>> This lecture is on the second part of descriptive geometry,
and among the topics we're going to talk about,
line line perpendicular to plane.
It says here a line perpendicular
to plane will appear perpendicular
to the edge of the plane.
Makes sense.
The plane is edge view, and the line is perpendicular to it,
it too should appear perpendicular in a view
where the edge view, where the plane is edge view.
A line perpendicular plane will also appear to length
in the axis of the plane.
So unless the line is to length as well,
when it appears perpendicular to the edge of the plane,
the two are not necessarily perpendicular.
So the quickest way, and most direct way, is very fine,
where the line is perpendicular to a plane, is to find a view
that shows the true length of the line,
automatically the plane should appear edge view,
and that edge view should be perpendicular to the line.
A line perpendicular to plane is perpendicular
to any line in the plane.
That makes sense.
And when two lines are perpendicular,
they will appear perpendicular to each other in a view
that shows at least one of them as a true line.
They don't have to be both true length, only one of them needs
to be true length, and the two should appear perpendicular.
Second topic is creating a line perpendicular to plane.
So in this case, we are given plane 1, 2, 3, front view here.
1, 2, 3 top view here, as well as point-0 front view
and top view, and we are asked to construct a line from point-0
that is perpendicular to plane 1, 2, 3.
So it says here all we have to do is construct a line
from point-0 perpendicular to the edge view.
So the first step is find the edge view of 1, 2, 3,
and the way you do that is you look for horizontal lines,
starting from point 3 horizontal line,
wherever that horizontal line intersects 1,
2, projected up here.
That point 4, when we connect 4 and 3, this line here,
and the top view becomes the true length
because it is the horizontal line.
Next thing we do is we take a reference plane perpendicular
to that true length that is 3, 4 in the top view.
Then project, make construction lines from 1,
2 and 3 perpendicular to the reference plane.
This reference plane is, of course, H on the left side here,
for the top view, and ARP1 on this side.
To find the locations of 1,
2 and 3 along their construction lines, we need to borrow
or measure the heights from this horizontal reference plane.
So this height of point 1 here is exactly the same
as this height here.
Same thing this height at point 2 is transferred here,
and height of point 3 transferred along its projection
to locate point 3, and if we did everything correctly, 1,
2 and 3 should line up perfectly.
And after that, we can also transfer point 0
in the auxiliary view by making this construction line
from point 0 perpendicular to reference plane, and borrowing
or measuring the height from H in the front view.
The next step is from point 0, recreate a line OP
that is perpendicular to 1, 2, 3.
And if OP is to be perpendicular,
truly perpendicular to the plane,
not only should OP appear perpendicular to edge view,
that OP has to be true length as well.
Once we find point P here, we can project it up back
to the top, so P can be anywhere along this construction line
created perpendicular to this reference plane.
But how do we know the location of point P along its projection?
There are two ways.
First is that point P has to be the intersection
of a perpendicular line coming from point O, perpendicular
to the construction lines, also perpendicular
to the true length 3, 4 in the top view.
Because we said that a line OP perpendicular
to the plane must be perpendicular
to any line in the plane.
So line 3, 4 here is true length, and therefore line OP,
which is perpendicular to line 3, 4 must appear perpendicular
to 3, 4 in the top view, because 3, 4 is true length.
Another way of looking at that is since OP here
in the auxiliary view is true length, what that means is
on the other side of this reference plane,
OP must be parallel to the reference plane
because to find the true length of the line, if you remember,
the reference plane has to be parallel to the line for it
to be true length on the other side.
So that defines our point P here.
Once you have that point P, you can start looking for point P
by creating this vertical construction line,
but we don't know exactly how far along this line.
So we use again the fact
that this horizontal reference plane here corresponds
to the horizontal reference plane here, so the distance
of P, P away from this H has to be the same as this distance.
So along this projection here, the location of P is
such that this height here should be exactly the same
height perpendicular distance from this H. And then finally,
we need to show correct visibility.
And the visibility of the top comes from the fact
that if you look down here, line OP is higher than line 1, 3.
Here is line 1, 3.
It's really, really low.
Showing the front view.
Which means that from the top line OP must be
completely visible.
Same thing for the visibility of the front view.
Any line that is closer to the front will be visible.
So we will compare OP versus 1, 2.
Which one, of OP, or 1, 2 is closer to the front?
Well, we look at the top view.
As we see here, it's OP, way closer to the front compared
to 1, 2, which is back here.
Therefore, OP has to be visible on the front view.
The next topic here is slope and slope direction of a plane.
It says here the slope of a plane is the angle
between the edge view of the plane,
and the horizontal reference plane, or any horizontal plane.
And the way to do that is you find the edge view off
of the top view, right off of the top view, because we have
to be simultaneously seeing a horizontal line
in order to find the slope.
So find the angle from the edge view
over the top view, as shown here.
So here is the front view of 1, 2, 3, the top view,
so we look for the edge view of the plane.
How do we do that?
Find this horizontal line here, 1, 2, let's call this O,
wherever O intersects 2, 3 connect 1 and 0 here,
that is a true length.
So what that means is, if this is a true length,
I'll need to take a reference plane here that is perpendicular
to that true length and that reference plane is H
on this side, and ALP1 on this side.
Then we create construction lines from points 1, 2,
3 perpendicular to the reference plane.
And to find the location of 2, 1 and 3,
you measure the heights, from the front view.
So this height of 2 is the same as that height of 1,
is the same as that, distance of 3 from H is the same as that,
and if we did everything correctly,
this will become actually the three point line.
Now, this is the edge view of the plane,
and notice that this line here corresponds
to the horizontal top plane.
So we create a line such as this,
parallel to this full line here, which is horizontal plane,
this line is horizontal, and therefore,
this angle here 41 degrees in this case is the angle
within the edge view of the plane, and the horizontal line
and that is the slope angle.
Now with respect to finding the direction of the slope,
or the slope direction, it says here direction is the direction
of the line or lines of steepest slope, and the lines
of steepest slope are perpendicular
to the direction of horizontal lines.
So what does that mean?
So since we are looking for directions,
and directions are always seen from the top view,
let's look at the top view of the plane.
This line 1, 0 is true length, in the top view,
because it is horizontal.
Okay? According to this, the slope direction is the direction
of a line perpendicular to that true length.
So here is the true length,
so we create this line here with an arrow, okay?
And we need to find a direction of that line,
why that line is perpendicular
to the horizontal line that is this 1, 0.
Now the question is why are we going southeast rather
than northwest?
Now how did we find southeast?
Here, if we draw north-south line here, okay,
the angle between that north-south line and this line
with an arrow head is 30 degrees and it is going southeast.
So how do we know that we have
to go southeast rather than northwest?
Well it has to be, the direction of the slope has to be going
from that higher point to the lower point.
Here is an illustration here of your slope direction,
or direction of slope.
The top here, this line here is horizontal.
And if you look at this plane here, this defines the direction
of the slope, and that line is perpendicular
to the horizontal line up here.
Now, we measure in this direction because,
according to this, the direction of the slope is measured
in the top view toward the low side of the plane.
So that means we need
to go downwards along the plane, rather than upwards.
In this case, the arrow should be going this way rather
than that way.
So going back here, how do we know that we're going downward
when we are going southeast?
So moving southeast in this direction means
that you're moving away from point 2.
As I move southeast along the plane, I am moving away
from point 2, and I know that point 2 is the highest point,
therefore moving southeast is moving down the plane.
If I go northwest, like in this direction here,
I'm actually moving closer and closer to point 2 in the plane.
That means I am moving up the plane.
The direction of the slope has to be measured
down the slope, down the plane.
The last part is piercing point, which is nothing but the line
of intersection, the point of intersection
between a line and the plane.
I was missing a word, "and" here.
It can be seen in the view that shows the edge view
of the plane, and is the apparent intersection
between the edge view and the line.
So all we have to do again is, this is from the 1, 2,
3 and line AB, top view of the plane in the line.
What we do is we look for the edge view of the plane
by creating this horizontal line O3, O projected to 1,
2 here will give us a true length here.
To find the edge of the plane we take the reference plane
perpendicular to the true length, or that simply
to taking this line of sight along the true length.
So once we have defined our reference plane here,
here is step 2, we project the points 2, 3, and 1,
and transfer their heights, as shown from the top view,
as always, and if we did everything correctly, 1,
2 and 3 will line up again along a straight line,
and that is the edge of 1, 2, 3.
After we do that, we also try to locate line AB
in the auxiliary view,
by creating this construction line perpendicular
to the reference plane, another construction line,
and transferring the heights of A and B
as shown from the front view.
Then connect A and B in the auxiliary view,
the apparent intersection of line AB
and the edge view is the piercing point, point B here.
Once you have the piercing point, we can easily project
that back into the top
and wherever this projection intersects AB,
will be the location of the piercing point in the top view,
and project that downward to the front, wherever it intersects AB
in the front will be your piercing point.
And the last step is visibility, again.
Let's look at the visibility of AB in the top view.
So right around here, line 1, 3 is trying to cover line AB.
Let's look at which one is higher.
Whichever of the two lines-1,3 versus AB,
is higher we will go from the top.
If you look at this line AB around this area
where they're trying to cover each other,
line AB is higher than-way higher than-1, 3.
Therefore, this side of AB should be visible.
Same thing to find the visibility of the front view,
you need to determine which of the two lines is closer
to the front.
So here, line 1, 2 is trying to cover line AB.
How do we know whether 1, 2 will be visible
or AB will be visible?
We need to know which one is closer to the front.
And how do we do that?
By looking at the top.
Right around where they're trying to cover each other,
we discover that AB here is closer to the front compared
to the corresponding 1, 2.
Therefore, from the front, AB on this side must be visible.
And then, on the other side of the piercing point,
it has to be hidden, because piercing point is
where the point where the line goes from one side
of the plane to the other.
Similarly, this one should be hidden.
By the way, the visibilities of the front and the top need
to be determined separately,
the two are not necessarily corresponding to each other.
In this case, they both are visible on the left side
of the piercing point.