>> This is the third part of the lecture on descriptive geometry.
The first topic we're going to talk
about is finding the piercing point of a line in a plane
by using the cutting plane method.
Previously we had a method of finding the piercing point
by simply looking at the edge view of the plane.
Now, this time here, we're not allowed to do an auxiliary view.
So what we do is we use what's called a cutting plane method.
OK. So we are given here the top and the front views
of plane 123 and line AB.
So the first thing we do is step 1, pretend that AB
in the top view is not just a line
but the edge view of a cutting plane.
So pretend that this cutting plane AB is infinite in extent
and it's at its edge view.
And pretend cutting plane AB intersects the actual plane 123
along line pq.
So pq here is the line of intersection
between the real plane 123
and the cutting plane, the pretend plane AB.
We can easily transfer the intersection of --
line of intersection -- between the view pq to the front view.
Point p is the intersection of the pretend plane with 123.
So we take this to the front view to line 13.
So that's point p. And then point q is the intersection
with line 23 so we take it down to line 23 in the front view.
And then we connect pq in the front view.
Now the intersection of this --
of pq with line AB is the piercing point.
And the reason for this is that since line AB is contained
in the cutting plane AB , the pretend edge view cutting plane
in the top view, the intersection of line AB
with plane 123, which is the piercing point,
should also be contained in the intersection of plane AB
with plane 123, which is line pq.
So the point of intersection, which is the piercing point,
has to be contained in the line of intersection.
After finding the piercing point, we can simply project it
up to the top view to find the corresponding location
of the piercing point in the top view.
Or we can actually repeat the process.
In this case, we're going to stop and start in the front view
and pretend that AB in the front view is not just a line
but the edge view of a cutting plane.
And this cutting plane, AB,
intersects the actual plane 123 along line xy.
We take x to edge 12 in the top and y to edge 13 in the top.
And connect xy in the top.
The line of -- this point of intersection of xy
with AB would be the piercing point.
And as a last step you need to show correct visibility
and I'm not going to go over that because we've looked
at that in the previous lecture --
how to show visibility of a line
after finding its piercing point.
Second topic we're going to talk
about is finding the intersection of two planes.
And this can be done in two ways.
The first way is by finding the edge view of one of the planes,
in which case you're going to need an auxiliary view.
The other method is by simply locating two piercing points
that can be found from line of intersection of a line from one
of the two planes into the other plane.
So what this means is you can use the cutting plane method.
Let's say that your two planes are 123 and ABC.
So we have 6 possible combinations.
We can test line AB versus plane 123, line BC versus plane 123,
line AC versus plane 123.
And then, if we didn't find two piercing points,
we can depress it and look at 12 as a line versus plane ABC,
23 versus ABC, and 13 versus ABC.
So basically 6 combinations and you --
you will eventually find 2 piercing points
and simply connect those 2 piercing points will be the line
of intersection.
The advantage, of course, is it's trial and error.
But the advantage is you don't need
to construct an auxiliary view.
And after finding the line of intersection
through the 2 piercing points, you always have
to show correct visibility.
The third topic in this lecture is finding true size of planes.
Now the way to see the true size of a plane is by taking a line
of sight that is perpendicular
to the edge view of the plane itself.
But knowing that the reference plane is always perpendicular
to the line of sight and if the line of sight is perpendicular
to the edge view, what we need to have now is a reference plane
that is parallel to the edge view.
So line of sight perpendicular to the edge view.
Line of sight is also perpendicular
to the reference plane.
Therefore, the edge view and the reference planes will have
to be parallel to each other.
The resulting view will show the true length of all of the edges
of the plane as well as the 2 angles between the edges.
So the steps are -- so our ultimate objective is
to have a reference plane that is parallel
to the edge view of the plan.
So the first step is find the edge of the plane
by finding the point view of any line in the plane.
You should start with horizontal line becomes true length
in the top.
And then we take a reference plane perpendicular
to the 2 lines in the top.
That will be the edge of the plane.
Once we have the edge of the plane,
we take a reference plane parallel to that edge view.
And the resulting view would be true size of the plane.
Find the angle between lines.
The angle between 2 lines if you are given 2 lines.
One, the view that was show the true angle
between 2 lines is one
that shows the true length of both lines.
And one way of doing that is
by creating a plane containing the 2 lines
and then finding the true size of that plane.
The steps are create a plane
by connecting the endpoints of the lines.
Then find the edge view of the plane, as we always do.
Then find the true size of the plane
by taking a reference plane that is parallel to that edge view
as we talked about on the previous page.
And that true size of the plane will also find the angle
between the lines -- show the angle between the lines.
Here's an example.
We are trying to find the angle between 2 lines, PQ and PR,
by using -- or by first finding the true size of the plane.
So we are given the front and top view of PQ --
this is PQ -- and PR -- this is PR.
This is the top and this is the front.
We're trying to find the true angle
between those lines so the angle QPR.
Of course, we cannot simply measure the angles on the lines
that we're seeing because we're not seeing the true size
of the plane.
We're not seeing the true length of both lines.
So what we do is -- over here --
we create a plane by connecting the ends Q and R.
And then once we have the plane PQR,
we try to find the edge of that plane.
And we have to find edge view is.
Find the horizontal line.
Create a horizontal line passing through point P wherever
that intersects QR will be up here.
And this will be true length.
Once we find this true length here, in the top view,
you take a reference plane that's perpendicular
to that true line, that reference plane, of course,
is H on this side and a RP1 on that side.
Then you project, you see construction lines
from the point's PQR, from the top view perpendicular
to the reference plane.
And then we transfer the height of Q here along its projection.
Height of P along its projection in line
with R. Along its projection.
If we did everything correctly, P, Q, and R,
should line up along a straight line.
And that's the edge view.
This is the edge view of PQR.
And then, once we have the edge view,
we take a second auxiliary reference plane -- here --
this orange one here -- which will be ARP1 on this side.
Take note that this whole view here is auxiliary view number 1
corresponding to ARP1.
And the other side is ARP2.
To find the location of P, Q, and R for ARP2,
or for the second auxiliary view, we project P, Q,
and R perpendicular to the reference plane.
And then we transfer the distance --
so this is our ARP1 -- this whole thing is our ARP1.
The distance of Q away from ARP1, this distance here.
That's viewed -- seen -- in the top view.
It's the same as the distance of Q away from this ARP1.
So this distance from this ARP1 in this view must be equal
to distance from this ARP1 from this view.
So we borrowed this distance here for Q here.
This is the distance of P --
this tiny little distance here --
is the distance of P. And this is the distance of R --
all the distances are measured away from ARP1 here
on the left side and away from ARP1 here into the top view.
And then we just simply connect Q, P, and R. Take note
that PQ is parallel to this reference plane.
Therefore, this is true length here.
PR is also parallel to this reference plane.
So PR is also true length,
which means that we're seeing the true size of the plane and,
therefore, this angle here, we measure it a protractor,
is the true angle of the two lines.
Finally, last topic is finding the shortest distance
from a point to a line.
So we're going to assume that we're given a line --
maybe 2 views, top view and front view --
and then a point shown in those 2 vies.
One method is what's called a line method.
You simply find a point view of the line
and then the shortest distance from the point
to the line is true length
in the view showing the point view of the line.
So the shortest distance would be perpendicular
to the line itself.
The plane method is one in which we create a plane as defined
by the line and the point.
And then we simply find the true size
of the plane containing the line and the point.
And that -- from that view, that shows us 3 sides.
All we have to do is to create a line
from the given point perpendicular to the line.
So basically find the true size of the plane
that contains the line and the point
and then make a perpendicular line
from the point into the line.