WEBVTT

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>> This lecture is on the second
part of descriptive geometry,

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and among the topics
we're going to talk about,

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line line perpendicular
to plane.

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It says here a line
perpendicular

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to plane will appear
perpendicular

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to the edge of the plane.

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Makes sense.

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The plane is edge view, and the
line is perpendicular to it,

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it too should appear
perpendicular in a view

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where the edge view, where
the plane is edge view.

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A line perpendicular plane
will also appear to length

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in the axis of the plane.

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So unless the line
is to length as well,

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when it appears perpendicular
to the edge of the plane,

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the two are not necessarily
perpendicular.

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So the quickest way, and most
direct way, is very fine,

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where the line is perpendicular
to a plane, is to find a view

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that shows the true
length of the line,

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automatically the plane
should appear edge view,

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and that edge view should be
perpendicular to the line.

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A line perpendicular to
plane is perpendicular

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to any line in the plane.

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That makes sense.

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And when two lines
are perpendicular,

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they will appear perpendicular
to each other in a view

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that shows at least one
of them as a true line.

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They don't have to be both true
length, only one of them needs

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to be true length, and the two
should appear perpendicular.

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Second topic is creating a
line perpendicular to plane.

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So in this case, we are given
plane 1, 2, 3, front view here.

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1, 2, 3 top view here, as
well as point-0 front view

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and top view, and we are asked
to construct a line from point-0

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that is perpendicular
to plane 1, 2, 3.

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So it says here all we have
to do is construct a line

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from point-0 perpendicular
to the edge view.

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So the first step is find
the edge view of 1, 2, 3,

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and the way you do that is
you look for horizontal lines,

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starting from point
3 horizontal line,

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wherever that horizontal
line intersects 1,

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2, projected up here.

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That point 4, when we connect
4 and 3, this line here,

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and the top view
becomes the true length

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because it is the
horizontal line.

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Next thing we do is we take a
reference plane perpendicular

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to that true length that
is 3, 4 in the top view.

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Then project, make
construction lines from 1,

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2 and 3 perpendicular
to the reference plane.

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This reference plane is, of
course, H on the left side here,

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for the top view, and
ARP1 on this side.

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To find the locations of 1,

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2 and 3 along their construction
lines, we need to borrow

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or measure the heights from
this horizontal reference plane.

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So this height of point 1
here is exactly the same

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as this height here.

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Same thing this height at
point 2 is transferred here,

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and height of point 3
transferred along its projection

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to locate point 3, and if we
did everything correctly, 1,

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2 and 3 should line
up perfectly.

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And after that, we can
also transfer point 0

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in the auxiliary view by
making this construction line

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from point 0 perpendicular to
reference plane, and borrowing

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or measuring the height
from H in the front view.

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The next step is from
point 0, recreate a line OP

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that is perpendicular
to 1, 2, 3.

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And if OP is to be
perpendicular,

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truly perpendicular
to the plane,

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not only should OP appear
perpendicular to edge view,

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that OP has to be
true length as well.

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Once we find point P here,
we can project it up back

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to the top, so P can be anywhere
along this construction line

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created perpendicular
to this reference plane.

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But how do we know the location
of point P along its projection?

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There are two ways.

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First is that point P has
to be the intersection

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of a perpendicular line coming
from point O, perpendicular

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to the construction
lines, also perpendicular

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to the true length
3, 4 in the top view.

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Because we said that a
line OP perpendicular

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to the plane must
be perpendicular

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to any line in the plane.

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So line 3, 4 here is true
length, and therefore line OP,

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which is perpendicular to line
3, 4 must appear perpendicular

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to 3, 4 in the top view,
because 3, 4 is true length.

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Another way of looking
at that is since OP here

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in the auxiliary view is true
length, what that means is

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on the other side of
this reference plane,

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OP must be parallel
to the reference plane

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because to find the true length
of the line, if you remember,

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the reference plane has to be
parallel to the line for it

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to be true length
on the other side.

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So that defines our
point P here.

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Once you have that point P, you
can start looking for point P

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by creating this vertical
construction line,

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but we don't know exactly
how far along this line.

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So we use again the fact

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that this horizontal reference
plane here corresponds

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to the horizontal reference
plane here, so the distance

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of P, P away from this H has to
be the same as this distance.

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So along this projection
here, the location of P is

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such that this height here
should be exactly the same

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height perpendicular distance
from this H. And then finally,

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we need to show correct
visibility.

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And the visibility of the
top comes from the fact

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that if you look down here, line
OP is higher than line 1, 3.

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Here is line 1, 3.

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It's really, really low.

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Showing the front view.

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Which means that from
the top line OP must be

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completely visible.

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Same thing for the
visibility of the front view.

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Any line that is closer to
the front will be visible.

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So we will compare
OP versus 1, 2.

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Which one, of OP, or 1,
2 is closer to the front?

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Well, we look at the top view.

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As we see here, it's OP, way
closer to the front compared

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to 1, 2, which is back here.

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Therefore, OP has to be
visible on the front view.

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The next topic here is slope
and slope direction of a plane.

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It says here the slope
of a plane is the angle

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between the edge
view of the plane,

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and the horizontal reference
plane, or any horizontal plane.

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And the way to do that is
you find the edge view off

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of the top view, right off of
the top view, because we have

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to be simultaneously
seeing a horizontal line

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in order to find the slope.

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So find the angle
from the edge view

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over the top view,
as shown here.

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So here is the front view
of 1, 2, 3, the top view,

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so we look for the
edge view of the plane.

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How do we do that?

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Find this horizontal line
here, 1, 2, let's call this O,

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wherever O intersects 2,
3 connect 1 and 0 here,

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that is a true length.

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So what that means is,
if this is a true length,

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I'll need to take a reference
plane here that is perpendicular

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to that true length and
that reference plane is H

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on this side, and
ALP1 on this side.

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Then we create construction
lines from points 1, 2,

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3 perpendicular to
the reference plane.

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And to find the location
of 2, 1 and 3,

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you measure the heights,
from the front view.

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So this height of 2 is the
same as that height of 1,

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is the same as that, distance
of 3 from H is the same as that,

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and if we did everything
correctly,

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this will become actually
the three point line.

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Now, this is the edge
view of the plane,

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and notice that this
line here corresponds

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to the horizontal top plane.

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So we create a line
such as this,

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parallel to this full line
here, which is horizontal plane,

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this line is horizontal,
and therefore,

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this angle here 41 degrees
in this case is the angle

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within the edge view of the
plane, and the horizontal line

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and that is the slope angle.

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Now with respect to finding
the direction of the slope,

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or the slope direction, it says
here direction is the direction

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of the line or lines of
steepest slope, and the lines

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of steepest slope
are perpendicular

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to the direction of
horizontal lines.

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So what does that mean?

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So since we are looking
for directions,

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and directions are always
seen from the top view,

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let's look at the top
view of the plane.

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This line 1, 0 is true
length, in the top view,

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because it is horizontal.

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Okay? According to this, the
slope direction is the direction

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of a line perpendicular
to that true length.

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So here is the true length,

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so we create this line
here with an arrow, okay?

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And we need to find a
direction of that line,

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why that line is perpendicular

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to the horizontal line
that is this 1, 0.

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Now the question is why are
we going southeast rather

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than northwest?

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Now how did we find southeast?

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Here, if we draw
north-south line here, okay,

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the angle between that
north-south line and this line

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with an arrow head is 30 degrees
and it is going southeast.

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So how do we know that we have

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to go southeast rather
than northwest?

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Well it has to be, the direction
of the slope has to be going

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from that higher point
to the lower point.

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Here is an illustration here
of your slope direction,

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or direction of slope.

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The top here, this line
here is horizontal.

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And if you look at this plane
here, this defines the direction

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of the slope, and that
line is perpendicular

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to the horizontal line up here.

00:10:19.166 --> 00:10:21.786 A:middle
Now, we measure in
this direction because,

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according to this, the direction
of the slope is measured

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in the top view toward
the low side of the plane.

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So that means we need

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to go downwards along the
plane, rather than upwards.

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In this case, the arrow should
be going this way rather

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than that way.

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So going back here, how do we
know that we're going downward

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when we are going southeast?

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So moving southeast in
this direction means

00:10:52.076 --> 00:10:54.026 A:middle
that you're moving
away from point 2.

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As I move southeast along
the plane, I am moving away

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from point 2, and I know that
point 2 is the highest point,

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therefore moving southeast
is moving down the plane.

00:11:07.676 --> 00:11:09.916 A:middle
If I go northwest, like
in this direction here,

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I'm actually moving closer and
closer to point 2 in the plane.

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That means I am moving
up the plane.

00:11:15.696 --> 00:11:17.666 A:middle
The direction of the
slope has to be measured

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down the slope, down the plane.

00:11:21.836 --> 00:11:25.906 A:middle
The last part is piercing point,
which is nothing but the line

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of intersection, the
point of intersection

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between a line and the plane.

00:11:30.326 --> 00:11:32.116 A:middle
I was missing a word,
"and" here.

00:11:32.816 --> 00:11:35.926 A:middle
It can be seen in the view
that shows the edge view

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of the plane, and is the
apparent intersection

00:11:38.486 --> 00:11:39.886 A:middle
between the edge
view and the line.

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So all we have to do again
is, this is from the 1, 2,

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3 and line AB, top view
of the plane in the line.

00:11:47.676 --> 00:11:49.956 A:middle
What we do is we look for
the edge view of the plane

00:11:51.016 --> 00:11:55.286 A:middle
by creating this horizontal
line O3, O projected to 1,

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2 here will give us
a true length here.

00:11:57.806 --> 00:12:01.056 A:middle
To find the edge of the plane
we take the reference plane

00:12:01.166 --> 00:12:03.506 A:middle
perpendicular to the true
length, or that simply

00:12:03.506 --> 00:12:06.436 A:middle
to taking this line of
sight along the true length.

00:12:07.086 --> 00:12:09.586 A:middle
So once we have defined
our reference plane here,

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here is step 2, we project
the points 2, 3, and 1,

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and transfer their heights,
as shown from the top view,

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as always, and if we did
everything correctly, 1,

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2 and 3 will line up again
along a straight line,

00:12:26.346 --> 00:12:27.836 A:middle
and that is the edge of 1, 2, 3.

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After we do that, we also
try to locate line AB

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in the auxiliary view,

00:12:34.696 --> 00:12:37.656 A:middle
by creating this
construction line perpendicular

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to the reference plane,
another construction line,

00:12:40.786 --> 00:12:44.166 A:middle
and transferring the
heights of A and B

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as shown from the front view.

00:12:46.176 --> 00:12:48.766 A:middle
Then connect A and B
in the auxiliary view,

00:12:48.766 --> 00:12:50.916 A:middle
the apparent intersection
of line AB

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and the edge view is the
piercing point, point B here.

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Once you have the piercing
point, we can easily project

00:12:57.736 --> 00:12:59.016 A:middle
that back into the top

00:12:59.596 --> 00:13:02.116 A:middle
and wherever this
projection intersects AB,

00:13:02.116 --> 00:13:05.276 A:middle
will be the location of the
piercing point in the top view,

00:13:05.276 --> 00:13:08.486 A:middle
and project that downward to the
front, wherever it intersects AB

00:13:08.486 --> 00:13:10.966 A:middle
in the front will be
your piercing point.

00:13:11.386 --> 00:13:13.836 A:middle
And the last step is
visibility, again.

00:13:14.926 --> 00:13:17.766 A:middle
Let's look at the visibility
of AB in the top view.

00:13:17.936 --> 00:13:26.996 A:middle
So right around here, line 1,
3 is trying to cover line AB.

00:13:26.996 --> 00:13:28.876 A:middle
Let's look at which
one is higher.

00:13:29.096 --> 00:13:31.726 A:middle
Whichever of the two
lines-1,3 versus AB,

00:13:31.726 --> 00:13:34.366 A:middle
is higher we will
go from the top.

00:13:34.366 --> 00:13:37.966 A:middle
If you look at this
line AB around this area

00:13:37.966 --> 00:13:39.156 A:middle
where they're trying
to cover each other,

00:13:39.156 --> 00:13:42.046 A:middle
line AB is higher
than-way higher than-1, 3.

00:13:42.456 --> 00:13:44.796 A:middle
Therefore, this side of
AB should be visible.

00:13:45.576 --> 00:13:49.066 A:middle
Same thing to find the
visibility of the front view,

00:13:49.166 --> 00:13:52.186 A:middle
you need to determine which
of the two lines is closer

00:13:52.186 --> 00:13:52.746 A:middle
to the front.

00:13:52.786 --> 00:13:56.996 A:middle
So here, line 1, 2 is
trying to cover line AB.

00:13:57.396 --> 00:14:00.196 A:middle
How do we know whether
1, 2 will be visible

00:14:00.196 --> 00:14:01.356 A:middle
or AB will be visible?

00:14:01.356 --> 00:14:03.616 A:middle
We need to know which one
is closer to the front.

00:14:04.176 --> 00:14:05.206 A:middle
And how do we do that?

00:14:05.386 --> 00:14:06.346 A:middle
By looking at the top.

00:14:07.126 --> 00:14:08.956 A:middle
Right around where they're
trying to cover each other,

00:14:08.956 --> 00:14:13.816 A:middle
we discover that AB here is
closer to the front compared

00:14:13.816 --> 00:14:15.676 A:middle
to the corresponding 1, 2.

00:14:16.716 --> 00:14:20.686 A:middle
Therefore, from the front, AB
on this side must be visible.

00:14:21.616 --> 00:14:23.756 A:middle
And then, on the other
side of the piercing point,

00:14:23.756 --> 00:14:26.176 A:middle
it has to be hidden,
because piercing point is

00:14:26.176 --> 00:14:29.236 A:middle
where the point where the
line goes from one side

00:14:29.236 --> 00:14:30.246 A:middle
of the plane to the other.

00:14:30.636 --> 00:14:32.616 A:middle
Similarly, this one
should be hidden.

00:14:32.996 --> 00:14:36.936 A:middle
By the way, the visibilities
of the front and the top need

00:14:36.936 --> 00:14:39.216 A:middle
to be determined separately,

00:14:39.626 --> 00:14:42.536 A:middle
the two are not necessarily
corresponding to each other.

00:14:42.536 --> 00:14:44.986 A:middle
In this case, they both are
visible on the left side

00:14:45.416 --> 00:14:47.656 A:middle
of the piercing point.