Solid Modelling Video Lecture - Mechanical Engineering

you
you
hello and welcome everybody again to the
lectures on computer graphics today we
are going to discuss methods to
represent solid objects how do you
represent solid object structures in
this field of computer graphics because
we have to know methods by which we
represent them or model them and then
position them in the first stage of the
viewing pipeline before we apply
normalizing transformations objects
which we see before us typically are
both natural and artificial objects and
there are various methods by which these
are represented depending upon the
application we sometimes represent to
need to represent solid objects
sometimes we need to represent soft
objects also but we will restrict our
discussion today to discussions on based
on methods to represent solid objects so
solid modeling is the discussion which
is the title of the discussion today and
out of the various categories the most
significant one being the constructive
solid geometry or CSG okay so if you
look into the slide the various methods
which are used to represent our model
solid objects are regularized boolean
set operations sweep representations
octrees CSG are constructive solid
geometry which is also one of the
methods of representing solids or
modeling solids then be reps are
boundary representations which you will
see I think mainly we will discuss this
first five methods depending upon the
scope and time available to us now the
there are methods of primitive
instancing
and fractal dimension which I leave for
users to read themselves which are the
fractal dimensions are methods by which
you represents our natural objects
structures not necessarily solids okay
so we first look into a method by which
a solid is represented
let us take an example of a simple cube
okay let us take the example of a
simple cubicle structures and which has
eight vertices as the at the endpoints
of the cube four on the bottom four on
the top or four on the left four on the
right the corresponding vertices are a b
c and d e f g and h and if you visualize
that the vertex a of this unit cube unit
cube means the length of this solid is
unity the length height and the bread
and if the vertex a is at the origin of
the coordinate system then the
corresponding vertices the the
coordinates of the word rest of the
vertices are also given on the left hand
side of you we assume that the line a B
lies on the x axis whereas the line ad
or eh lie on the y axis and
correspondingly the line let us say AE
or bf they lie parallel to these their
axis so these are the three orthogonal
axis on which the cube is lying and the
vertex a is at the origin of the
coordinate system so based on this
concept you can visualize the left-hand
side the table which talks about the
coordinates of the different vertices on
the left-hand side so a set of eight
vertices can represent a solid in this
case a cube let us say so you you have
to store basically the set of vertices
and that is information sufficient in
this case to represent a a cube but let
us say if we talk of an arbitrary
objects it is not always possible to
represent the solid object structure
with the help of vertices only in this
of course you need to store vertices all
right but you also need to find out
which of the vertices are connected by
lines and which of the lines form a
particular polygonal shape or a square
shape in this particular case of a cube
so let us know is connected to the
vertex P at 1 0 0 by a line correct this
as you can see there is no mention of
the line joining what this is ANC
because it doesn't exist in the picture
ok so you can see
lines such as a CDV they don't exist on
the right hand side column if you count
the number of lines how many lines do
you think exist in a queue you know four
and the bottom four on the top and the
four along the side so there should be
12 lines or twelve edges which exist in
a cube and you can count on the
right-hand side that there are actually
12 lines specified so these twelve lines
tell you which of these vertices are
connected okay that is one way by which
another way by which the solid object
also can be represented and that means
you not only need to specify the
vertices but you also need to specify
the land lines which connect the
corresponding vertices okay so this is
one way by which a solid can be
represented with the help of vertices
and lines of course but this is not a
very complete representation we need
actually some more information in terms
of trying to know which of these lines
form an enclosed polygon okay so solid
we know is bound by surfaces solid is
borne by surfaces so we need to also
define the polygons of the vertices
which form this solid it must be also a
valid representation if you take any
solid in front of you let us say you
take the furniture of a classroom I
check the chairs and the tables and in
some cases you have the board you may
have computer systems be used for
practical laboratory classes you may
have apparatus electrical and mechanical
instruments being used if you look at
those artificial solid objects typically
any such solid object preferably it's
easy for you to visualize a box type of
a structure or a or a cube for that
matter the most simplest case it is
always borne by surfaces so you need to
actually on the top a level of
representation define the surfaces which
enclosed a particular solid okay and
when you talk of surfaces in this case
of course we are assuming very
simplistic planar surfaces or planes
which bind the particular solid okay and
any solid for that matter and when we
talk of planar surfaces or planes which
bind which build which a constitute the
surfaces to boundary which bound the
solid then of course you need the
a representation to represent those
surfaces or planar surfaces and planar
surfaces can be represented using
polygons okay using polygons we know
that in the case of a cube of course the
polygons were simply squares of unit
length but it could be any arbitrary
solid which you can visualize which
should be bound by surfaces and so we
need to hence define polygons with the
help of vertices to which you should
represent a surface so let me again
clarify that a solid is borne by
surfaces there are let us say M number
of surfaces which bind a particular
solid each of the surfaces will be
represented by polygons so each of those
polygons will have let us say n
different vertices or lines which define
that polygon and since as you have seen
in the previous slide that once you
define the vertices with the help of the
vertices you can define the lines okay
we join the corresponding vertices and
then you can pick up a set of lines or
edges which considered a polygon those
polygons actually define these surfaces
which bind the solid so I hope you can
understand that at the top you are
talking of surfaces then the edges are
lines of which define these polygons all
surfaces and then of course vertices
which define the corresponding line are
the edges which join a set of word this
is a pair of vertices to form a line so
very no 11 our vertices then lines or
edges and then the polygon and surfaces
to represent a solid structure or you
can look from the other way round from
top to bottom that solid structure is
represented with the help of polygonal
surfaces polygons define with the help
of lines or vertices and at the very low
level you have the vertices which define
these lines as well so so if you look
back a so I repeat these sentences once
again a solidly is borne by surfaces so
we need to also define the polygons of
the vertices which form the solid I hope
this statement is very clear now and of
course then we also need to check
whether it is a valid representation of
a solid because a solid is bound okay
there is no open space for a solid but
typically when we see a solid structure
so it should be bound by surfaces on all
sides left right top
from front and back and so and so forth
cube is a typical example we cannot
visualize that a solid exists like a
single plane you cannot define a solid
with a single plane because it is not
binding any particular volume or a solid
a volume or a other structure okay so we
will see methods by which we need to
validate the representations of solids
and one such method we will see with the
help of the first representation which
we will look is the regularized boolean
set operations in the solid so before we
move on to valid representations are
methods to validate we will look into
another representations of solids and
methods by which you can create
different type of solids from simple
structures let us look at this simple
example rigorous boolean set operations
are of three categories you basically
define a set of operators which operates
on solid structures and the three
operators which simply are used are
called the Union intersection and the
difference the corresponding symbols are
given on the right hand side this is the
symbol for the Union intersections it's
a simple set set set based operation
operations on sets okay and the
difference between the two okay so
boolean intersections of say cubes
because we know actually right now that
how to represent a cube at least because
we can define the cubes with help of
about eight surfaces each such surface
will be defined with the help of fold
polygonal edges are squares so the
number of the surfaces which you have
although are six the number of edges are
lines which we have seen for a polygon
are twelve and the number of vertices
are also eight so remember these figures
this will come back I repeat again for a
cube we have six surfaces twelve edges
are lines and eight vertices KC remember
these figures and with the help of once
you define this cube we will say that
with these operations of Union
intersections and difference mainly the
intersection let us say of say cubes you
may produce other types of solids you
can also produce planes lines points and
what we call as non objects and we will
see with an example what do you mean by
this
okay let us see some examples of
regularized boolean set operations but
we'll just see the examples of boolean
set operations first and then we'll see
what do you mean by the term
regularization on these boolean set
operations okay let us take two cubes
yeah the same cube which you have seen a
couple of slides back earlier now I have
taken to such cubes each of this cube
are of the same size and dimension they
have each of them have I repeat again 8
vertices and 6 surfaces and 12 lines
which constitute each of these cubes
remember these values once again and let
us say there are two such cubes
represent by two different colors of
identical size a and B if that is so
then we look at the intersection of a
and B now depending upon the position of
a with respect to B you will get
different values of a intersection B you
have to visualize that this intersection
can be obtained by placing this one cube
within the other where there is a
overlap of one portion of the solid
portion or the volume covered by a and
also with B so the a intersection B
which you are seeing is a common part of
the volume occupied by both a and B when
a and B are not only placed adjacent but
one is a little bit inside the other so
if both a and B occupy the same volume
that means they occupy the same space
they're in position but it is the same
object now if you start sliding one with
respect to the other one and assume both
a and B in the same place remember we
are working in a virtual environment
okay computer graphics is supposed to
produce a virtual reality or a virtual
environment and although in the in
realistically in the true world in which
we live not in the virtual world we
cannot put one solid object within
another one unsay unless you have a
vacuum inside if you have a solid
structure say a rock or a table we
cannot put another object like a chair
or a pen inside that where the surface
is intersect it not possible but
but this is a virtual scenario a virtual
reality we are living in a virtual world
with computer graphics and here it's
possible to have one solid penetrating
inside the zone of the other one that's
virtually possible whether it is valid
or not we will see later on but right
now I assume that it is so if it is
possible let us say if a and B are two
separate cubes are shown in this slide
a couple of minutes earlier the last
slide which you have seen if we place a
and B in such a manner that both of them
occupy identical are same volumes in the
coordinate system that means the we have
one of the vertices at the origin of the
coordinate system of both these cubes
and the corresponding three apses which
come out of that particular vertex let
us align with the XY and z axis so that
is the case then both a and B occupy the
same position and in that particular
case the intersection of a and B will
produce the same output however if we
take a b and start sliding it with
respect to B along any of the one
orthogonal axis XY and z then what will
happen is if you don't take the of
course the be completely out of a
thereafter is selling a little bit we
will have some common volume which will
be or some volume in 3d space which will
be not a cube but a rectangular
parallelepiped type of a structure which
will be common to both these volumes a
and B common to both these a and volume
C and this is the structure which is
shown in this slide here so what I have
done is basically put B on a at the same
position and started sliding a little
bit the more you slide lesser will be
the width of this intersection and when
you have just read a little bit and
there is a huge amount of volume which
is common to a and B then the
intersection volume also will be very
very large so you have to visualize this
that the intersection a and B is a
common volume between both a and B now
if you keep on sliding it quite a lot
what will happen is at some point of
time a and B will be adjacent to each
other adjacent means there'll be just
one common plane which is just common to
both of the volumes a and B and the
common interception actually could
result in a plane depending that means
depending upon the relative
positions of a and B these two solids
the intersection which is also a solid
output in general will depend on the
relative positions of a and B okay the a
and B relative persons dictate the
intersection volume I repeat when a and
B occupy the same place you have the
same output of the structure of both a
and B starts sliding you start of
intersection volumes a rectangular
parallelepiped coming out the more
amount of slide which you give lesser is
the common volume which comes out and at
one point of time when both a and B are
just adjacent where there is just one
plane in between the two solids a and B
you will have just you have to visualize
this that you have just one common plane
it's possible that I can position a and
B in such a manner that there could be
just one particular line that just could
be one particular line common to a and B
or even one particular point common to
NB just one of the vertices could be
shared or common between the two
structures visually there is an object a
or a solid in this case a cube and I put
another solid B of the same structure
and such a manner that there is just one
vertex which is common to both these
volumes so depending upon the relative
location of a and B you will generally
have intersecting solids which is the
common volume between these two solids a
and B you could have a resultant plane
you could have a common volume as a
resultant line or a vertex point or the
other option is when two solids a and B
are separated wide apart where there is
nothing common between the two you could
result in a what is the common space
which will be resultant you will be
having a null object you can have a null
object as an output when and we do not
share anything in common in terms of
volume surfaces lines or even points so
that is what was the point in the last
slide in the statement which we made
that intersection produces volumes it
could produce planes it could produce
lines it could produce a point or it
could also produce a null object in this
particular case the example shown I have
just shown the example where the example
shows that you have a common
intersecting volume like a rectangular
parallelepiped but you can visualize if
I keep sliding B
more with respect to a the common volley
will start sinking in size and at some
point I have a plane I can have a line
or a particular we look at an operation
called a difference between a and B a
minus B is the difference operation here
what I have done is I have taken the
solid a and I have taken this solid B
and positioned in such a manner that the
lower right volume portion of a is the
same common volume as the top left
portion of P and if you do that and
subtract from a the the common part or
the basically the intersection between a
and B a subtract that portion you result
in a difference operation as you can see
that white volume portion is the part of
being which is basically common or
through a and if you you will result in
this structure a minus B as a difference
operation produced by this so this is
another operation the difference
operation you can visualize a difference
operation as you produce an intersection
first between the two solids a and B and
subtract that common part from the
structure of a okay so this is how you
can produce another type of a structure
using the difference operation so I hope
you will be able to visualize again I
repeat again the relative positions of a
and B will dictate the resultant volume
which you will get in the difference
operation a minus B as you would have
got in the intersection volume in of a
the relative positions of a and B as was
dictating the intersection in volume
results and volume due to the
intersection operation a and B also the
relative positions will also dictate
what you will get by the difference
operation so these are the two
operations of course you can visualize
what is going to have going to happen in
the case of a union Union will be the
aggregate sum of the volumes okay so if
you have a and the B next to that or
with the common part or just even
adjacent the total volume integrated
which what you will have as
and we put together and of course again
the resultant volume which you will get
a resultant solid structure or the
volume which you will get as the result
of a union operation basically is
dictated by the relative locations of
both a and B okay so you have seen three
operations auto I have illustrated only
two if you have understood these two the
Union operation is easy to visualize
which into regular structure so with the
help of these cubes you can generate
other solids and then you can actually
keep generating more and more type of
complex structures with the help of this
as for example in this case you could
use intersection B along with the a
minus B are the difference between a and
me to produce more complex structures by
positioning them relative with respect
to each other and then creating either
Union difference Union difference or
intersections there the three boolean
set operations on solid structures in
this case of course you have seen we
started with cubes but as you can see
that the resulting structures which you
are getting are solids but they are not
cubes any more so you can create more
and more complex structures with the
help of these boolean set operations and
we move into constructive solid geometry
we will see how with the help of these
boolean set operations we can create a
few more complex structures okay know
what is the process of regularization
okay and the question is why do you need
regularization in a solid okay you as I
said before with the help of operations
such as intersection you could have
hanging points lines or even surfaces of
solids coming out due to boolean set
operations I was talking about this
operation call intersection where you
put a and B at this end in such a manner
that there is a common plane or it could
have a common line or you can have a
common point if that is the case that's
not a solid because a line or a single
surface or definitely a point cannot
enclose volume it cannot bound the
particular solid cannot create a solid
with just with the help of one or
even to to surfaces or lines or points
cannot create a solid or he cannot
define a solid with respect to with the
help of just one or two points lines or
surfaces and in reality of course there
is no question of a single surface or a
single point or a line hanging in the
free space and you need to represent
such a typical because we talk when you
are talking of solids there we talk of
the these are structures or volumes
which are bound by surfaces okay
definitely need more than one or two
surfaces will mind so if there is a
result of any operation boolean set
operations mostly the difference
operation or the let us say the
intersection operation may produce an
isolated point or a line or a surface
you don't need to keep that because
either if it exists along with a solid
structure or without it cannot exist in
practice so you need some regularization
operations on boolean set operations to
remove these hanging points lines and
surfaces to create an ideal solid which
is bound by surfaces okay so we need
these operations and that's what we are
talking about regularization here and if
you look at the structure here I have
produced a operator star B where I will
say the star indicate that it's a
regularization operation whereas in the
right hand side if you see I have used
an operation a op P where this op
operator basically is a is a boolean set
operation without regularization so that
so what we have discussed earlier are
operators
intersection Union difference which are
operators boolean set operators but
without regularization because these
produce hanging points and lines and
surfaces and we need to implement
regularization on this which will help
us to produce what we call as
regularized boolean set operations so
let us see with this example let us say
the left hand side is an object
structure which is produced with the
help of some operator which is non
negative and we have a solid structure
but in addition we have a hanging line
or it could be a hanging
surface and also we have an isolated
point here on this object so we need to
remove these two and get only this
object structure it may exist with a
hole there's absolutely no problem of of
a solid structure having a hole but it
should not have this hanging line or a
hanging surface or anyone of course in
this case I am asking you to visualize
this in 2d but you can visualize this
object also in 3d that as if this is a
cylindrical concentric cylinder okay
there is a hole in between so we define
an operation called an interior on the
resultant operation on this object
produced by a operated on B and that
produces this object on the left hand
side and then we have this interior
operation this interior operation is the
one which helps to remove what we call
as boundaries of objects boundaries are
are certain surfaces which I will say is
common to the to the set of points
defining in the object and the
complement of the object complement of
the object in the space outside the
object so this is the surface which will
be I will say as common to the object
and its complement to the object so that
is what if I use to define a boundary
then an interior operation is an
operation which will throw away these
boundary points and that will help us as
you can see to throw all boundary points
including the boundary of the object
itself and of course the hanging plane
on this line goes and of course what
will happen is these isolated point will
go but it might create a small hole here
because we are looking at interior
operations which is this object - the
bond is so any points are all lines
which are defined as boundaries on this
object are removed to create this entity
at point so of course we may have two
holes but this object is with this
boundary and now what we do is we
implement a closer operation after the
interior operation which will restore
this boundary back both the interior and
the external boundary is so the closure
operation is the operation after the
integer operation which defines the
boundary and defines an exterior I am
sorry
it takes the solid structure and if
the boundary includes the boundary that
is the boundary on the solid structure
see if you if you have a small hole
which is just the boundary points around
and if it doesn't exist in the structure
those boundary points will be added and
it will be created as an interior parts
so this closure operation will include
boundary points it allows us close this
small hole or gap which exists due to
this isolated point here so the
regularized boolean operation as you can
see is a successive operations of an
interior operation and a closer
operation I would request you to go to
books and see a few more examples of
regularized boolean set operations this
is picked up from the example from the
book by Foley and John etcetera and so I
repeat again regularize boolean set
operations followed by interior and
closure I repeat again boolean set
operations Union intersections are
difference we had seen the three
examples and followed by interior and
closer operations implements what we
call as the regularized boolean set
operations so this is how we create
solids which is a valid solid now
because they're solid will not have
hanging points which will not have
hanging lines okay in the object and the
object will be closed by the boundary
bounding planes so this is an example of
how you implement regularized boolean
set operations we will move to the next
representation of a solid which is the
sweep representation the sweep
representation is a method by which you
can generate solids within of 2d
outliers or 2d structures okay in this
particular case I imagined you to
visualize the left hand side structure
which is a circle consisting of a set of
points on the circumference so define a
set of points on the circumference of a
circle which is not a solid it's just a
circle it's just a 2d arc a circle so
you have just the circumference and you
have a set of points in there for the
time being you assume that these points
are uniformly spaced and then what you
have to visualize is I take an axis a
vertical axis passing through the centre
of the circle
and I now rotate this 2d circular arc in
3d about these axis I hope you remember
3d transformations of rotation okay
when you talk of rotation we defined an
axis of rotation and then an amount of
rotation about that axis so now what we
do is we have taken it 2d is a circular
arc it's just a set of points around the
circumference
that's what could be present this circle
you can visualize that these are the
points generated by the bresenhem's
midpoint circle algorithm
remember midpoint circle algorithm
midpoint ellipse midpoint line so out of
those three think about the midpoint
circle algorithm so if as if you have
generated set of 2d points lying on the
circumference of the circle and now you
visualize an axis a vertical axis theta
z axis about which you will load it this
particular structure and then what you
do is you rotate it by a finite amount
of steps ok so you take the circular arc
I am rotated by a small amount Delta
Theta and then get these new points in
3d with the new location just toward the
set of new points which this points on
the arc of the circle will create ok
rotate it by another amount the same
amount Delta Theta again and then store
all these amounts if you visualize what
will happen is these points as you keep
rotating of course the you need to
rotate just by about PI radians or 180
degrees in steps of course of a small
amount that small amount could be 5
degrees 10 degrees or even 30 degrees as
large as that but typically to get
better resolution it is better to have
incremental a small amount of rotation
by teacher it could be as low as about
one degree or even half a degree let us
say so what it means is now although you
have taken points on the circumference
of the circle which is in 2d you have to
visualize that this circle lies in a
plane in 3d which could be the ZX plane
or zy plane assume as at Y plane and the
circle is rang in this
to buy things because is it Trudy and
when you rotate this circle in 3d just
by a small amount delta theta a small
rotation which you give what will happen
to these points and the circle is if
you'll go and lie in a new set of it
will create a new set of points in 3d
which will be different from its
original position so store these points
and these new points which will be
generated are new vertices of which will
be used to represent this field again
you rotate by another amount and keep
doing this till you are rotated in steps
as you can see here by by about 180
degrees up to 180 degrees
not in steps of monetary you need to
increment by Delta Theta 2 Delta Theta
two dresses and so on in steps and you
stop when the total amount of rotation
which you have given to these 2d
structure is is monitoring this what
will result in at each point after
rotation by Delta Theta you need to
store these 3d coordinates of these
points which you have created on the
circumference of the circle and when you
create that and you store those points
all those points after all those
rotational incremental notations when
you store you are basically got the
points on the surface of a sphere you
have got the points of the surface of
the sphere if you look back this will be
the resultant sphere which will be
generated by giving this 2d circle a
sweep or a rotation about it three
dimensional vertical axis of course you
can rotate about any arbitrary axis it
could be a horizontal axis or it could
be an inclined axis does not matter you
will still you're creating an operation
which I have just described is all the
sweep operation and with the help of the
sweep operation what you have created is
a set of 3d points a CTO a set of 3d
points on the surface of an object and
these set of 3d points are nothing but
the vertices of those polygons which
will enclose this particular solid in
this case you have generated a sphere
from a circle so you need to sweep you
need to sweep any arbitrary in it
general case when you talk of a general
case you can visualize any arbitrary
two-dimensional structure or a polygon
for that matter and when you talk of a
sweep you can you can rotate with
respect to a particular
three-dimensional axis you can translate
it you can give it any sort of
transformations which we have studied it
in under three-dimensional transforms
category okay you can give you a
combination of transformations typically
a rotation and a translation a rotation
and a scale and or you can actually
sweep it along any arbitrary line or a
curve or an arc or just simply apply
rotation which you have just seen which
in this case which helps you to create a
a sphere from a circle a sphere from a
circle is the best simple example for
you to visualize it is something like if
we can create a circular ring and hang
it with the help of a rope and just
simply give it a twist you can almost
visualize that that day when when the
ring spins which will create an
impression of a volumetric structure of
a sphere and and the bounding surface in
the spherical surface will be traversed
by the arc of that particular ring which
you will be spinning about that axis I
repeat again imagine a ring created by a
simple iron thin iron rod or even a
stiff rope type of a structure if you
can visualize but an iron ring let us
say take a large iron ring now with the
help of a iron wire created being
suspended from a string and then give it
a rotation okay if you give a a very
fast rotation or a spin it will it will
create an impression of a spherical
structure which you'll get what it
basically means that the reality which
will happen in computer graphics
literature for sweep in this case to
create a surface from a circle what you
are doing you are taking points West
Ham's algorithm for midpoint circle
algorithm
you remember that take points on that
arc and at each step give it incremental
notation and at each point of rotation
you just note down or store the
points on the the 3d points on the arc
of the of the circle in 3d because at
each point when rotating in 3d you will
get different set of point XY Y I is
that I for each point X I Y Z died after
rotate it will go to a new point X I
prime Y prime Z Y prime give it another
rotation that point goes to X Y double
prime X Y I double prime Y double Prime
and so on and this is true for all the
points I on the circumference of a
circle as you keep on rotating this is
what you will create and here I have
shown the arc of rotation each rotation
could be in any direction there could be
any arbitrary axis I repeat in this case
I have taken a vertical axis but you can
take any axis and this is the structure
of the sphere which you will be
generated if you look at this structure
you would have seen this type of a
structure but many places not only in
computer graphics literature but you
could have also seen that in the case of
the map of a globe when you talk of
latitudes and longitudes in a global map
or in a globe type of a structure then
what you will basically have is
latitudes and longitudes latitudes and
longitudes are parallel lines parallel
to the equator which never meet and then
you have the vertical arcs which
basically intersect at the North Pole
and the South Pole of the particular
solid ok so this is what is created by
with the help of the sweep
representation for a circle in the case
of generating a sphere and the the
vertical lines the circles intersect at
the North in the South Pole in the case
of a globe and of course you have
horizontal lines concentric reducing
circles which keep reducing the radius
as you move towards the North or the
South Pole imagine the case of a globe
grow that's a very simple example to
visualize this with the help of a globe
structure and that is what is an ideal
example of it sweep representation to
generate a sphere out of a circle now
the question is what should be the
radius of the circle which you choose to
generate the sphere and how much is that
value of Delta Theta which you have to
select to give it incremental rotations
as you
upon sleeping well that depends upon two
aspects what is the size of the
structure which you are generating that
means the radius of the sphere is in
this case simply going to detect the
radius of the circle which you need to
choose that is very simple and
straightforward the other aspect is what
is this Delta Theta
well the lesser you choose the amount of
the Delta Theta value you will get more
amount of set of 3d points which will
enclose the sphere or more number of
polygons which will enclose the
spherical surface if you have if you
choose a larger value of Delta Theta you
will have lesser amount of points and
hence lines and lengths polygons
polygonal surfaces which define the
spherical surface the more higher degree
of resolution you want the greater
degree of approximation you want because
what is happening in this particular
case is if you even take a pen let us
look back into the slide if you look
into a particular small area a set of
four vertices which will enclose low
keep watching my mouse at it as it keeps
pointing to four different vertices
which will enclose and give you a
particular polygonal shape those four
vertices will actually create a
polygonal shape so it will create a
plane and at that region that plane is
going to approximate that part of the
small spherical structure or the part of
that sphere or the surface of the sphere
at that region will be approximated by
that polygon and polygon is always a
plane and all parts of the sphere is you
know is a curved object which never
planner spheric sphere surface is always
a case where it is never the planar
surface it is always curved so at any
point of time you are always
approximating a section or a small part
of a curved structure in this case of
course a spherical surface by a planar
shape or a plane or that polygon okay
now how close you want to have your
approximation depends on how much you
can afford because you can take a very
small amount of Delta Theta
what will in
happen is this polygonal approximation
area will become smaller and smaller and
you are having a greater degree of
approximation but at what cost the cost
is that you have to store larger and
larger number of vertices and lines
which are used to represent this solid
in the case of a sphere so if you want a
very good approximation that means the
more closer to the curved object which
you need to represent more smoother
approximation which you want to do then
you have to go for smaller amount of
Delta Theta larger values smaller amount
of Delta Theta for the sweep which means
larger number of points and lines you
need to store for D to represent that
object structure whereas if you can't
afford if you cannot afford a larger
space for storage for your solid
structure because you need to use the
solid structure for various other
operations like create various other
operations complex structures with the
help of boolean set operations then you
need to of course find out what is the
shading so that you can shade the
surface with certain color we will see
these when we talked of shading models
or visible surface determination
algorithms we'll talk of those and they
could become very complex and you need a
lot of computation time when the number
of the polygonal surfaces increases
quite a lot so this storage is a
requirement and that also enhances the
computational time whether you can
afford that is the another question so
all these butyl dictate the amount of
smoothness or coarseness when you talk
of coarseness I am talking in this
particular example larger values of
Delta Theta which will create lesser
number of points and lines to represent
and store for this solid object and in
that case that will reduce the number of
computations required for creating
higher-level complex structures or even
sharing this particular object but the
object structures will require very
coarse it will be a very crude
approximation low level approximation of
this curved object if you can afford it
if you want it it is fine but if you
want if we can afford greater amount of
storage if you are if you have
computational
a lot of available if we can afford a
lot of computational time in your system
then you will have to have a very good
approximation done based on your
application and then you have to have
smaller values of delta theta this is
true for any shape we will see for other
examples of sleep representation I am
just asking you to concentrate on this
particular case where I have generated a
surface it's a volume there from effort
I'm sorry a sphere has been generated by
sweeping a two-dimensional structure
which is in this case a circle a circle
is cheap to generator here in this
particular case and if you can afford a
larger amount of complexity in terms of
storage space as well as computational
time as the requirement says then in
your application you need to have a very
good approximation of the curved object
and that can only done with smaller step
sizes of your delta theta in this
particular case so it's a it's a
compromise between the the approximation
or the finest the resolution you need to
represent the object and the
computational time requirement so you
better choose whether you can go for a
final resolution and then choose smaller
values of delta theta if you can't
afford it then you make your delta theta
much more larger the course
approximation so we moved forward to
look into other applications of sweep
representation and I have just a few
more examples for you generated to
visualize the sweep representation but I
hope I have sufficiently described how
do you represent an object using stream
this is these are some more examples of
sweep representation there are three
different structures being talked about
in the first left example I have taken a
simple triangle and the sweep is just a
straight line and axis in 3d along which
I translate this object okay so that
generates the stream so you can
visualize that I have these three
structures the sub structure 2d
structure is a triangle with three
vertices and as I keep sweeping along a
particular direction or a vector in 3d I
can do it in steps fine steps more
complex structure in terms of the
were of space requirement and I can have
larger D placements where it does you
know in a core structure unless a number
of points to store I can afford to do
this in this particular which type of a
structure generated by a triangle
because I am approximating planar
surfaces when I am representing planar
surfaces with the help of a strip in
fact I can go to a very coarse
approximation where I can take afford to
take the last first position of the
triangle and the last question of the
triangle and I don't take any positions
in between so I sweep from a starting
position to F on I to to the finishing
position or the end position and I can
still define the structure because in
this case I am representing a solid
which is bound only by planar surfaces
or plates when I am representing curved
surfaces curved structures a surface is
bound by curved surfaces rather than
planar once I have to go in for a finer
sweep representation a finer
representation and then I have to sweep
in steps maybe in finer steps if the
curvature is more for this solid if it
is if the curvature is slowly moving or
it's a plane the step size could be
larger so come back again here as I was
saying that in this particular case of
the wedge I can have the starting point
and finishing one that is just
sufficient that is not true for these
structures of course in the structure on
the right hand side which is still the
suite direction is is a line I can have
the starting point and finishing point
but that is not the position that's not
the case in this arbitrary structure as
you can see a bony type of a structure
were along when I am sweeping I am not
only sweeping along a curved line but I
am also giving the structure a scale so
it is a simple example where you can
visualize I can create a cone also with
the help of a circular structure when I
sweep along the along an axis which
passes to the center of the center of
this square assuming an axis passing
through the center of the square I sweep
along that I will get a cylinder but
when I scale up or down I will get a
conical structure
yes in this case I can generate various
types of solids with the help of this
structure these are the third case
example the largest one in these cases
an arbitrary structure chosen and
dragged or swept along an arbitrary
curve and asthma and and when I am
sweeping along this curved line
I am also giving it a scale I am
increasing it larger and larger giving
it a zoom I am expanding it out and that
is what I create okay these are certain
examples again as I was taking you can
create a pyramidal structure with
respect to this with the help of this
square structure when I am sweeping I
give it a zoom I can create a simple
corn a simple this structure is a very
simple example of it like a conical
ice-cream where I can have this but the
structure is a cone where I can take a
circular structure sweep it along an
axis which passes to the center of the
circle and when I am sleeping I give it
also a scale or Ezra and I can also have
an arbitrary part for this sweep that is
also possible nobody says that you have
to sweep only around about an axis and
then sweep a circle to create a sphere
sweep it again on an axis passing
through the center to create a cylinder
or a pyramid or a cone and all that to
create very irregular structures I need
to have two combinations a combination
of a part or a sweep which could be very
arbitrary not only along a particular
axis it could be audited it could be an
arbitrary helical structure which you
have just seen and when I am sweeping
along that particular structure I give
the structure is a scale so that you
know the The Vicar I can give that to
produce more arbitrary structure so they
so a very general example is this in
this case as you can see in this slide
that that the sweep could be an
arbitrary part in this case a helical
structure and that - not very smooth it
could be very arbitrary and you can
visualize what sort of a structure which
unit I asked you to imagine I am
requesting you to imagine a structure
which will be generated when I am taking
a circular a circle in 2d the circle
take a circle in 2d and then sweep
about another circular arc in 3d let us
say the circle is in the XY plane okay
assume an XY plane the front in your
monitor at the expert parallax is going
towards you okay an XY plane and assume
a circle in this XY plane and I take a
circular arc based on which I am
sweeping and that is in this red X plane
that is towards you X's on the right
hand side Y is the vertical axis let us
say so assume assume a circular sweeping
arc the path of this sweet to lie on the
Z explained and the circle in XY plane
so if I provided a sweep along this you
can visualize that I have a ring type of
a structure but I get a structure which
is called thyroid or a torus okay so
that is a structure which I can so I can
generate various types of models using
sweep both regular and in fact very
highly irregular structures the only
thing which I must have is I should be
able to represent as arbitrary solid
with the help of a sweeping arc or the
path of the sweep you have to define and
also you must define the structure which
you need to shape in that winter for the
structure which you need to shape then
the path along which you need to sleep
and then when you are sweeping the
structure along the path whether you
give other transformations like scale
okay the the path could be a simple
translational path or a curvilinear path
based on a rotation or a combination of
rotation and translation it could all
happen and you can provide scale and
shear as you move around so this is the
combination of these three 3d
transformations which we have seen
earlier is all coming back which is
necessary now to provide this sweep
representations okay so this is how you
provide sweep ripple I had just given
one example with the help of the circle
and the sphere but I have given also
other illustrations of pyramid the corn
the wedge and I have of course visually
ask you to imagine the case of a torus
and of course other arbitrary structures
one or two examples I had given so this
is how you generate and what
you store as an output of this
representation is the case when you
store the XYZ word this is generated by
this weight power that's dumped so that
is how do you generate solids with the
rest of the Alpha Team in the remaining
time available to us in this lecture as
we continue solid modeling and
constructive solid geometry discussions
today as well as in the next class we
will go to the next representation which
is probably the most popular method of
representing solids called boundary
representations or B reps boundary
representations are B reps we talk of an
object which is defined in terms of
surface boundaries what this is edges
and face this is nothing of course new
because when we talked of a solid solid
is also defined and it is bound by
surfaces and we the surfaces are again
defined in terms of edges or faces and
edges and faces are again defined with
respect with the help of vertices so we
know that okay but this representation
is different than the case of sweeps so
curved surfaces are always approximated
with the help of polygons so we talk of
piecewise linear or planner
approximations of curved surfaces with
the help of boundaries and this is very
commonly used in practice so this is
nothing new we talked of this solid
object in the case of a cube which is
defined with the help of surfaces or
there are surface which bind this
particular solid on the top we of course
in the cube we had two surfaces one on
the top and the bottom two on the left
and right two on the front there were
about six surfaces and there were eight
vertices if you remember and there were
twelve lines which define so I try to
imagine bond representations closer to
that particular case of a cube where the
cube was bound by certain surfaces
polygons lines edges are what is we use
the term faces also here to represent
this and and unless in the case of
unlike in the case of sweep
representation so where we used a sweep
to represent curved objects here in the
case of boundary representations we
always go for a piecewise linear or
Planner approximation sweep also does
that I must admit here but always in
this case we are talking about polygons
which approximate curved surfaces as
well okay so we use planner polygonal
boundaries but may also use convex
polygons or triangles to represent in
the case of be reps and we define a new
term based on which we on polygons where
we define the term called polyhedron a
polyhedron is a solid that is bounded by
a set of polygons whose edges are each a
member of an even number of polygons
this is interesting we will see and we
will discuss of additional constraints
later on in the class but we will wind
up the lecture here today with the
discussion of what is called a
polyhedron so a there's a new term which
we use to define a solid under the
category of V reps or bonded
representations of solids where we say
that a polyhedron is a solid of course
it has to be bounded by certain surfaces
it's a solid that is bounded only by a
set of polygons and those polygons are
again defined with the help of edges and
those edges are each a member of an even
number of polygons so we say that we can
have edges all right which define the
polygons polygons are the planes or
faces which bind the solid but in these
case we put a first restriction which we
put is that each edge of such polygons
should be shared or it must be common to
more than one not only more than one but
we should have an even number of
polygons we will see how that constraint
can be holds good with the help of
examples when we visit the next lecture
on the solid modeling after this but
here we again wind up the let lecture
today with the definition of a
polyhedron once again where we say that
polyhedron is a solid that is bounded by
a set of polygons which whose edges are
each a member of n remember this even
number of polygons edges are each a
member of an even number of polygons and
that is the first constraint we put on
boundary representations first
constraint we put on boundary
representations we will see other
constraints with respect to the edges
that is the lines of the polygons and
also the surfaces of the polygons and
the vertices we
put additional constraints to define
what we call remember the beginning of
the lecture a valid solid average solid
should have a valid solid should have
certain constraints based on which we
can say that separates on it object and
not an open or enclosed one and that
constraints will define with the help of
planes lines edges and vertices the
first constraint which we have seen
today for the polygon is that an edge
must have or must be shared by an even
number of polygons we look into
additional constraints later on in the
next class when we meet to discuss
further on boundary representations for
representing solids thank you very much