WEBVTT

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>> This is the third part of the
lecture on descriptive geometry.

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The first topic we're
going to talk

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about is finding the piercing
point of a line in a plane

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by using the cutting
plane method.

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Previously we had a method
of finding the piercing point

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by simply looking at the
edge view of the plane.

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Now, this time here, we're not
allowed to do an auxiliary view.

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So what we do is we use what's
called a cutting plane method.

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OK. So we are given here
the top and the front views

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of plane 123 and line AB.

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So the first thing we do
is step 1, pretend that AB

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in the top view is
not just a line

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but the edge view
of a cutting plane.

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So pretend that this cutting
plane AB is infinite in extent

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and it's at its edge view.

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And pretend cutting plane AB
intersects the actual plane 123

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along line pq.

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So pq here is the
line of intersection

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between the real plane 123

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and the cutting plane,
the pretend plane AB.

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We can easily transfer
the intersection of --

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line of intersection -- between
the view pq to the front view.

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Point p is the intersection
of the pretend plane with 123.

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So we take this to the
front view to line 13.

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So that's point p. And then
point q is the intersection

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with line 23 so we take it down
to line 23 in the front view.

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And then we connect
pq in the front view.

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Now the intersection of this --

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of pq with line AB is
the piercing point.

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And the reason for this is
that since line AB is contained

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in the cutting plane AB , the
pretend edge view cutting plane

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in the top view, the
intersection of line AB

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with plane 123, which
is the piercing point,

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should also be contained in
the intersection of plane AB

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with plane 123, which
is line pq.

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So the point of intersection,
which is the piercing point,

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has to be contained in
the line of intersection.

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After finding the piercing
point, we can simply project it

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up to the top view to find
the corresponding location

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of the piercing point
in the top view.

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Or we can actually
repeat the process.

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In this case, we're going to
stop and start in the front view

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and pretend that AB in the
front view is not just a line

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but the edge view
of a cutting plane.

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And this cutting plane, AB,

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intersects the actual
plane 123 along line xy.

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We take x to edge 12 in the top
and y to edge 13 in the top.

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And connect xy in the top.

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The line of -- this point
of intersection of xy

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with AB would be
the piercing point.

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And as a last step you need
to show correct visibility

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and I'm not going to go over
that because we've looked

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at that in the previous
lecture --

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how to show visibility of a line

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after finding its
piercing point.

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Second topic we're going to talk

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about is finding the
intersection of two planes.

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And this can be done
in two ways.

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The first way is by finding the
edge view of one of the planes,

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in which case you're going
to need an auxiliary view.

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The other method is by simply
locating two piercing points

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that can be found from line of
intersection of a line from one

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of the two planes
into the other plane.

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So what this means is you can
use the cutting plane method.

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Let's say that your two
planes are 123 and ABC.

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So we have 6 possible
combinations.

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We can test line AB versus plane
123, line BC versus plane 123,

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line AC versus plane 123.

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And then, if we didn't
find two piercing points,

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we can depress it and look at
12 as a line versus plane ABC,

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23 versus ABC, and
13 versus ABC.

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So basically 6 combinations
and you --

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you will eventually
find 2 piercing points

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and simply connect those 2
piercing points will be the line

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of intersection.

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The advantage, of course,
is it's trial and error.

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But the advantage
is you don't need

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to construct an auxiliary view.

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And after finding the
line of intersection

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through the 2 piercing
points, you always have

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to show correct visibility.

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The third topic in this lecture
is finding true size of planes.

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Now the way to see the true size
of a plane is by taking a line

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of sight that is perpendicular

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to the edge view of
the plane itself.

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But knowing that the reference
plane is always perpendicular

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to the line of sight and if the
line of sight is perpendicular

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to the edge view, what we need
to have now is a reference plane

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that is parallel
to the edge view.

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So line of sight
perpendicular to the edge view.

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Line of sight is
also perpendicular

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to the reference plane.

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Therefore, the edge view and
the reference planes will have

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to be parallel to each other.

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The resulting view will show the
true length of all of the edges

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of the plane as well as the
2 angles between the edges.

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So the steps are -- so
our ultimate objective is

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to have a reference
plane that is parallel

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to the edge view of the plan.

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So the first step is find
the edge of the plane

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by finding the point view
of any line in the plane.

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You should start with horizontal
line becomes true length

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in the top.

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And then we take a
reference plane perpendicular

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to the 2 lines in the top.

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That will be the
edge of the plane.

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Once we have the
edge of the plane,

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we take a reference plane
parallel to that edge view.

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And the resulting view would
be true size of the plane.

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Find the angle between lines.

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The angle between 2 lines
if you are given 2 lines.

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One, the view that was
show the true angle

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between 2 lines is one

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that shows the true
length of both lines.

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And one way of doing that is

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by creating a plane
containing the 2 lines

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and then finding the
true size of that plane.

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The steps are create a plane

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by connecting the
endpoints of the lines.

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Then find the edge view of
the plane, as we always do.

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Then find the true
size of the plane

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by taking a reference plane that
is parallel to that edge view

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as we talked about
on the previous page.

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And that true size of the
plane will also find the angle

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between the lines -- show
the angle between the lines.

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Here's an example.

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We are trying to find the angle
between 2 lines, PQ and PR,

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by using -- or by first finding
the true size of the plane.

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So we are given the front
and top view of PQ --

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this is PQ -- and
PR -- this is PR.

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This is the top and
this is the front.

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We're trying to find
the true angle

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between those lines
so the angle QPR.

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Of course, we cannot simply
measure the angles on the lines

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that we're seeing because
we're not seeing the true size

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of the plane.

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We're not seeing the true
length of both lines.

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So what we do is -- over here --

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we create a plane by
connecting the ends Q and R.

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And then once we
have the plane PQR,

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we try to find the
edge of that plane.

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And we have to find
edge view is.

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Find the horizontal line.

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Create a horizontal line
passing through point P wherever

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that intersects QR
will be up here.

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And this will be true length.

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Once we find this true
length here, in the top view,

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you take a reference
plane that's perpendicular

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to that true line, that
reference plane, of course,

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is H on this side and
a RP1 on that side.

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Then you project, you
see construction lines

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from the point's PQR, from
the top view perpendicular

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to the reference plane.

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And then we transfer the height
of Q here along its projection.

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Height of P along its
projection in line

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with R. Along its projection.

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If we did everything
correctly, P, Q, and R,

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should line up along
a straight line.

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And that's the edge view.

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This is the edge view of PQR.

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And then, once we
have the edge view,

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we take a second auxiliary
reference plane -- here --

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this orange one here -- which
will be ARP1 on this side.

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Take note that this whole view
here is auxiliary view number 1

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corresponding to ARP1.

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And the other side is ARP2.

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To find the location of
P, Q, and R for ARP2,

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or for the second auxiliary
view, we project P, Q,

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and R perpendicular to
the reference plane.

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And then we transfer
the distance --

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so this is our ARP1 -- this
whole thing is our ARP1.

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The distance of Q away from
ARP1, this distance here.

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That's viewed -- seen
-- in the top view.

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It's the same as the distance
of Q away from this ARP1.

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So this distance from this
ARP1 in this view must be equal

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to distance from this
ARP1 from this view.

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So we borrowed this
distance here for Q here.

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This is the distance of P --

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this tiny little
distance here --

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is the distance of P. And
this is the distance of R --

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all the distances are
measured away from ARP1 here

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on the left side and away from
ARP1 here into the top view.

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And then we just simply
connect Q, P, and R. Take note

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that PQ is parallel to
this reference plane.

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Therefore, this is
true length here.

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PR is also parallel to
this reference plane.

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So PR is also true length,

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which means that we're seeing
the true size of the plane and,

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therefore, this angle here,
we measure it a protractor,

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is the true angle
of the two lines.

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Finally, last topic is
finding the shortest distance

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from a point to a line.

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So we're going to assume
that we're given a line --

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maybe 2 views, top
view and front view --

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and then a point
shown in those 2 vies.

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One method is what's
called a line method.

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You simply find a
point view of the line

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and then the shortest
distance from the point

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to the line is true length

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in the view showing the
point view of the line.

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So the shortest distance
would be perpendicular

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to the line itself.

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The plane method is one in which
we create a plane as defined

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by the line and the point.

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And then we simply
find the true size

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of the plane containing
the line and the point.

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And that -- from that view,
that shows us 3 sides.

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All we have to do
is to create a line

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from the given point
perpendicular to the line.

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So basically find the
true size of the plane

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that contains the
line and the point

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and then make a perpendicular
line

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from the point into the line.