>> This lecture is on the first part of descriptive geometry
and you remember this figure here.
This was the figure that we did for one of our labs
on secondary auxiliary views.
We were given the front here and the right side view
and you were given this primary auxiliary reference plane here,
the secondary auxiliary reference plane here.
The first reference plane gave us this auxiliary,
primary auxiliary, view.
And then the secondary auxiliary view is here.
It turns out to be the true size of plane ABC.
So the question is, "How did we know how
to position these two auxiliary reference planes in order
to get the true size of plane ABC?"
And that's one of the questions that we're going to be answering
in this topic called descriptive geometry.
It involves the projection of three dimensional figures
on a two dimensional plane of the paper in a manner
that allows geometric manipulation
to determine lengths, angles, shapes,
areas and other quantitative information by means of graphic.
So we're going to use both the graphic projections specifically
in order to extract useful quantitative information
from the drawings.
The auxiliary views are used to show things like true length
and point of views of lines, true sizes and edge views
of planes, angles between two lines, between two planes,
and between the line and the plane.
This is based -- The method we're going to use is based
on the Folding-Line Model which we can use in order
to visualize the resulting auxiliary views.
Starting from the simplest entity, we can look at in 3D,
a point is nothing but a location in space.
Of course no dimension, just a location.
And to specify the location you're going
to need three coordinates, X, Y, Z, to define a point in space.
So for instance this could be the height H below the
top plane.
The width, number of units to the left
of the right profile plane P. And also the depth
which is how far behind you are from the frontal plane.
In order to define all three of the dimensions, height,
width and depth, we're going to need at least two views.
Okay. Here's an illustration of --
This seems to be an oblique perspective of the glass box
that we're trying to illustrate using the folding line method.
Here's point two, okay, and here's the projection
of point two in the front view, top view
and the right side view.
And it's illustrating the three dimensions in height
which is measured in how many units you are below the top
horizontal plane.
There's the dimension of width, how many units you are
to the left of the profile plane here.
And this depth here, how many units you are behind the
frontal plane.
And take note that for each fold in the --
corresponding to the projection planes,
you have your planes labeled horizontal here
for the top view, F for frontal, P for profile.
Going to lines now which are nothing but straight paths
between two points, a line can appear in a view
as either foreshortened, true length or point view.
The first set of top and front views here, one, two,
is shown both in the front view here and top view
as foreshortened, meaning it appears as a line,
but it's shorter than its actual true length.
And the second example here, line three, four, is horizontal,
HL, because it's parallel to the fold line that corresponds
to the top plane here.
So this line here, three, four,
three and four have the same height.
Therefore it's horizontal and a horizontal line when viewed
from the top like this is true length or TL in this case.
The last combination of front
and top views illustrates line five, six, which happens
to be perpendicular to the frontal plane.
And, as a result, this line five, six becomes true length
in the top view, but a point view in the frontal plane,
meaning five and six both correspond to the same point.
Classification of lines according to their orientations
with respect to the principal projection planes of horizontal,
frontal and profile planes.
We have a line classified as principal, principal line.
It was parallel to at least one of the principal planes.
An oblique line is one that's neither parallel nor
perpendicular to any of the principal planes.
Horizontal line is a principal line.
It's parallel to the top horizontal plane and,
as a result, appears true length in the top view.
A frontal line is parallel to the frontal plane
and appears true length in the front view.
Similarly, a profile line is parallel to the profile plane
and appears true length in the side views.
Here is an illustration of the horizontal line.
One, two is horizontal.
They have -- One and two have the same height.
As a result, it's true length in the top view.
Three, four is frontal.
Why? Because three and four have the same depth
or distance behind the frontal plane either viewed from the top
or here at the right side view.
As a result, the frontal line three,
four appears true length in the front view.
And five, six is an example of a profile line.
It's parallel to the profile plane, the right profile plane.
As a result, it's true length in the right side view.
One of the things we can find using descriptive geometry is
the true length of a given line.
We can either use Pythagorean Theorem using true length
diagram or by using an auxiliary view
to find the true length of the lines.
Using Pythagorean Theorem,
really Pythagorean Theorem in 3D.
Those of you who have taken physics and Calc 3, you know
that this line here, that it
in 3D generally can have three components,
one along X which you call the width,
one along Z which you call the depth
and Y which we call height.
If you look at the projection of the plane in the XY plane
of the -- in the XY plane, this length here is equal
to the square root of X squared plus Y squared
by Pythagorean Theorem.
And now if you look at this triangle over here now,
it is also a right triangle where this is one of the legs
and this is the other leg, is Z --
If we use the Pythagorean Theorem,
we see that the true length of the line here
from point one here to point two
in 3D space is just the square root of the sum
of the squares of the XYZ components.
Here is a graphic where you're finding the true length.
Here's the top view and here's the front view.
What you do is you create your answers here
to create your true line diagram,
a vertical axis here and the horizontal.
And what you do is you measure the height of the line.
That is the vertical distance from point one to point two.
And plot it on the vertical side here.
And then measure the horizontal extent
or the horizontal projection
which can be seen from the top view.
That's H in the horizontal.
And just connect this to points here to find the true length
of the line, measure the true length of the line.
Another way of doing -- finding the true length of the line,
of a line, is by using an auxiliary view.
Given, for instance, say the top view of AB here
and the front view of AB, we're asked to find an auxiliary view
that would show the true length of line AB.
Now for you to see the true length
of a line, visualize this.
If you have a line in 3D space,
the only way you can see the length of that line is
if your line of sight is perpendicular
to the line itself.
So the line of sight needs to be perpendicular to the line,
but we know that the line of sight is always perpendicular
to the projection planes or the reference plane which means
that if two things are perpendicular to the line
of sight, okay, those two things must be parallel to each other.
Therefore the reference plane must be parallel to the line
that you're trying to find the true length of.
What you need to do is, as shown here --
You take a reference plane here, parallel to line AB.
I did that on the top view.
And then construct these green lines
which are projection lines projected perpendicular
to the reference plane here.
So we have labeled the fold line here.
It's between the top view and the front view,
so H and F, H and F here.
The fold line here is between the horizontal or the top view
and the primary auxiliary view.
So this is the auxiliary reference plane number one.
This is the fold line between the two.
And what we do is to locate the position
of A along this construction line, the depth of point A
which represents the perpendicular distance
from H here, the top plane, is the same as the depth
as seen in the front view.
So our current central view is the top view corresponding
to the horizontal plane.
Perpendicular distance away from H as seen
from the front view should be exactly the same
as the perpendicular distance away from H
as seen in auxiliary view.
So, as we can see here, A is at the height
of 1.5 below the top plane
or H. Therefore it's also 1.5 units away from the H here.
Similarly, B is anywhere along this green projection line,
but it's -- a height of 0.5 below H. Therefore it is going
to be 0.5 away from H. So the procedure is, just to review,
take this reference plane parallel to AB,
create these construction lines perpendicular
to that reference plane that was created parallel to AB --
By the way, the position of this auxiliary reference plane is
arbitrary as long as it's parallel to AB.
It could be slightly farther here or slightly closer to AB.
It wouldn't make a difference.
And then to find the locations of A and B we need to measure
or borrow the heights or distances away
from H here shown in the front view.
Direction of a line.
When you're looking at a map,
the map is really the top view of the terrain.
So direction of a line is always shown in the top view.
And there's two common ways in engineering
to give the direction of a line.
One is using the compass bearing which is nothing
but an angle measured from the north or the south,
depending on which one's closer, measured towards the east
or the west between 0 and 90 degrees.
Or as the azimuth usually used by civil engineers.
It's the angle measured always from the north.
At least in the northern hemisphere we use north
as the reference.
In the southern hemisphere,
their azimuths are measured from the south.
And it's an angle that's measured from 0
to 360 degrees clockwise.
So, for instance, line AB -- So with A as the reference,
I have four lines, AB, AC, over here AD and AE.
And we are given these angles here.
Based on these angles we can find the compass bearing
and azimuth.
So AB, here's AB.
It's closer to the north.
So we start measuring from the north.
And it's towards the east so we --
We're going to need this angle here which happens
to be the complement of 65.
Therefore the compass bearing of AB is north 25 degrees.
90 minus 65 east.
North 25 degrees, east.
Whereas its azimuth is right here
from the north and clockwise.
It just says north 25 degrees.
No east or west because it's always measured clockwise.
AC, on the other hand, is closer again to the north,
but it goes towards the west by 35 degrees.
So it's north, 35 degrees west which means that the azimuth --
We need to find this angle measured
from the north all the way to where AC is.
That gives you 360 minus 35 which 325.
So it's north 325 degrees.
Same thing for AD and AE.
So hopefully you'll see why for AD it's south,
because it's closer to the south, we need the angle here
from the south towards the east.
And that's a complement of 60 which is 30 whereas
for E it's also measured from the south, starts closer
to the south, but we need this angle here towards the west.
So it's going to be the complement of 38
which is 52 south, 52 degrees west,
whereas for the azimuth it's always measured from the north,
clockwise towards AE here.
That's why it's 132.
If a line is going directly east, we don't say it's north,
90 degrees east or south 90 degrees.
We simply say it's due north -- due east, rather.
We don't ....
Here's an illustration of a line.
If a line is not perfectly horizontal, such as this one is,
what you need to do is two appears to be higher than three.
What you need to do is project two and three
in to a horizontal plane which basically is your top view.
That's why we look for directions in the top view.
And the way you give directions,
you always use the higher point as your reference.
So in giving the direction of line two, three,
you need to start from point two and move towards three
because two is the higher end of the length.
In this case, directional line two, three is north,
45 degrees east or azimuth of north 45 degrees.
Slope of a line is the measure of the steepness
or inclination with respect to the horizontal horizontally,
can be specified using a slope which --
The slope which is rise over run, you see that in algebra.
So the rise of the vertical distance
and the run would be what we call "H" here,
or the horizontal projection of the line.
Which -- It can also be expressed in grade which is --
percent grade which is the slope --
numerical value of the slope times 100 percent.
Or it's also common to give you the actual angle.
So, as an example, a slope of 3
over 4, (I know a 3-4-5 triangle).
So 4 is along the horizontal and three's the vertical.
The slope would be 3 over 4.
The percent grade of that would be 75 percent
and the slope angle would be equal to the arctangent of 3
over 4 which is about 36.87 degrees.
So the slope angle is the angle between --
You have to be measuring the true length of the line relative
to the horizontal plane.
And to do this graphically, first you need
to find the true length of the top view
and then the resulting view will show the angle
with the true length and the edge view
of the horizontal plane which corresponds
to the fold line between H and ARP.
Here's an illustration here.
It's the same exact line we looked at earlier,
AB in the front, AB in the top.
What we did was we looked for the true length of the line
by taking this reference plane parallel to AB in the top view.
If you want to find the slope angle,
you have to do the reference plane,
auxiliary reference plane number one, parallel to the top view.
Here is the true length of the line AB,
which means that if this is true length,
this line here is parallel to the horizontal plane
because it's parallel to H
which is the fold line folded from the top.
So what that means is this angle here measured
at 17 degrees is the slope angle of the line
because it's the angle meeting the true length of AB
and this line here which happens to be horizontal
because it's parallel to H. Point view of a line.
So to find the point view of a line we need to make our line
of sight parallel to the true length of the line itself.
And the procedure is first find the true length of the line.
The way you find the true length of a line,
as we just discussed this earlier,
take a reference plane parallel to the line.
Then project the end point perpendicular
to the reference plane.
Label all the reference planes.
This distance away from the previous corresponding
reference plane.
Then take the reference plane -- the next reference plane,
perpendicular to the true length project, and transfer.
That should be the point view of that.
I will illustrate this in the next lecture.
Edge view of a plane involves finding the point view
of any line in the plane.
So imagine a plane.
If you can find any line in the plane and find the point view
of that line, you will get the view
that is edge view of the plane.
So the procedure is find a line in the plane
that is already shown as true length.
And the easiest way to do that is find a line that's horizontal
because if it's horizontal it will be true length
in the top view.
Then take a reference plane perpendicular
to the true length in the top view.
Project the points of the plane.
Take measurements of height from the front view.
And then if you did your procedure correctly,
the points will line up into an edge view.
Here's an illustration.
Here's step one.
We are given the front view here of 1, 2, 3, the top view of 1,
2, 3, and we're asked to find the edge view of the plane,
a view that will show line 1, 2,
3 with an edge or a straight line.
So you need to have 1, 2, and 3 lined up perfectly.
The way you do that is step one.
Find a horizontal line, 1, 0.
How did I find 0?
I created this line here parallel
to the horizontal plane here.
Wherever this horizontal line intersects 2, 3,
I will call that point 0.
Project that point 0 in to the top view.
It should intersect 2, 3 here at the same vertical projection.
Therefore 1, 0, the top view which is here,
should be true length.
And the next thing we need to do is take the line
of sight parallel to that true length and that's equivalent
to taking a reference plane perpendicular
to the true length.
So as soon as you find 1, 0, its true length in the top view,
take a reference plane here labeled horizontal on the left
and 1 or primary auxiliary view on the other side.
This will be our reference plane perpendicular
to the true length.
And then what you need to do is find locations of point 0 and 1,
0 and 1, in the primary auxiliary view.
1 and 0 must be anywhere along this projection.
How do I find how far they are
from the fold line here which is H?
I borrow the depth of --
the height of 1 as measured here in the front view.
This distance H here should be the same
as this distance H here.
And, as you can see, the distance --
The height of 0 is the same as height of 1.
Okay. And 1 and 0 are in the same projection which means
that 1 and 0 will end up being a point view because they are
in the same projection here
because I made this projection plane perpendicular to line --
the true length of line 1, 0.
And they are the same height here.
So the same distance away from this fold line.
I'll do a similar thing for point 2 and point 3 here.
So I construct the projection of point 2 here,
the projection of point 3 here.
Both projection lines perpendicular
to the fold line here.
To find point 2 here, I'll borrow the height
or distance away from this H here as seen from the top
as being the same as this distance of 2 away from this H
as seen in the auxiliary view.
Same thing with point 3 here.
This height here is the same as this height.
If I did everything correctly, the 3 points,
1 here, 0 and 1 are here.
It's the same location as this.
If I did everything correctly, 3, 1 and 2 should line
up along a straight line
and that is the edge view of plane 1, 2, 3.