FOREWORD.
If
I’ll try to show what this book is, I’ll be putting myself between a rock and a
hard place, so instead I would rather show what this book is not.
First
and foremost is not a scientific compendium of elaborate formulae, footnotes
and referrals, theoretically discussing the fundamental laws and principles of
solid modeling.
Second,
it is not a step by step check-list taking the reader through all the
intricacies of Solid Modeling. Such a thing is feasible only for a specific
software package at a time, and we don’t want to be particular about any
package.
Third,
is not a compendium of “all things 3d”.
And in general, is not a book
you read in bed at the end of a long day at work.
On the other hand, it will surely
make you sleepy if you read it in bed!
OK, we saw what it isn’t.
Let
see what it is, or rather what is intended to be.
Is
supposed to be a guide into the field of solid modeling for people intending to
learn the ropes just right now as well as people who are already “in”, using it
(the 3d modeling software) but did not have the time to learn the ropes.
We
know that there are a lot of people in both these categories out there.
And
no blame is cast on any of them; we know too well that a lot of people in the
field were forced into using very complex tools (like solid modeling) without
having time to learn the fundamentals. The need to get to speed is always
prevalent in such cases.
From
his own experience, the author knows that a lot of people already using Solid
Modeling Software have gotten there without being asked. Somebody else made the
decision for them.
I
mean, if the “management” decided that Solid Modeling has to be implemented by
the end of the year, you got your share of headaches scrambling to learn it on
the fly. Unless you rather quit and go to a company that still uses drafting
boards.
Learning
on the fly gets you someplace, but is very probable that the basics were never
properly learned. Exactly like in math, if you didn’t grasp too well the
fractions, you’re bound to be lost in calculations when the “calculus” starts,
because you’re missing the fundaments.
This
book comes close to be a “take me by the hand” guide, but for very good reasons
doesn’t go totally there.
The
reasons are that –as we mentioned before- due to the specifics of each
commercial software dealing in this subject matter, it is impossible to get
into a unique step by step routine. Each software dealing with solid modeling
has its own set of tools and commands, each requires more or less steps to
accomplish a simple task like sketching, so, unless the author would have gone
mad and tabulated each step according to each software, like here:
Software Work plane Sketch Feature
XXX already set click to sketch Pick after sketch
YYY chose WP start after choosing feature chose before sketch
ZZZ chose and confirm chose 2d or 3d sketch after
sketch ready
orientation
… …. ….. ….
Besides,
more likely the author would have gotten in trouble with the software producers
for favoring one package against another.
For these reasons, this book
has taken the aspect of a brush-up guide for the fundaments of Computer Aided
Design with emphasis on 3D.
We tried very hard to explain for all levels
of knowledge on what foundation 3D resides, and what are the minimal
prerequisites to get to a good comfort level while working on a model.
To
return to the main thread, we tried here to brush up some forgotten (or never
well known) basic notions about elements of geometry as used in conjunction
with solid modeling: coordinates, position, point, planes, axes, and so on.
These are the fundaments.
They
should be enough to make the reader more comfortable when navigating through a
maze of work planes or when having to create a new one from scratch.
For
instance, if you knew that a plane could be defined by only three points or a
line and a point, you’ll get to create a custom plane in a jiffy, even if it is
skewed and offset like hell.
And
as we mentioned before, we can’t show in this book command by command or mouse
click after mouse click sequences needed to create a solid model. It will be as
large as an encyclopedia, and not universal, not two software have many things
in common.
We
did, however, went through a lot of trouble explaining how to create all the
features used in solid modeling, from sketch to the finished solid.
Here is what the reader will
find in this book:
Chapter one.
(16 illustrations)
a short practical introduction
to the basic geometric notions like:
point;
line;
origin;
position;
ordinates;
vectors;
systems
of coordinates;
planes;
projections;
axes.
Chapter 2.
(16 illustrations)
Another short and practical
brush-up for the following notions:
Work
plane;
Work
axis;
Sketching
tools;
Solidifying
tools;
Open
loop profiles;
Power
of smart dimensioning.
Chapter 3.
(72 illustrations)
Dedicated exclusively to “how
to” create Extrusions, with lots of
examples.
Chapter 4.
(47 illstrations)
Dedicated exclusively to “how
to” create Revolved features, with
lots of examples.
Chapter 5.
(34 illustrations)
Dedicated exclusively to “how
to” create Sweeps, with lots of
examples.
Chapter 6.
(23 illustrations)
Dedicated exclusively to “how
to” create Lofts, with examples.
Chapter 7.
(19 illustrations)
Dedicated exclusively to “how
to” create Shells, with examples.
Chapter 8.
(16 illustrations)
Also dedicated entirely to “how
to” create Helix features.
Chapter 9.
(25 illustrations)
Dedicated entirely
to the creation of Ribs in Solid
Modeling.
Chapter 10.
(35 illustrations)
This chapter will
be shared among lesser features as: Fillet,
Chamfer and Surface Draft.
Chapter 11.
(69 illustrations)
Here we put
everything together and create Assemblies.
Chapter 12.
(15 illustrations)
Here, we go back
from 3 dimensions to two: Drafting
from models.
Chapter 13.
(10 illustrations)
If you’re
superstitious, don’t touch it, it is mainly quizzes.
Chapter 14 (and last, thanks God)!
(7 illustrations)
Deals with
editing: how to edit sketches, features, part browsers.
CHAPTER ONE.
BASICS OF THE BASICS.
I’m
calling this chapter the “Basics of the basics” because in it we will cover the
first elements on which we will build our knowledge; these are elements
essential to the art of three dimensional design and drawing. And even beyond.
Some
will argue that nowadays, the software you will likely use to design in 3d does
not require you to know all these basic elements in detail. And that is true to
a point, but there are times when you have to solve a tricky situation to which
your software does not provide an easy solution. If you find yourself in such a
situation and you don’t know your basics, you will be in pain.
What
I’m conveying here is: it is not strictly necessary for you to know what the
definition of a point or a plane is in order to create a model; with modern
software you just pick your work plane and go. But, if you need to create an
out of ordinary work plane and you don’t know what defines a plane, you’ll wish
you knew the basics, or be left fishing.
On
the other hand, there will always be a lot of people out there craving to know
how to do solid modeling but not having the slightest idea about these
concepts. True, you can open the software and little by little, through trial
and error, you’ll succeed in creating a model, but your knowledge is very
shaky. If you are faced with a problem that involves the knowledge of the
fundamentals, you’re lost, because you never knew them.
In
any event, either you knew the fundamentals or you just want to know them, here
is the opportunity to brush up or to learn something new.
So, the
best place to start is the beginning, as though you never heard any of these
ideas before.
The
elements from which we build our solid modeling knowledge are, in no particular
order, the following geometric entities:
-point
-origin
-position
-ordinate
-vector
-projection
-work axis
-work plane
-sketch
-feature.
About the last entry, FEATURE, there are some fine points to
emphasize:
This word is used in 3D
modeling in three different situations, describing three things very tight
related, such as:
1) Feature
is the tool that allows the designer to transform a sketch into a simple solid
or a part of a more complex one.
For
instance, to obtain an extrusion, you have to use the feature (tool)
“extrusion” in order to obtain an extruded solid. This is also called a
“feature” if is only a part of a solid.
2) Feature
is also (as we just said above) an element of a model, obtained from a sketch
thru one of the features (tools for solidification) available.
3) Feature
is in the third place the name of the operation through any sketch is
transformed into a solid element.
To recap: using a feature
(tool) we obtain a feature (solid
element) through a feature
(operation).
Some
of these elements are used all the time, others are used only now and
then. But one way or another they are
used, and to brush up on them is not going to hurt anybody.
POINT.
The most minute of all, the point is an element of geometry we need to
know in order to be able to define other elements like: direction, distance,
line, center, and plane. One is inclined to say the point is the very base of
geometry.
The
point is defined in The Webster Dictionary as:
“An element in geometry having
only a position, but not size, shape, or extension”.
Let
try to find a few examples of points:
The
intersection of two or more lines creates a point, because it has a position
(at the intersection), but no other attributes.
The
center of a circle is also a point, positioned “at the center”.
The
intersection of three or more orthogonal planes is a point too.
The
beginning and the end of a line are each a point.
Hence, a line can be defined by two points.
And so we already covered the line as well.
ORIGIN.
The
word origin is taken here in the sense of the point of inception, the point
from which we define relations with other points.
The
origin can be situated anywhere in space, the only condition for it to exist is
a location.
The
center of the earth, the origin of spatial computations, has a unique location,
the center of the earth.
The
North Pole also has a definite location.
The
intersection of the Equator and the
No
confusion there, all these points are very precisely located.
Remember:
an origin is the starting point for a system that helps to define or locate
other points in space.
Most
of the time, we use the word ORIGIN to indicate the intersection of the three
work axes (and planes too) at the left-bottom corner of our work sheet, but we
must be aware that this is only a convention and an origin can be situated
wherever we deem it necessary.
THE WORK AXES.
These
are the workhorses in our field, for they determine almost everything:
orientation of the work planes, direction of plus and minus, degrees of
freedom, even the ORIGIN, which is always located at their intersection.
Well,
depends on what we look at first, like the egg or the hen. Some people will
argue that the Origin determines the position of the three axes, which is also
true.
Better
let the Einstein-s of this world decide this one, for us, it works both ways.
And
we must thank whoever decided that only three of such elements are enough to do
our work. One more and everybody would have gone nuts.
By
universal convention, it is established that the three axes of the system are
intersecting perpendicularly on each other at the origin, and extend in both
sides of the origin.
A)
The X axis
goes towards the right (east) for positive values and left (west) for negative
values.
B)
The Y axis,
goes up (North) for positive values, down (South) for negative values
C)
The Z axis
goes away from the plane formed by x and y for positive values and stabs the
x-y plan for negative values.
Here,
a note has to be added:
If
the reader is more comfortable with indicating directions by using the dial of
a watch, then: 12 is North, 3 is East, 6 is South and 9 is West.
Also,
the above arrangement for the three axes is only one of many ways it is
conventionally represented, you may see them turned upside down and inside out,
and it doesn’t matter. This is the plain vanilla flavor.
This
is the most common representation of our basic system of coordinates.
Take
a look at Fig.1.1, and try to visualize and better, memorize, this concept
which in my humble opinion is the bible of 3D. If you don’t assimilate this
foundation block now, you’ll most likely suffer from an ulcer latter.
Fig.1.1 The three work axes.
PLANE.
A
plane, according again to our trusty Webster source, is:
“A flat surface that wholly
contains every straight line joining any two points on it”. Well, it’s like
scratching your left cheek with your right hand, very straightforward, isn’t
it? Anyway, it’s an earful.
Let
tray an easier approach;
A
PLANE is to be considered as a flat surface –like a flat sheet of paper- made
from infinity of lines glued parallel to each other. This is not to say that
the lines must be only parallel to each other, no, they can be intertwined or
interlaced as to create a fabric like surface, but without thickness. Since the
points don’t have any dimension, so do these infinite lines. Hence, the plane
is also
a-dimensional, A PLANE HAS NO THICKNESS, and
no boundaries, goes to infinity in all the directions.
What
are the ways of defining a plane, what makes a plane different from others?
A plane is uniquely defined by
any one of these:
-
3 points;
-
a line and a point;
-a
closed surface.
Let
ponder the first notion that a plane is defined by three points:
Take
a pencil and a small piece of cardboard and keeping the pencil with the point
up, try to make the cardboard stay put on it. If you have a lot of patience,
maybe you’ll find the center of gravity of the cardboard and succeed, but most
of the times you will need at least two other points to hold that cardboard in
position.
Now
take two pencils and try the same trick, the cardboard will tip one way or the
other and fall.
If
you now have three pencils standing tips up and lay the cardboard on all three
of them, the piece of cardboard will rest comfortably on these three points (it
may even thank you). This is why all the chairs and tables and cabinets, etc,
have at least 3 legs, it is the minimum required to obtain a supporting plane.
Now,
if you move just one of the resting points a tiny bit up or down, the plane
readjusts to lay on all three points again and in the process becomes a
different plane, defined by one new point and two old ones. Imagine how many
times you can do this without duplicating a plane? If you can imagine Infinity,
you’re good!
The
second way of defining a plane is by indicating a line and a point.
If we go back at our experiment
with the pencil and cardboard and try to balance the cardboard on the length of
a pencil this time, we’ll find that it is possible but tough. The 3d point is
not truly defined yet.
But if we lay the cardboard on
the pencil held horizontally and sustain one of its edges with a finger or
another pencil, the cardboard is happy again and won’t fall. The horizontal
pencil is our line, while the other is our point. A plane is defined by a line
and a point as I said before.
The
same idea with moving the point up and down applies here too, with the same
magic results; there is infinity of planes that can be created this way.
The
third way and the most obvious is the closed surface, like a circle, square,
rectangle, triangle, etc.
If
we take our tired cardboard and lay it on the mouth of our coffee mug, it will
happily stay there forever if we don’t remove it to drink the coffee.
Why? Because it lays on a
closed surface, the circle defined by the mouth of our mug. We said circle
because, from personal experience we saw that square mouthed mugs are very
rare.
Again,
the three ways to define a plane are: 3 points, a line and one point, and a
closed surface.
A
line and an angle will do the trick too, but is a duplicate of the second
axiom.
And
again, remember: the plane has no thickness, I can stuff a zillion planes in
the space of a thousand of an inch, maybe more if I get very motivated. And I’m
not different from him or you!
And the plane stretches to
infinity in all directions.
Don’t
forget these, please.
THE WORK PLANES.
These
are our workhorses, we use them constantly; as a mater of fact, you cannot
start working on a solid model until you do not select a work plane first.
They
follow the same principles established for planes in general; what make them
work planes is a simple convention, yet another one governing our field of mechanical
engineering, the one that establishes the three planes defining the Cartesian
system:
1) The x-y plane, also called the paper
plane is the plane lying flat on the drafting table and on which the top view
of a part must reside. Or, the surface of the screen on the computer running
your drafting software. This is the plane “you look at”, either because is laid
on the top of your desk, on the computer’s screen or hung on a wall like a map.
Maps
are conventional representations of geographical entities, and they also are
governed by conventions: North will be always on top of a map, south at the
bottom, east to your right and west to your left. Maps are also divided in
little squares of a grid, which facilitates the location of any point on the
map by the number of squares from south to north and west to east.
As
your x-y plane hangs from the wall, or lays flat on the table the origin is
located at the left-bottom corner of the grid.
This
is the universally accepted convention for an x-y work plane, it is the plane
you do your work on, having its origin on the left-bottom corner.
Take
a look at Fig. 1.2. You’ll see that the
XY plane is in fact stretching in all directions to infinity and the origin is
located somewhere arbitrarily in the middle of it. So, then, why say that the
origin of this plane is at the left-bottom corner? Because, in most usual
conditions, the only part of the XY plane we ever use and see is the positive
portion, bordered by the positive Y and positive X axis.
The reality is, as we said
before the plane stretches all the way to infinity in all directions.
Fig.
1.2 the X-Y plane.
2) The Y-Z plane, is defined by the y and z axis, as shown in Fig. 1.3
Fig.
1.3 the Y-Z plane.
3)
The X-Z plane is defined by the x and z axes and Fig. 1.4 depicts this
plane.
Fig.
1.4 the X-Z plane.
When added together in one picture, the three planes will
look like Fig.1.5 below. The picture doesn’t show the entire extent of the
planes for clarity, shows only the most used part of them, but remember, they
stretch all the way to infinity, wherever that is.
Fig.1.5 The 3 work planes.
To
get a clearer picture of this, you can imagine yourself standing in a room,
your feet on the X-Z plane, facing the X-Y plane, and having the Z-Y plane at
your left. Again, this is only one of the ways the 3 planes could be oriented
in relation to the World coordinate system.
Fig.
1.6 A more intuitive way to look at the work
planes.
SYSTEMS OF COORDINATES.