasses that form the muscles. This applies to the hands. I draw your attention to the white polygons. they come from under the pectoral muscle and go around the deltoid.

With this article I begin a series of lessons on organic 3D modeling. This article is specifically about the principles of modeling, i.e. absolutely does not depend on the features of your (any) 3D package. The series of articles will cover the following topics:

form,
proportions,
poles,
topology
and much more.

There are a huge number of modeling methods and all of them have their advantages and disadvantages, so there is no such thing as "The best modeling method".

The reason why I took this path forms- she works. I also always wanted to become a sculptor. Before going into detail, I like to sketch out a rough shape. It is because of this that I have achieved so much and that is why I decided to write this article to help beginners in organic 3D modeling and show them the shape before they start doing anything.

The first thing I started with was the shape of the head and I got frustrated because I tried to make it without any reference information (no references- from English reference), only using your imagination. Instead of sketching out a rough shape, my mind was busy with questions like: "How many cuts are needed? Why? Where and When?"

I was worried not only about my head, but also about my eyes, nose and mouth (and I hadn't even gotten to them yet). My brain was confused and I was completely at a loss as to how to create this head... until one day when I managed to sketch out a basic boxing head and behold... see the moment of truth! I was so excited that I decided to do it again! And then again and again, until I got tired of it and was exhausted.

Looking back, it seems so basic and simple to me. All that was needed was to create a box and make a couple of cuts and edits!

However, if it is so simple, then why did I struggle with it for so long? Can we all do this without the problems I experienced? Well, my answer is YES! But only if you mean it the right mindset. For example, I didn’t have one when I first started.

What I realized now is that when we learn 3D modeling, then we are just We don’t teach 3D at all! What we're really doing is looking for the right "mindset." So when you experience difficulties in something, it does not mean that you lack skills or knowledge. This is all because you don't have the right mindset to do what you're trying to do.

Once you have rewired your mind, your rational mind will take over and you will begin to do things naturally. So that's the first thing we have to try to rebuild - the mindset.

Mentality

Drawing a profile (contour): connecting dots

This little example will help you change your mindset.

First, just look at this image. Now we will draw a profile using dots and connect them. If you only had two points (on your forehead and chin) to connect them. How would you do it? Answer: from the forehead to the chin, because there is simply no other way.

However, if you increase the number of points, they will not only allow you shape the profile more exactly, but they will also allow it to be done in many ways, and this already leads to style formation(artistic).

This is very important to keep in mind when you need to make cuts or know where to finish them.

Key Cut (KR) and Fill Cut (FC).

At first it was very difficult for me to understand where and how many cuts I should make when creating a particular shape. So I was looking for an analogy for this process. This analogy turned out to be Animation.

Animation has a concept Key Personnel(KK). In short, this is characteristic poses character in a certain moment of time. This concept also includes Intermediate Frames(PrK), which fill time intervals between Key Personnel.

This not only speeds up the process, but also makes it easier. The more Intermediate Frames (fill cuts) you have, the smoother and more precise the movement will be.

If you are an animator, then you have the power to control the number of PRKs. This is very similar to cutting polygons in 3D.

Drawing a large number of PCs and managing them all is a very tedious job. The same goes for moving a large number of vertices in 3D - it is very labor-intensive.

The idea behind the CD is joints. When a modeler sketches out a rough shape, he always starts with the KR, which always looks rough. If the editor you're using supports dice, then use it to figure it out. Bend/twist the bones at the joints to see your gross form in the poses.

Once all the CDs are ready, you have two options:

Smooth the model.
Sometimes I create a CR, and then simply let the code responsible for dividing the model into a larger number of polygons (subdivision) complete all the CRs for me. The downside is that it doesn't look realistic. So the next step is to use a soft highlight to correct the shape. Sometimes this can save a lot of time (but it depends on what you're modeling).
Add ZR manually.
In most cases, I prefer manual work, since this way I can control the number of points and their location.

Please note that this concept of Key and Fill cuts is not only useful for creating shapes, but also for detailing your mesh. KR and ZR created using partitioning are one of the ways to optimize the mesh (buttocks, thighs, etc.). Also, sometimes a Fill Cut can become a Key Cut depending on how you look at it. You are the creator, so everything is in your power.

What's also important is that this concept also works great for topology/loops (Key and Fill Loops).

Main and Fill is a very interesting concept because it can be applied to almost anything! Next time you look at the topology mesh, try to find the Key Loop, since every head has at least one of these.

Based on what I saw, there are such head topologies:

C-loop
X-loop
E-loop
And a bunch of others

I'll talk about all this later, but for now let's focus on the form.

Rounding

This is the most common mistake of all beginners. They create Key Cuts, and then Fill in between them, and leave it all without the slightest change. If you don’t round your GR, the result will be square (unnatural, inorganic) and you’ll have to work hard to fix it later. If, every time you create the next Filling Cut, you correctly fit it to the shape, then you will save yourself from constantly reworking the mesh.

Following the lines of the form (body lines, smoothness of lines).

Another common mistake is NOT following the smooth lines of the subject. Remember, this is an organic simulation, so try to think organically. When sketching body parts such as a tail or a body that curves, try to imagine a curving cylinder. And create blocks accordingly.

Fear, haste and doubt

This is a mental level of challenge when you are just starting out in 3D modeling.

Every time you do something for the first time, you experience great difficulty. The point is, don't give up! Everyone goes through this. It is rare to meet a person who has gone through this initial stage and does not talk about how he suffered.

So here’s my advice: take it easy, slow down, there’s no rush here. Try spending a month or two playing with your form. Start with objects that allow you to make a lot of mistakes, such as creatures. And just practice. If it turns out crap, delete it and start over.

At first, everything will work out slowly for you, but as you do similar tasks, your speed will increase all the time. This is exactly why we need practice, to do everything better and faster.

When you create a model for the first time, it can be a very fun process. All because of the "look at the whole."

Take, for example, the human figure. Let's say you start with the torso and use extrude to stretch it out. If you don't have legs and arms/head yet, then it all looks very comical. To make "it" look human, you must complete all the remaining body parts.

So there is no need to lose interest due to the terrible result of not having all the pieces in place. You just need to extrude all the body parts and place them in the right places, only then “it” will start to look like a human figure.

Practice

Simulation subject

First let's talk about the modeling subject.

If you are doing character modeling, then you will obviously start at the head and work your way down. A simplified head, torso, and then arms and legs. After a few weeks you will realize that the head is the simplest part of the body, since it is just one block, entirely visible from one point. And all you need to model is to zoom in and out of it (the head).

Other body parts (arms, legs) will be more challenging because they require you to rotate and zoom the model in the viewport. And since you're new to 3D, chances are you're not used to making full use of rotation, spin, pan, and zoom in viewports.

At first, to avoid unnecessary difficulties, use references. And once you get the hang of it, try modeling from memory.

Creating a hand from memory for the first time is hard. So try to use reference pictures/photos first and memory later.

Why do it from memory at all? Just to see if your understanding of the shape of the hand (or whatever object you're creating) has improved.

If you model different creatures, then the situation is the same here. Start with the head, then the body, and then everything below. Don't limit yourself by modeling just one part. Jump from one part to another (I, for example, do this), so you (thanks to changing the type of activity) will constantly maintain interest in this process.

Extrude.

Before you start extruding parts like arms and legs, you should know that there are only two ways to do it. This has to do with how to model the angle.

Method A is, of course, faster, but you will still, sooner or later, come to method B. You can also convert A to B using the Polarization method (more on this later). Also note line shape (red).

I have seen many variations of Method A for creating realistic human hand. While method B is suitable for unrealistic characters, for example, cartoons and the like.

If you find it difficult to rotate every time you extrude, then use Method A. But it doesn't really matter which method you choose, since you can convert one topology to another as you go.

This concludes the first part of the article. You can ask questions if something is unclear.

Let me conclude with a few the best.

This is my translation of an excellent series of posts from SomeArtist on subdivisionmodeling.com (which were deleted because the forum ceased to exist).

Subscribe to blog updates(Here ).

P.S. The barbarian turtle in the title picture was made by American Jesse Sandifer. The simulation was performed entirely in Mudbox, then the whole scene was assembled into 3ds Max and visualized by forces Vray. Photoshop used for texturing and post-processing. For other types of character, as well as a discussion of the work, read

Mathematical Structures and Modeling 2000, no. 6, p. 107-114

UDC 530.12:531.18

TIME AND TOPOLOGY OF THE HUMAN BODY

The Philosopher Kant declared the time is given us a piori, i.e. is defined for person from birth. Has it relation with the topology and geometry of the human body? In Minkowsky space-time the four-dimensional topology of the human body is trivial and diffeomorphic to R = x B, where BcRl Such topology allows to perceive the sensations by consecutively any point of the body. If body has other four-dimensional topology which is not diffeomorphic to R, then there exists full collapse of memory in an effort to observe the sensation consecutively. Hence, other topology of the body means the absence of time in that form to which we have become accustomed.

This article was written with the aim of a comprehensive study of the consequences of the theory of absolute space-time. It is known that the material body is described in the theory of relativity by a set of world lines, but physics is not interested in the human body. Let us, however, try to find out how the pseudo-Euclidean geometry of space-time correlates with the four-dimensional topology of a body that a living organism can have in the absolute Minkowski World of Events.

1. The illusion of time

Human life occurs in time. We organize the events that happen to us by dating them. We know thoroughly that the past in our lives is something that is irretrievably gone, and the future that awaits us is unknown because it has not yet arrived. But we know that death awaits us ahead.

At birth a person receives a body. From the point of view of mathematics, life is a four-dimensional region R, which has a topological structure diffeomorphic D1 xB, where D1 is a one-dimensional disk, a period of time that a person is destined to live, and B is his body in three-dimensional space, the topology of which is simplified in Fig. 1. The modern theory of space and time suggests that the World of Events is a so-called four-dimensional pseudo-Euclidean space V4, called space-time. An event is a point in space-time V4. The life path of an elementary material object is a curve, a world line, in the V4 Event World. Therefore, a person’s life as the totality of all events occurring in his life is a smooth embedding h: D1 x B -> V4. World line

Email: [email protected] Omsk State University

Topology- a rather beautiful, sonorous word, very popular in some non-mathematical circles, interested me back in the 9th grade. Of course, I didn’t have an exact idea, however, I suspected that everything was tied to geometry.

The words and text were selected in such a way that everything was “intuitively clear.” The result is a complete lack of mathematical literacy.

What is topology ? I’ll say right away that there are at least two terms “Topology” - one of them simply denotes a certain mathematical structure, the second one carries with it a whole science. This science consists of studying the properties of an object that will not change when it is deformed.

Illustrative example 1. Bagel cup.

We see that the mug, through continuous deformations, turns into a donut (in common parlance, a “two-dimensional torus”). It was noted that topology studies what remains unchanged under such deformations. In this case, the number of “holes” in the object remains unchanged - there is only one. For now we’ll leave it as it is, we’ll figure it out a little later)

Illustrative example 2. Topological man.

By continuous deformations, a person (see picture) can unravel his fingers - a fact. It's not immediately obvious, but you can guess. But if our topological man had the foresight to put a watch on one hand, then our task will become impossible.

Let's be clear

So, I hope a couple of examples brought some clarity to what is happening.
Let's try to formalize all this in a childish way.
We will assume that we are working with plasticine figures, and plasticine can stretch, compress, while gluing different points and tearing are prohibited. Homeomorphic are figures that are transformed into each other by continuous deformations described a little earlier.

A very useful case is a sphere with handles. A sphere can have 0 handles - then it’s just a sphere, maybe one - then it’s a donut (in common parlance, a “two-dimensional torus”), etc.
So why does a sphere with handles stand out among other figures? Everything is very simple - any figure is homeomorphic to a sphere with a certain number of handles. That is, in essence, we have nothing else O_o Any three-dimensional object is structured like a sphere with a certain number of handles. Be it a cup, spoon, fork (spoon=fork!), computer mouse, person.

This is a fairly meaningful theorem that has been proven. Not by us and not now. More precisely, it has been proven for a much more general situation. Let me explain: we limited ourselves to considering figures molded from plasticine and without cavities. This entails the following troubles:
1) we can’t get a non-orientable surface (Klein bottle, Möbius strip, projective plane),
2) we limit ourselves to two-dimensional surfaces (n/a: sphere - two-dimensional surface),
3) we cannot obtain surfaces, figures extending to infinity (of course we can imagine this, but no amount of plasticine will be enough).