Optimizing topology for a game Character
Creating a humanoid game character for animationMåns Mäkinen

Computer Graphic Arts, bachelor's level2018Luleå University of Technology
Department of Arts, Communication and Education

Preface
This thesis summarizes my bachelor’s degree in Computer Graphics at Luleå
University of Technology in Skellefteå, Sweden.
I would like to thank my classmates for these three years of studies and fun, I
would also like to thank LTU and my instructors Arash Vahdat, Håkan Wallin,
Samuel Lundsten, Stefan Berglund, Fredrik Tall and Aron Strömgren for making
this education possible.
Måns Mäkinen

Abstract

This thesis will go through how to create good topology for a humanoid
character that will be animated for a game.
First, I will go through some things to keep in mind and why they are important. I
will then create two different models, one when I’m following the principles ihave gone through in the thesis and one when I don’t do that and then compare
the results to anatomy references.
The two models will have the same weight paints as far as possible and will play
the same animation.
The purpose is to be able to use this thesis as a help with that your supposed tothink about when you create your characters topology and how to avoid simple
mistakes.

Sammanfattning
Denna rapport kommer gå igenom hur man skapar bra topologi för en humanoid
karaktär som ska animeras för ett spel.
Först kommer jag ta upp lite olika saker man ska tänka på och varför det är
viktigt. Jag kommer sedan skapa två olika modeller, en när jag mer följer det som
jag har gått igenom och en när jag inte gör det och sedan jämföra dem med
anatomi referenser.
De två karaktärerna kommer ha så lika weight paints som möjligt och köra
samma animation.
Syftet är att man ska kunna ha den här rapporten som hjälp med vad man ska
tänka på när man skapar sin karaktärtopologi och hur man kan undvika simpla
misstag.

Table of contents
12Preamble

1.1

Background2

1.2

Problem2

1.3

Question Formulation2

1.4

Purpose2

1.5

Delimitations

2

Theory
2.1

3

5

3

Word list

3

2.1.1

Topology

3

2.1.2

Edge loop & Edgeflow3

2.2

Topology3

2.3

Creating a character

4

2.3.1

Shoulders

4

2.3.2

Knees and elbows

6

2.3.3

Body

8

Method
3.1

4

2

11

Test

11

3.1.1

Shoulders

12

3.1.2

Knees and elbows

12

3.1.3

Body

12

3.2

Skinning test sequence

13

3.3

Method critique

18

Results

19

4.1

Shoulders19

4.2

Knees and elbow

20

4.3

Body

22

Discussion

23

5.1

Shoulders23

5.2

Knee and elbow

24

5.3

Body

26

6

Conclusion

27

7

Bibliography

29

8

Appendices30

1

1 Preamble
1.1 Background
Creating good topology is something that has always been a hard part of being acharacter artist. There are a lot of things you need to think about when creating
your character or prop. How to make the character deformations look like a real
person and how to do this on a low polygon budget.

1.2 Problem
A big problem when creating a game character is how to create the topology forthe best results when animating. Essentially how to make the character look likea real human when its moving and how to create the illusion of muscles under
the skin.

1.3 Question Formulation
What should you think about when creating a game character?

1.4 Purpose
The purpose of this thesis is to show some ways to create good topology for ahumanoid video game character centered around animation and mention why
this matter, what you should think about. The idea is that this thesis will be asimple guide of some basic guidelines to be used when creating a character.

1.5 Delimitations
The thesis will be limited to just the arms, legs and the body and not head, hands
or feet. It will also only focus on human or humanoid characters. Also limited to aquite low polygon budget to easier see the deformations of individual polygons.

22 Theory
2.1 Word list
2.1.1 Topology
Topology is how the layout of the model looks. Where and how all the edges andvertices are placed to create the mesh surface. Good topology is essential if you
want good deformation and fast framerates. A bad topology can in some cases
create rendering problems.
Mastering creating good topology takes time to learn. It usually requires a lot of
trial and error to learn what happens when your topology is bad.1
2.1.2 Edge loop & Edgeflow
An edge loop is a series of connected faces or edges that runs completely aroundan object back to a starting point.2 Edgeflow are how the edge loops flow on yourmodel.3

2.2 Topology
As mentioned in the word list topology is the layout of the polygons the 3d modelin build by and how they flow.
A good game topology should center around creating a good silhouette, defineedge loops for better deformation, minimize the stretching of textures and thechanges between Vertexnormals and make it easy to create good UV-seams. Oneof the main purpose of a good topology is to define the silhouette of the model, if
the polygons don’t add to the silhouette don’t add them depending on thepolygon budget your working after.4
When thinking about the deformation of a model a good topology has its vertices
and edges in places to give the model a good bending, compression andstretching when the model is skinned or morphed.5 A good topology maintains it
volume when bent and doesn’t collapse. 6
A bad topology generally lack definition and don’t have an edgeflow that flows
with the model. A good topology has an edgeflow that defines the flow of themodel and creates a better shape of the model often with less polygons. It also
has anatomy in mind to get the flow and deformations as close to a real human as
possible.7

1 Polycount, (2017), “Topology”, Polycount2 Wings 3D, (2018), “Edge Loops”, Wings 3D
3 Niko Mäkelä, (2010), “Surface flow - why it matters”, CGmascot
4 Polycount, (2017), “Topology”, Polycount5 Polycount, (2017), “Topology”, Polycount6 Polycount, (2017), “Limb Topology”, Polycount7 Niko Mäkelä, (2010), “Surface flow - why it matters”, CGmascot3

When animating a bad topology, it looks unnatural and deforms in unnaturalways compared to a real human.8

2.3 Creating a character2.3.1 Shoulders

Figure 2.1: Shoulder topology example. (Brian Tindall 2009)

When creating the shoulder geometry there are a lot of things you want toconsider with how you structure your edgeflow and where you place yourpolygons. There are two main problem areas, these are the armpit and over thearm socket. If you don’t have enough geometry in the armpit it will stretch when
the characters arms are above the head and if you have to much it might deform
in a unnatural way when the characters arms are down.

Figure 2.2: Topology review shoulder. (Jonathan Williamson 2010)

The shoulder edge loops should go down on the chest and around the back. To
define the shoulder, you need to use some tricky edge loops as the green one in
the picture above. This type of loop is called a x-loop. It’s two loops going
opposite direction forming an x. It helps define the shoulder to create a nice lookand work as a great way to attach the arm to the body geometry. 9
8 Niko Mäkelä, (2010), “Surface flow - why it matters”, CGmascot

9 Niko Mäkelä, (2010), “Modeling for animation - Body”, CGmascot

4Figure 2.3: diagram of a x-loop. (Niko Mäkelä 2010)

When creating the character especially the shoulders you want to think about
which pose you place the character in. To place your character in a T-pose was
the standard before but because a character’s arm is seldom in the upper half, it
now more common to use a relaxed pose also called A-pose. With a A-pose all thelimbs are between the two extremes making it easier to rig as it helps the IK
solver determine the preferred angle and helps with texture stretching on theshoulders when the arms go down all the way.10

Figure 2.4: T-pose vs A-pose (A. Wiro 2004)

In the image it shows that when in A-pose it’s easier to get more subdivisions on
the shoulder. More geometry means that each edge doesn’t need to move as
much when extending creating less stretching.
Some riggers prefer a pose that looks like the character are sitting on a
motorcycle to minimize stretching problems in the shoulder area. 11

10 Cineversity, (2018), “T Pose Wiki”, Cineversity
11 Ferdinand Engländer, (2015), “The T-pose – all about this mighty blueprint”, Animatorisland

52.3.2 Knees and elbows
Knees and elbows work in a similar fashion as each other. They are both quite
easy to create and might work with just two edge rings.

Figure 2.5: Limb topology (Polycount, 2015)

The problem with only using two edge rings as the green cylinder shows is thatwhen bent the hollow of the knee or the armpit might deform improperly andcollapse.
A better way is to use two tris instead of just quads, this creates a bigger quad in
the hollow of the knee that retains the limbs volume. This is shown on the yellow
cylinder in fig 2.5. 12

12Polycount, (2017), “Limb Topology”, Polycount6

Figure 2.6: arm muscles moving (Uldis Zarins 2014)

When a real arm or leg is moving it’s the muscles underneath that tighten andrelax as shown in fig 2.6. When creating an arm or leg in 3d we want to have thesame effect. This is quite hard to create for a video game character so theimportant part to consider is that the limbs don’t collapse and lose mass.13

13 Polycount (2017) “Limb Topology”, Polycount7

Figure 2.7: Leg topology. (Polycount, 2015)

Figure 2.7 shows a leg created using the same method shown on the cylinders infigure 2.5, it shows quite clearly a collapse of the edge loop on the B-leg. This
makes the leg look more natural and move like a real leg.
When defining the kneecap and elbow an x-loop might be useful. The image
below shows a knee with an x-loop with much mass left in to define the knee cap.
To define the kneecap isn’t as much of a trouble area as it doesn’t deform as
much as the hollow of the knee.14

Figure 2.8: Knee with x-loop (Niko Mäkelä 2010)

2.3.3 Body
The body is mainly how to connect all the other topologies of the character.
There are some things that you want to consider when creating the body.

14 Niko Mäkelä (2010) - “Modeling for animation - Body”, CGmascot

8

Figure 2.9: Body edgeflow. (Niko Mäkelä 2010)

When creating the body, it’s important to take in to consideration the main
moving masses of human/humanoid. You want to create edge loops that follows
the bigger muscles under the skin, it can get needlessly complex considering how
many muscles there are in the human body, but you only really need loop for themain muscle groups or body masses to get good deformations and to get theright idea of the look. 15

15 Niko Mäkelä, (2010), “Modeling for animation - Body”, CGmascot

9Figure 2.10: Male torso from realistic to simplified (Uldis Zarins 2014)

To create a good shape of the chest and back you will need edge rings goingaround the torso. As stated in the shoulder part one problem area is where thearm connects to the torso. You want to have edge loops going from the chest overthe shoulders. 16
Figure 2.10 show the basic forms of the human male body and some simple flowof it. When creating our topology, we need to have these forms in mind to get theright edgeflow.

16 Niko Mäkelä, (2010), “Modeling for animation - Body”, CGmascot

10

3 Method
3.1 Test
To test the theories described in the chapter above I decided to re-top acharacter topology in two different ways. One is to take all the things written
above into consideration when creating the model and the other not taking this
into consideration. Some parts became quite similar such as the chest, shoulders
and pelvis.
They were both modeled in A-pose to have more natural shoulders with lessstretch and to help the solvers know how the limbs would bend. The characters
are done on a quite low polygon budget since I didn’t want to waste time creatingcomplex models just for simple tests, that's also why the head and hands are as
simple as they are.
The models have a quite low polygon count since it’s easier to see how a single
polygon deforms when each polygon is bigger. This might create some problems
as well. The two characters have around 1900 tris each.
The two models then where rigged and weight painted before running the same
skinning test animation from mixamo.

Figure 3.1: showing my two test meshes with circles showing the differences (Me 2018)11

3.1.1 Shoulders
Both models follow the same theory with having edge loops going around thechest and up on the shoulders with a x loop to define the shoulder.

Figure 3.2: the shoulder x-loop (Me 2018)

3.1.2 Knees and elbows
The two models use the two different methods mentioned in the literature
chapter. Model B use the triangles on the backside of the knees and the backsideof the elbow. While model A use normal edge loops.
3.1.3 Body
Both bodies mainly follow the guidelines made by Mäkela with how the edge
loops flow on the model. Model B’s edge loops follow the shape of the rib cage
better than model A’s does. This makes the bending of the torso look more
realistic and follow the shape of the human body.

12

Figure 3.3: Edgeflow of my models (Me 2018)

3.2 Skinning test sequence
The following figures are still frames of the two models (A in red and B in blue
doing the same skinning sequence from mixamo. The skinning test sequence are
55 frames long and made to see how the skinning looks on the character.

Figure 3.4: Frame 01 from skinning sequence (Me 2018)13

Figure 3.5: Frame 04 from skinning sequence (Me 2018)

Figure 3.6: Frame 08 from skinning sequence (Me 2018)

Figure 3.7: Frame 12 from skinning sequence (Me 2018)

Figure 3.8: Frame 15 from skinning sequence (Me 2018)14

Figure 3.9: Frame 18 from skinning sequence (Me 2018)

Figure 3.10: Frame 21 from skinning sequence (Me 2018)

Figure 3.11: Frame 23 from skinning sequence(Me 2018)15

Figure 3.12: Frame 26 from skinning sequence (Me 2018)

Figure 3.13: Frame 30 from skinning sequence (Me 2018)

Figure 3.14: Frame 33 from skinning sequence (Me 2018)16

Figure 3.15: Frame 38 from skinning sequence (Me 2018)

Figure 3.16: Frame 41 from skinning sequence (Me 2018)

Figure 3.17: Frame 47 from skinning sequence (Me 2018)17

Figure 3.18: Frame 55 from skinning sequence (Me 2018)

3.3 Method critiqueSome of the things mentioned would have worked differently if more time were
put into weight painting since it was not the focus of the thesis I didn’t go too far
in to it, it might have had a hole other outcome if someone else would haverigged it and done the weight paint.
There are also some trouble areas know that would have been fixed better with ahigher polygon budget. Since adding more polygons create less stretching oneach polygon.
This test also only includes one body type as it might be some differences whencreating topology for females or heavier/skinnier humans.

18

4 Results
4.1 Shoulders

Figure 4.1: shoulder movement (Uldis Zarins 2014)

Figure 4.2: Shoulder comparison front (Me 2018)19

Figure 4.3: Shoulder comparison back (Me 2018)

The shoulder of both my characters have the same edgeflow. The edgeflow of theshoulder isn’t the best and has some problems. The geometry under the arm
seams to follow the arm more than it does in the reference images in figure 4.1

4.2 Knees and elbow

Figure 4.4: my models bending down (Me 2018)20

Figure 4.5: my models standing up (Me 2018)

Figure 4.6: Knee bending (Uldis Zarins 2014)

The knees on model A (red) in figure 4.4 the hollow of the knee collapses
creating geometry going inside itself. The leg on model B (blue) looks much more
like the real leg in figure 4.6.

Figure 4.7: Elbows bending (Me 2018)

21

The same is true about the elbows as well.

4.3 Body

Figure 4.8: Chest of the two characters (Me 2018)

Figure 4.9: Backs of the two characters (Me 2018)

Figure 4.10: Male arms out and up (Uldis Zarins 2014)22

Here we can see that the muscles under the arms extends when the arms arelifted up. On my characters the armpit is a problem since it has a duller angle
then the model in figure 4.x who has a very hard angle under his arms.
There are also some troubles with the twist on the over arm since the rig
mixamo uses don’t have any twist joints.

5 Discussion
The original question was “what should you think about when creating a gamecharacter?”. During the research of this thesis I have conclude that it’s quite a lot
and it’s quite hard to get information about it other than from forum post or
pictures people post as examples. When comparing my result with real photos
some problem areas occur.

5.1 Shoulders
The skinning sequence didn’t have an extreme pose with arms straight up so it’s
hard to see to a hundred percent how the shoulder would react to that.

Figure 5.1 & 5.2: Shoulder movement from front and back (Me 2018)

Figure 5.3: Shoulder movements (Uldis Zarins 2014)

When the arms are straight down the shoulder compare quite well with the reallife model’s shoulder. The shoulder with this topology overall follows the
movement and shape of the real shoulder quite well.
23

5.2 Knee and elbow
The knee is a real trouble area. When comparing the two models with each otherand reference pictures it’s easy to see why the method with just edge loops usedon the red model isn’t something you want.

Figure 5.4 & 5.5: Model A bending knees (Me 2018)

The hollow of the knee collapses when bent creating a sharp edge. This looks
unnatural compared to a real human. When comparing figure 5.4 and figure 5.6
it’s easy to tell that the legs in 5.4 doesn’t look natural.

Figure 5.6: Knee bending (Uldis Zarins 2014)

When comparing the legs in figure 5.5 with figure 5.6 we can see that they keep
their form much better and we can see that we don’t get the hard edge where theedge loop collapse. The problem with this design is the 45-degree angle that
occurs with the edge loops in the hollow of the knee. This might be a problem
with how I set up the edge loops since in figure 5.4 by polycount this isn’t as bigof a problem.

24

Figure 5.7: Leg topology. (Polycount, 2015)

This might be a problem with either how I set up the edge loops or with theweight paint.
The elbow work in much the same ways as the knee and has some similar
problems.

Figure 5.8: Elbow bending (Me 2018)25

Figure 5.9: Arm bending (Uldis Zarins 2014)

The problem with the elbow is that when it’s bent it loses some of it shape. A realelbow as shown in figure 5.7 has a quite sharp outer angle when bent. This mightalso be a problem with the weight paint on my models. The inner angle of theelbow on model A much collapses much like the knee creating sharp edges andunrealistic shapes compared to the references in figure 5.7. The inner angle ofthe elbow on model B looks much closer to the real-life references.

5.3 Body

Figure 5.10: Backs of the two characters (Me 2018)

Figure 5.11: Back muscles (Uldis Zarins 2014)26

Comparing my characters backs in figure 5.10 with the back of a real model infigure 5.11 we see that the last edge loop going from the shoulders better follow
the muscles on model B create in better shapes. The lowest edge loop on bothmodels in figure 5.10 could be improved to better follow the lines of the rib cage
and the back muscles.

Figure 5.12: My models chest (Me 2018)

Figure 5.13: Male arms to the side and arms above head (Uldis Zarins 2014)

Comparing my model’s chests in figure 5.12 to the real chest in figure 5.13 we
can see that the edge loops follow the lines of the chest and the rib cage quite
well. The deformations are not the greatest the side of the rib cage gets a little
too triangular when the arms are raised. This can though be seen in the pictures
where the arms are just to the sides as well. My models have bigger muscles
under the arms then the model in figure 5.12. So, this might just be a problem
with the person in the picture and my models having different body types. The
person in figure 5.11 gets a closer result to my models.

6 Conclusion
There are a lot to be learn of how you set up the topology of your character and a
lot to keep in mind. the methods I bring up in this thesis are a good beginning but
it’s only the tip of the iceberg. You learn the most by trial and error as stated
before in the thesis.
One thing to take in to consideration as well are the number of polygons you’reusing, in my experiments I was quite sparse with my polygon budget. So, some of27

the methods in this thesis might not work on characters with higher polygon
budget but most of the mindset should be the same.
It’s good to learn the basis of anatomy and consider the way the character is
supposed to move. The subject of topology is a subject that are hard to find exact
information about as most people have their own way of doing it that works forthem. So, you need to find what works best for you, but a good basis is to follow
the shape and the flow of the human body.
There are more areas that I didn’t bring up in this thesis in details, such as thepelvis and face. These areas are quite complex and needs a lot of consideration
when creating their topology.
To continue the work, one might want to create more variations on thecharacters to learn more of what works and what doesn’t work also take into
consideration the head, hands and pelvis.

28

7 Bibliography
1 Polycount, (2017), “Topology”,Polycount, 07/05-2018

http://wiki.polycount.com/wiki/Topology
2 Wings 3D, (2018), “Edge Loops”, Wings 3D, 03/05-2018

http://www.wings3d.com/?page_id=766
3 Niko Mäkelä, (2010),“Surface flow - why it matters”, CGmascot, 07/05-2018

http://www.cgmascot.com/design/surface-flow-matters/
4 Polycount, (2017), “Topology”,Polycount, 07/05-2018

http://wiki.polycount.com/wiki/Topology
5 Polycount, (2017), “Topology”,Polycount, 07/05-2018

http://wiki.polycount.com/wiki/Topology
6 Polycount, (2017), “Limb Topology”, Polycount, 07/05-2018,

http://wiki.polycount.com/wiki/Limb_Topology
7 Niko Mäkelä, (2010),“Surface flow - why it matters”, CGmascot, 07/05-2018

http://www.cgmascot.com/design/surface-flow-matters/
8 Niko Mäkelä, (2010),“Surface flow - why it matters”, CGmascot, 07/05-2018

http://www.cgmascot.com/design/surface-flow-matters/
9 Niko Mäkelä, (2010), “Modeling for animation - Body”, CGmascot, 07/05-2018,

http://www.cgmascot.com/design/modeling-for-animation-body/
10 Cineversity, (2018), “T Pose Wiki”, Cineversity, 04/05-2018,

http://www.cineversity.com/wiki/T_Pose/
11 Ferdinand Engländer, (2015), “The T-pose – all about this mighty blueprint”, Animatorisland,

04/05-2018,https://www.animatorisland.com/the-t-pose-all-about-the-mighty-blueprint/
12 Polycount, (2017), “Limb Topology”, Polycount, 07/05-2018,

http://wiki.polycount.com/wiki/Limb_Topology
13 Polycount, (2017) “Limb Topology”, Polycount, 07/05-2018,

http://wiki.polycount.com/wiki/Limb_Topology
14 Niko Mäkelä, (2010), “Modeling for animation - Body”, CGmascot, 07/05-2018,

http://www.cgmascot.com/design/modeling-for-animation-body/
15 Niko Mäkelä, (2010), “Modeling for animation - Body”, CGmascot, 07/05-2018,

http://www.cgmascot.com/design/modeling-for-animation-body/
16 Niko Mäkelä, (2010), “Modeling for animation - Body”, CGmascot, 07/05-2018,

http://www.cgmascot.com/design/modeling-for-animation-body/29

8 Appendices
ImagesFigure 2.1: By Brian Tindall - http://www.hippydrome.com/ArmsShldrRot.html
Figure 2.2: By Jonathan Williamson - http://wiki.polycount.com/wiki/ShoulderTopology
Figure 2.3: By Niko Mäkelä - http://www.cgmascot.com/design/modeling-for-animation-body/
Figure 2.4: By A.Wiro - https://www.3dm3.com/tutorials/female_body/
Figure 2.5: By Polycount - http://wiki.polycount.com/wiki/Limb_Topology
Figure 2.6: By Uldis Zaris - “Anatomy for sculptors” P. 156
Figure 2.7: By Polycount - http://wiki.polycount.com/wiki/Limb_Topology
Figure 2.8: By Niko Mäkelä - http://www.cgmascot.com/design/modeling-for-animation-body/
Figure 2.9: By Niko Mäkelä - http://www.cgmascot.com/design/modeling-for-animation-body/
Figure 2.10: By Uldis Zaris - “Anatomy for sculptors” P. 22
Figure 3.1: By Me - illustration on screenshots from maya
Figure 3.2: By Me - screenshot from maya
Figure 3.3: By Me - illustration on screenshots from maya
Figure 3.4: By Me - screenshot from maya of animation sequence
Figure 3.5: By Me - screenshot from maya of animation sequence
Figure 3.6: By Me - screenshot from maya of animation sequence
Figure 3.7: By Me - screenshot from maya of animation sequence
Figure 3.8: By Me - screenshot from maya of animation sequence
Figure 3.9: By Me - screenshot from maya of animation sequence
Figure 3.10: By Me - screenshot from maya of animation sequence
Figure 3.11: By Me - screenshot from maya of animation sequence
Figure 3.12: By Me - screenshot from maya of animation sequence
Figure 3.13: By Me - screenshot from maya of animation sequence
Figure 3.14: By Me - screenshot from maya of animation sequence
Figure 3.15: By Me - screenshot from maya of animation sequence
Figure 3.16: By Me - screenshot from maya of animation sequence
Figure 3.17: By Me - screenshot from maya of animation sequence
Figure 3.18: By Me - screenshot from maya of animation sequence
Figure 4.1: By Uldis Zaris - “Anatomy for sculptors” P. 46
Figure 4.2: By Me - screenshots from maya
Figure 4.3: By Me - screenshots from maya
Figure 4.4: By Me - screenshots from maya
Figure 4.5: By Me - screenshots from maya
Figure 4.6: By Uldis Zaris - “Anatomy for sculptors” P. 207
Figure 4.7: By Me - screenshots from maya
Figure 4.8: By Me - screenshots from maya
Figure 4.9: By Me - screenshots from maya
Figure 4.10: By Uldis Zaris - “Anatomy for sculptors” P. 75 & 79 edited by me
Figure 5.1: By Me - screenshot from maya
Figure 5.2: By Me - screenshot from maya
Figure 5.3: By Uldis Zaris - “Anatomy for sculptors” P. 46
Figure 5.4: By Me - screenshot from maya
Figure 5.5: By Me - screenshot from maya
Figure 5.6: By Uldis Zaris - “Anatomy for sculptors” P. 207
Figure 5.7: By Polycount - http://wiki.polycount.com/wiki/Limb_Topology
Figure 5.8: By Me - screenshot from maya
Figure 5.9: By Uldis Zaris - “Anatomy for sculptors” P. 156 edited by me
Figure 5.10: By Me - screenshot from maya
Figure 5.11: By Uldis Zaris - “Anatomy for sculptors” P. 48
Figure 5.12: By Me - screenshot from maya
Figure 5.13: By Uldis Zaris - “Anatomy for sculptors” P. 75 & 79 edited by me30