WEBVTT

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>> This is the lecture
for Week 2.

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We're going to talk about
engineering geometry,

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which is very important
in graphics.

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It involves geometric
elements and forms used

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in engineering design.

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In order to be able to represent
things accurately in graphics,

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we need to be able to
describe locations.

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Shapes. Size.

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And operation of
parts and products.

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Very important in
graphics obviously in order

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to clearly communicate our
ideas and design intent.

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Let's start with the basic, the
coordinate system, the 2-D space

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that you learned in Algebra 1.

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The Cartesian coordinate system.

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And also the polar coordinate
system that you learned in trig,

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if you have taken trig.

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Okay, so in two-dimensional
space the location of a point

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such as this point here,
okay, can be expressed

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by you saying the
Cartesian coordinate system,

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which consists of two mutually
perpendicular lines, okay.

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Horizontal is X. Vertical is
Y. To the right is positive X,

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upwards positive Y. Okay.

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And intersection at
coordinate [zero,

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zero] that's called the origin.

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And anywhere here
with positive X

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and Y coordinates is
called quadrant one.

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Quadrant two is when X is
negative and Y is positive.

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When both are negative,
it's quadrant three.

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And this is quadrant four.

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So to locate the position of
a point such as this guy here,

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we specify the X coordinate.

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In this case it's 3.

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And the Y coordinate
in this case is 5.

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So to locate the
point, you measure 3,

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which is positive, to the right.

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And 5 upwards, okay.

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On the other hand, in the
polar coordinate system,

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here is still your
X and your Y, okay.

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But to locate the point,
say this point here.

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Instead of giving the
X and Y coordinates,

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we give one distance "r"
and the angle theta measured

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from the positive X. So
the polar coordinates

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of this point are distance
"r" and angle theta.

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Theta is measured
clockwise, counterclockwise

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from the positive X. A positive
angle is counterclockwise.

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Extending it to the
3-D coordinate space,

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we have also the three
Cartesian coordinate system.

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Instead of having just
X and Y, we have X, Y,

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Z. And they still intersect

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at that single point
called the origin.

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And the three axes are
truly perpendicular.

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And the orientations are
defined by the right-hand rule.

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I will illustrate that later.

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And the location of points
can be specified by using

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or creating rectangular prisms,
as I will illustrate later.

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In addition to Cartesian
coordinate system,

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3-D also has the cylindrical
coordinate system, okay.

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Which is the 3-D counterpart

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of polar wherein you
specify one distance,

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the radius of the cylinder.

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And another, and angle
in the X-Y plane.

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And another distance
in the Z direction.

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So Cartesian coordinates
here, three distances.

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X, Y, Z. Cylindrical,
you have two distances.

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R and Z. And one angle.

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On the other hand, the spherical
coordinate system is one

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in which you have only one
distance, okay, and two angles.

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So three distances
here for Cartesian.

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Two distances and one
angle for cylindrical.

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And two angles and one
distance for spherical.

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We also need to distinguish

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between absolute
coordinate system.

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Absolute coordinates are
always measured relative

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to your origin zero or zero.

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Or zero, zero, zero in 3-D.

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Whereas, relative coordinates
are coordinates measured

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to some other point,
temporary point.

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May be the previous one
rather than the origin.

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World coordinate system is
attached to the world, okay,

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as opposed to a local coordinate
system, which you can move

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to ease your construction
of your drawing.

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A local coordinate
system, for instance,

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is also called user-coordinate
system.

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It's defined by the user.

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It's orientation and
location can be defined

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at any point in time.

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Here's an illustration, the
Cartesian coordinate system.

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Here is the positive
X horizontal.

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Positive Y vertical.

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Positive Z is also horizontal.

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The three are mutually
perpendicular.

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90 degrees, 90 degrees.

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And the location of
any points, which,

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and axis, this point here.

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X is zero.

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Y is 4. And Z is zero.

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Now, the location of any point
in 3-D space is obtained by,

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established by creating
a rectangular parallel.

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In fact, like our prism, okay.

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So, for instance,
here this point here

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with coordinates X is 4.

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Y is 3. And Z is 2 is
obtained as follows.

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You start from the
origin, and then you move

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in the X direction by four.

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Because the X coordinate's 4.

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So we need to move
one, two, three, four.

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And then the Y coordinate's 3.

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So we move upwards, the
positive Y direction, by three.

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One, two, three.

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Takes us to this point [4,3].

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Still, Z is zero.

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And then to account for
a Z coordinate of 2,

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we move in the positive
Z direction of one, two.

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Taking it to this corner
of this rectangular prism.

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Here's an illustration of
the right-hand rule, okay.

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To verify whether a given
coordinate system is

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right-handed, you should
be able to align your thumb

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to the positive X. Your
index to the positive Y.

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And your middle finger to the
positive Z. If you can do that,

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then it's a right-handed
coordinate.

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Of course, using
your right hand.

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Another way, by using
your right hand.

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It's a right-handed coordinate
system if you align your thumb

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in the positive Z direction.

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And you're able to curl your
fingers from the X to the Y.

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So positive Z for your
thumb, you should be able

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to curl your fingers from X to
Y. Or, if you align your thumb

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in the positive X, you should
be able to curl from Y to Z.

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On the other hand, if
you align your finger,

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your thumb in the positive
Y, the curling will be from Z

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to X. Cylindrical coordinate
system, okay, as I said,

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gives you two distances.

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In this case, 4.5 and 7.

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And one angle.

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In this case, 60 degrees, okay.

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Now, focusing on this
circle here that is

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in the X-Y plane, okay.

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It's this point here,
okay, with 4.5 and 60.

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It's just like giving you
the polar coordinates.

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One distance, the
radius of the circle.

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And one angle from the
positive X axis, 60.

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Then we turn the polar
coordinates into 3-D

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by simply giving
you a Z component

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or a Z distance parallel
to the Z axis.

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So two distances and one angle.

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As opposed to spherical
coordinate system,

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you only have one distance.

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The radius of the sphere.

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And two angles, okay.

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One angle.

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In this case, 60 is
measured from the X-Y plane

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to the vertical plane containing
your point right here.

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And then the third coordinate
is another angle, 20 degrees.

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Which is the angle
from the X axis

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to where you are
in the vertical.

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So this is commonly
called also the latitude.

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Longitude.

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And this is the radius
of your globe.

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You can look at this as
the Earth or the globe.

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And just to summarize, one
distance and two angles.

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So Cartesian has
three distances.

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X, Y, Z. Cylindrical
has two distances.

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Radius of the cylinder
and the Z. And one angle.

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Whereas, spherical
has two angles.

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Latitude, longitude.

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And the radius, only
one distance.

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Basic geometric elements
that we need

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to understand before we
do further construction.

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We all know what a point is.

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Specified in 3D space
by its location or coordinates.

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Line is the shortest
distance between two points.

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And we understand lines that are
parallel, they never intersect.

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They are the same direction
or slope orientation.

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Lines that are perpendicular
make an angle 90 degrees

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relative to each other.

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Intersecting lines are lines
that have a common point.

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Tangent line.

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A line is tangent to a circle

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if it touches the
circle at only one point.

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Curved lines, okay,
single versus double.

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A single curve is one in which
the entire curve can be drawn

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on two-dimensional
plane of your paper.

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And then a double curve is one

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that cannot be drawn
in the paper.

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An example of a double
curve would be a helix.

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Because, as it curves, it's also
moving up in the Z direction.

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Circle is a set of points
of the same distance

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from a given point
called the center.

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Some elements of the circle.

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The radius is the distance
from the center to any point

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in the circle, on the circle.

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Diameter is twice the circle.

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Circumference is 2 pi
R or diameter times pi.

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This is called a chord, okay.

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This is an arc.

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Semicircle is 1/2 of a circle.

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Quadrant is 1/4.

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Here's a sector,
a slice of pizza.

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Here's a segment, a
piece of the pizza.

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Tangent touches the
circle at only one point.

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A tangent line is always going

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to be perpendicular
to a radial line.

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Secant line, on the other hand,

00:10:48.696 --> 00:10:50.606 A:middle
intersects the circle
at two points.

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When two circles have the
same circle, the same center,

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they are called concentric.

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If they have different center,
they are called excentric.

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A circle circumscribes
a polygon if it's,

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if the polygon is inside.

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So it's outside, the
circle's outside the polygon.

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Inside, if the circle is
inside the polygon is called

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inscribed circle.

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More geometric elements.

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These conic sections you learned
in algebra, algebra two maybe.

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Formed by an intersection
of a plane

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with a right circular cone.

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And you understand how these
are formed with a parabola.

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We have the hyperbola.

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Ellipse. And, of course,

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a circle is just a
special type of ellipse.

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Okay, here's an illustration

00:11:37.106 --> 00:11:39.136 A:middle
of why they're called
conic sections.

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Here's your cone, okay.

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It's a right circular cone

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because the axis is
perfectly vertical, okay.

00:11:48.906 --> 00:11:50.156 A:middle
And we have a cutting plane.

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Here's our cutting plane.

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Depending on the orientation of
the cutting plane, in this case,

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the intersection of
the cutting plane

00:11:58.336 --> 00:12:02.756 A:middle
with a cone is an ellipse, okay.

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Further illustration here.

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It's the same cone.

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The cone actually
has two sides, okay.

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And here's the equation
of the cone.

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I actually plotted this

00:12:12.086 --> 00:12:15.456 A:middle
in 3-D space using a program
called graphing calculator.

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Now, if the orientation of the
cutting plane is as shown here,

00:12:19.356 --> 00:12:22.256 A:middle
okay, this cutting plane
intersects the upper part

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of the cone along
the curve, okay.

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And the lower part of the
cone along another curve.

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So there's actually two
parts of the curve, okay,

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corresponding to
here is one part.

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And here is the other part.

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And this gets us what we
call the hyperbola, okay.

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If I rotate the cutting
plane now,

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such as the cutting
plane is parallel

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to now this inclined
surface of the cone, okay.

00:12:48.016 --> 00:12:51.996 A:middle
What you see now is that
the cutting plane only cuts

00:12:51.996 --> 00:12:54.476 A:middle
or intersects the cone
along a single curve.

00:12:54.876 --> 00:12:58.796 A:middle
And that curve, single
curve here is your parabola.

00:12:58.916 --> 00:13:02.806 A:middle
So hyperbola here
becomes a parabola.

00:13:02.806 --> 00:13:07.806 A:middle
If I rotate it even further,
it intersects along this curve

00:13:07.806 --> 00:13:10.856 A:middle
that is formed that
is now an ellipse.

00:13:11.496 --> 00:13:14.646 A:middle
If I rotate it further so
that the cutting plane is now

00:13:14.756 --> 00:13:17.986 A:middle
perpendicular to the
axis of the cone, okay,

00:13:18.266 --> 00:13:21.276 A:middle
the curve of intersection
is now a circle.

00:13:21.276 --> 00:13:24.246 A:middle
So just by changing
the orientation

00:13:25.366 --> 00:13:29.726 A:middle
of the cutting plane you can
go from hyperbola to a parabola

00:13:30.246 --> 00:13:32.096 A:middle
to an ellipse to a circle.

00:13:32.466 --> 00:13:34.716 A:middle
Those are called conic sections.

00:13:35.906 --> 00:13:38.006 A:middle
Some quadrilaterals.

00:13:38.176 --> 00:13:41.806 A:middle
Quadrilaterals are
polygons of four sides.

00:13:41.876 --> 00:13:44.556 A:middle
You, of course, know the
square and the rectangle.

00:13:44.736 --> 00:13:45.746 A:middle
The rhombus is also.

00:13:46.266 --> 00:13:46.986 A:middle
Parallelogram.

00:13:46.986 --> 00:13:48.846 A:middle
The trapezoid and so on.

00:13:48.846 --> 00:13:50.776 A:middle
Polygons are named
according to the number

00:13:50.776 --> 00:13:52.366 A:middle
of sides or corners they have.

00:13:52.756 --> 00:13:53.866 A:middle
Triangle. Square.

00:13:53.956 --> 00:13:55.006 A:middle
Penta is five.

00:13:55.416 --> 00:13:56.196 A:middle
Hex is six.

00:13:56.636 --> 00:13:59.206 A:middle
And so on and so forth.

00:14:00.026 --> 00:14:00.806 A:middle
Polyhedra.

00:14:01.076 --> 00:14:05.376 A:middle
Polyhedra are named according to
the number of faces they have.

00:14:05.376 --> 00:14:06.706 A:middle
This is called tetrahedron.

00:14:06.706 --> 00:14:07.746 A:middle
Tetra is four.

00:14:08.146 --> 00:14:10.106 A:middle
So you have four
sides, four triangles.

00:14:10.106 --> 00:14:11.436 A:middle
One, two, three, four.

00:14:11.496 --> 00:14:13.396 A:middle
So there's four faces not sides.

00:14:14.136 --> 00:14:15.306 A:middle
This is hexahedron.

00:14:15.436 --> 00:14:16.396 A:middle
Hex means six.

00:14:16.396 --> 00:14:19.396 A:middle
Because the cube
actually has six faces.

00:14:19.396 --> 00:14:20.216 A:middle
Front. Back.

00:14:20.376 --> 00:14:20.776 A:middle
Right. Left.

00:14:20.776 --> 00:14:22.376 A:middle
Top and bottom.

00:14:23.326 --> 00:14:29.366 A:middle
Here's an octahedron
because it has eight faces.

00:14:30.086 --> 00:14:34.446 A:middle
Prism, on the other
hand, is a 3-D object generated

00:14:34.646 --> 00:14:39.206 A:middle
by taking a cross-sectional
area and sweeping

00:14:40.156 --> 00:14:43.006 A:middle
or giving it a height, okay.

00:14:43.176 --> 00:14:45.746 A:middle
It's called a right
prism if the direction

00:14:45.746 --> 00:14:52.506 A:middle
of sweep is perpendicular to the
original cross-sectional area.

00:14:52.506 --> 00:14:55.386 A:middle
It's oblique if it's
not, if the direction

00:14:55.386 --> 00:14:56.766 A:middle
of sweep is not perpendicular

00:14:57.436 --> 00:15:00.646 A:middle
to the original cross-sectional
area.

00:15:00.766 --> 00:15:02.536 A:middle
Pyramid, okay.

00:15:02.756 --> 00:15:05.226 A:middle
Instead of sweeping the
cross-sectional area,

00:15:05.226 --> 00:15:08.726 A:middle
you actually draw this line
from a central point here

00:15:09.266 --> 00:15:11.016 A:middle
into the cross-sectional area.

00:15:11.236 --> 00:15:15.406 A:middle
It's right pyramid if
this axis is perpendicular

00:15:15.406 --> 00:15:17.616 A:middle
to the original cross-sectional
area.

00:15:18.236 --> 00:15:21.446 A:middle
If it's along, if it's
not perpendicular,

00:15:21.446 --> 00:15:23.436 A:middle
the axis is not perpendicular,
it's called oblique.

00:15:24.286 --> 00:15:26.876 A:middle
And, if you cut off part
of, it's called truncated.

00:15:28.486 --> 00:15:32.276 A:middle
Very important, this class,
one of the main objectives,

00:15:32.796 --> 00:15:36.186 A:middle
goals of this class is to help
develop your visualization

00:15:36.186 --> 00:15:37.506 A:middle
skills, okay.

00:15:37.506 --> 00:15:40.556 A:middle
And visualization is an
interactive dynamic process

00:15:40.956 --> 00:15:44.816 A:middle
that involves the mind, eyes
and some physical stimulus,

00:15:44.936 --> 00:15:48.886 A:middle
either the object you're looking
at or the image you're drawing.

00:15:49.136 --> 00:15:51.856 A:middle
This is similar to the
sketching that we talked about.

00:15:52.386 --> 00:15:53.826 A:middle
There's an interactive process

00:15:53.826 --> 00:15:58.816 A:middle
between representing
with your hand.

00:15:59.626 --> 00:16:05.876 A:middle
Imaging when looking, and
imaging using your mind.

00:16:06.086 --> 00:16:07.896 A:middle
So eye-mind coordination.

00:16:07.986 --> 00:16:11.766 A:middle
Seeing, imaging and
representing.

00:16:12.896 --> 00:16:17.246 A:middle
Some more terminologies

00:16:17.246 --> 00:16:18.906 A:middle
so we will understand
each other better

00:16:18.906 --> 00:16:20.146 A:middle
as we talk about graphics.

00:16:20.246 --> 00:16:24.436 A:middle
Edges. Intersections between
two faces of an object.

00:16:24.436 --> 00:16:27.476 A:middle
Faces. Areas of uniform
gradual changing,

00:16:27.786 --> 00:16:31.986 A:middle
gradually changing lightness
and are always bounded by edges.

00:16:31.986 --> 00:16:36.406 A:middle
Limiting element, okay, is
for curved surfaces, okay.

00:16:36.906 --> 00:16:39.096 A:middle
And they represent the
farthest outside feature.

00:16:39.096 --> 00:16:41.386 A:middle
I will illustrate that
in the next slide.

00:16:41.666 --> 00:16:45.806 A:middle
Vertex is intersection
of two or more edges.

00:16:45.806 --> 00:16:47.286 A:middle
Here's an illustration.

00:16:47.286 --> 00:16:50.476 A:middle
Here is a vertex where
these three edges --

00:16:50.476 --> 00:16:52.566 A:middle
one, two, three -- intersect.

00:16:53.136 --> 00:16:55.746 A:middle
Here's an edge because
it's an intersection

00:16:55.786 --> 00:16:58.096 A:middle
between these two faces, okay.

00:16:58.496 --> 00:16:59.806 A:middle
Here's also an edge here.

00:16:59.806 --> 00:17:01.376 A:middle
This circle here
represents an edge

00:17:01.376 --> 00:17:04.706 A:middle
because it's an intersection
between this right face

00:17:05.096 --> 00:17:07.476 A:middle
and the cylindrical surface.

00:17:07.476 --> 00:17:11.046 A:middle
These two lines here, one
over here and one over here,

00:17:11.226 --> 00:17:12.496 A:middle
are called the limiting
elements.

00:17:12.496 --> 00:17:18.856 A:middle
These lines here do not actually
exist in the physical world.

00:17:18.856 --> 00:17:22.496 A:middle
If I rotate the cylinder you
would not see a line here.

00:17:22.496 --> 00:17:26.316 A:middle
The, what this line's
representing are the farthermost

00:17:26.316 --> 00:17:30.426 A:middle
farthest extent of
the curved surface.

00:17:31.416 --> 00:17:34.706 A:middle
Those are called
limiting elements.

00:17:34.706 --> 00:17:36.186 A:middle
In order to help you visualize,

00:17:36.346 --> 00:17:38.856 A:middle
develop your visualization
skills,

00:17:38.856 --> 00:17:41.396 A:middle
there's some techniques
you can use.

00:17:41.836 --> 00:17:44.706 A:middle
By looking at objects,
complicated objects,

00:17:44.706 --> 00:17:47.306 A:middle
that's nothing but
combinations of simpler objects.

00:17:48.626 --> 00:17:51.346 A:middle
By looking at cutting
planes, similar to what we did

00:17:51.346 --> 00:17:52.416 A:middle
for the conic sections.

00:17:52.826 --> 00:17:56.456 A:middle
And changing the orientation,
it's called normal plane

00:17:56.456 --> 00:17:59.246 A:middle
if it's parallel to one of
the principal directions.

00:18:00.586 --> 00:18:02.866 A:middle
You can rotate it
about a single axis

00:18:02.866 --> 00:18:03.936 A:middle
to give inclined plane.

00:18:03.966 --> 00:18:07.526 A:middle
Or rotate it again gives you
an oblique cutting plane.

00:18:07.956 --> 00:18:10.556 A:middle
You can also look for
planes of symmetry.

00:18:11.446 --> 00:18:13.706 A:middle
And also look at what's
called a development.

00:18:13.806 --> 00:18:20.676 A:middle
Meaning try to slice the skin
of the object very, very thin.

00:18:20.676 --> 00:18:21.916 A:middle
And then try to flatten it

00:18:21.956 --> 00:18:24.326 A:middle
on two-dimensional
plane of the paper.

00:18:24.826 --> 00:18:27.576 A:middle
Here's an illustration
of visualization

00:18:27.576 --> 00:18:29.216 A:middle
by combining solid objects.

00:18:29.246 --> 00:18:32.736 A:middle
So this object here, you
can look at as a combination

00:18:32.736 --> 00:18:36.306 A:middle
of a box and a tiny
little cylinder, okay.

00:18:36.306 --> 00:18:38.916 A:middle
And, if you do this combination
in your visualization,

00:18:38.916 --> 00:18:43.906 A:middle
it's very important to keep
in mind the relative location

00:18:43.906 --> 00:18:45.446 A:middle
and orientation of
the two boxes.

00:18:46.056 --> 00:18:49.666 A:middle
So these two are correct
representations of this.

00:18:50.596 --> 00:18:54.376 A:middle
Because the location of
the cylinder relative

00:18:54.376 --> 00:18:57.726 A:middle
to box are correct,
orientation location.

00:18:57.726 --> 00:19:01.756 A:middle
This is not correct because the
orientation location relative

00:19:01.816 --> 00:19:05.686 A:middle
to each other is not correct.

00:19:06.266 --> 00:19:11.206 A:middle
Here's another way of combining
the two basic solids, the box

00:19:11.206 --> 00:19:16.486 A:middle
and the cylinder, by
removing or subtracting.

00:19:16.486 --> 00:19:21.376 A:middle
If you look at this object
here, these three objects here.

00:19:21.376 --> 00:19:22.676 A:middle
At first glance you might think

00:19:22.676 --> 00:19:24.536 A:middle
that they are very,
very different, okay.

00:19:24.856 --> 00:19:29.166 A:middle
But one way to visualize them
is that they're fairly similar.

00:19:29.816 --> 00:19:32.706 A:middle
That involves the
subtraction or removing

00:19:33.056 --> 00:19:36.096 A:middle
of a smaller box
from the bigger box.

00:19:36.336 --> 00:19:38.606 A:middle
The only difference
is that, okay,

00:19:38.606 --> 00:19:41.916 A:middle
for the first one you have
a tiny little skinny box.

00:19:42.366 --> 00:19:44.586 A:middle
And then for the second
one a slightly bigger box.

00:19:44.586 --> 00:19:47.166 A:middle
And for the third one
an even bigger box.

00:19:47.556 --> 00:19:50.506 A:middle
So you can look at it,
these three as being similar

00:19:50.506 --> 00:19:52.186 A:middle
in the way that they
are being constructed

00:19:52.186 --> 00:19:55.156 A:middle
by removing one solid
from another.

00:19:55.516 --> 00:19:56.266 A:middle
Same thing here.

00:19:56.726 --> 00:19:59.096 A:middle
These three objects
here can be viewed

00:19:59.096 --> 00:20:03.456 A:middle
as removing these progressively
getting larger wedges

00:20:03.456 --> 00:20:06.506 A:middle
from the original box.

00:20:07.216 --> 00:20:07.836 A:middle
Same thing here.

00:20:08.886 --> 00:20:12.086 A:middle
These three objects
here are obtained

00:20:12.086 --> 00:20:16.406 A:middle
by removing a pyramid
from the box, okay.

00:20:17.036 --> 00:20:20.066 A:middle
It's just a matter of
increasing the size

00:20:20.066 --> 00:20:23.376 A:middle
of the pyramid that
you're removing.

00:20:23.616 --> 00:20:27.106 A:middle
Okay, another illustration
of combining simpler objects

00:20:27.106 --> 00:20:33.106 A:middle
into a more complex object is
looking at this compound object

00:20:33.106 --> 00:20:37.376 A:middle
as the combination of a big
box, about this size here.

00:20:39.426 --> 00:20:45.006 A:middle
And then from that big box you
subtract a cube, the pink one.

00:20:45.816 --> 00:20:48.706 A:middle
And then subtract this
tiny little cylinder

00:20:48.706 --> 00:20:49.816 A:middle
to give you that hole.

00:20:49.816 --> 00:20:53.326 A:middle
And then subtract from the
corner this wedge here, okay.

00:20:53.786 --> 00:20:54.716 A:middle
Giving you this.

00:20:54.976 --> 00:20:59.036 A:middle
And then adding the
yellow rectangle to account

00:20:59.036 --> 00:21:02.596 A:middle
for the protruding part
of this compound object.

00:21:03.806 --> 00:21:07.236 A:middle
Okay, just like we did
for conic sections.

00:21:07.236 --> 00:21:11.466 A:middle
One way to visualize what the
object looks like in 3-D is

00:21:11.466 --> 00:21:13.326 A:middle
by looking at cutting planes.

00:21:14.396 --> 00:21:17.626 A:middle
A cutting plane is said to be
normal if it's parallel to one

00:21:17.626 --> 00:21:19.766 A:middle
of the three principal planes.

00:21:19.986 --> 00:21:22.726 A:middle
So look at this rectangular box.

00:21:22.806 --> 00:21:26.926 A:middle
The three principal planes
are front, frontal plane.

00:21:27.576 --> 00:21:28.916 A:middle
Top, horizontal plane.

00:21:29.226 --> 00:21:31.716 A:middle
And right, profile plane, okay.

00:21:32.216 --> 00:21:34.956 A:middle
A cutting plane that
is parallel to one

00:21:34.956 --> 00:21:37.946 A:middle
of those three principal planes
is called a normal plane.

00:21:37.946 --> 00:21:41.226 A:middle
This case it's normal because
the cutting plane is parallel

00:21:41.226 --> 00:21:43.656 A:middle
to the right profile
plane, okay.

00:21:43.656 --> 00:21:48.366 A:middle
If I rotate that, say,
relative to the X axis

00:21:48.366 --> 00:21:49.966 A:middle
and the Z axis here, okay.

00:21:49.966 --> 00:21:52.916 A:middle
So it was originally parallel
to the right profile plane.

00:21:52.916 --> 00:21:57.436 A:middle
I rotate it slightly
relative to the Z axis.

00:21:57.746 --> 00:22:03.506 A:middle
The plane becomes an
incline plane, okay.

00:22:03.506 --> 00:22:08.026 A:middle
So you can use, vary, you
can vary the orientation

00:22:08.026 --> 00:22:12.746 A:middle
of your incline cutting plane
and visualize the curves

00:22:12.746 --> 00:22:15.906 A:middle
of intersection like we did
for the conic sections in order

00:22:15.906 --> 00:22:18.546 A:middle
to develop your visualization
skills.

00:22:18.726 --> 00:22:20.706 A:middle
If I rotated the plane again,

00:22:21.496 --> 00:22:27.996 A:middle
it becomes about another
axis, about Y axis here.

00:22:28.316 --> 00:22:30.236 A:middle
It becomes an oblique
plane, okay.

00:22:31.096 --> 00:22:34.576 A:middle
So an oblique plane is
not perpendicular to any

00:22:34.646 --> 00:22:35.976 A:middle
of the three principal plains.

00:22:36.106 --> 00:22:38.506 A:middle
Here the incline plane, here,

00:22:38.726 --> 00:22:41.426 A:middle
it's still perpendicular
to one of the planes.

00:22:41.426 --> 00:22:43.526 A:middle
It's still perpendicular
to the frontal plane.

00:22:44.416 --> 00:22:49.306 A:middle
So just by changing the
angle of rotation, okay,

00:22:49.306 --> 00:22:52.896 A:middle
and trying to visualize what
the intersection would be can

00:22:52.896 --> 00:22:54.576 A:middle
develop your visualization.

00:22:54.816 --> 00:23:01.166 A:middle
Here it, from a circle, when
the plane is horizontal,

00:23:01.166 --> 00:23:04.616 A:middle
to an ellipse to an even
bigger ellipse of higher eccentricity.

00:23:06.326 --> 00:23:11.586 A:middle
Another way of developing
your visualization skills is

00:23:11.656 --> 00:23:13.216 A:middle
by looking for planes
of symmetry.

00:23:13.216 --> 00:23:16.136 A:middle
So, for instance, this
cylinder here, okay,

00:23:16.136 --> 00:23:18.806 A:middle
is perfectly symmetrical
with respect

00:23:18.806 --> 00:23:21.926 A:middle
to this vertical plane here
that passes through its center.

00:23:22.036 --> 00:23:24.606 A:middle
Because whatever you
see on the front,

00:23:24.606 --> 00:23:28.146 A:middle
in front of the plane is
exactly a mirror reflection

00:23:28.146 --> 00:23:29.536 A:middle
of what's behind it.

00:23:30.316 --> 00:23:36.876 A:middle
It's also, that same cylinder
also has this horizontal plane

00:23:36.876 --> 00:23:37.476 A:middle
of symmetry.

00:23:38.176 --> 00:23:41.576 A:middle
Because whatever you find
above it is a mirror image

00:23:41.576 --> 00:23:44.846 A:middle
of what's under it or below it.

00:23:45.546 --> 00:23:47.306 A:middle
Now, development.

00:23:47.486 --> 00:23:51.136 A:middle
So imagine moving your cutting
plane so that it's as close

00:23:51.136 --> 00:23:53.926 A:middle
as possible, your normal
cutting plane so it's as close

00:23:53.926 --> 00:23:56.106 A:middle
as possible to the surface.

00:23:56.106 --> 00:23:57.826 A:middle
So you're slicing only a very,

00:23:57.826 --> 00:24:03.036 A:middle
very thin skin of
your 3-D object.

00:24:03.036 --> 00:24:06.516 A:middle
You do that for the right
profile plane, the top

00:24:06.706 --> 00:24:08.606 A:middle
and the horizontal, okay.

00:24:08.656 --> 00:24:10.636 A:middle
So slice, slice,
slice very thinly.

00:24:10.636 --> 00:24:20.596 A:middle
And then imagine flattening
the skin of your 3-D object.

00:24:20.746 --> 00:24:22.276 A:middle
And that's what you
call development.

00:24:22.276 --> 00:24:25.686 A:middle
How you can get from 2-D
-- like this cardboard box,

00:24:25.686 --> 00:24:28.726 A:middle
it's flat initially
-- into a 3-D object.

00:24:28.856 --> 00:24:32.116 A:middle
In one of your homework
exercises you'll actually be

00:24:32.116 --> 00:24:35.446 A:middle
doing this, cutting
paper in order

00:24:35.446 --> 00:24:41.376 A:middle
to generate 3-D boxes
and prisms, okay.

00:24:41.736 --> 00:24:45.866 A:middle
You can also try to
visualize the resulting image.

00:24:45.866 --> 00:24:50.706 A:middle
If you take a projection
plane or an image plane, okay,

00:24:50.766 --> 00:24:55.716 A:middle
and you project the features
you have onto this image plane.

00:24:55.776 --> 00:24:58.296 A:middle
So this image plane we
have is perfectly parallel

00:24:58.296 --> 00:25:02.246 A:middle
to the frontal plane of
the 3-D object, okay.

00:25:02.636 --> 00:25:05.346 A:middle
And the projection lines
are perfectly perpendicular

00:25:05.556 --> 00:25:06.716 A:middle
to the image.

00:25:07.396 --> 00:25:10.376 A:middle
As a result, okay,
as it says here,

00:25:10.556 --> 00:25:12.336 A:middle
we only capture 2 dimensions.

00:25:12.426 --> 00:25:15.036 A:middle
The height of the object
and the width of the object.

00:25:15.096 --> 00:25:17.046 A:middle
But you lose the depth.

00:25:17.156 --> 00:25:19.336 A:middle
Which makes sense
because an object

00:25:19.426 --> 00:25:21.746 A:middle
in 3-D space actually
has three dimensions.

00:25:22.496 --> 00:25:24.406 A:middle
The height the width
and the depth.

00:25:24.846 --> 00:25:27.966 A:middle
And you're trying to represent
an image in the plane,

00:25:28.476 --> 00:25:31.136 A:middle
only has two dimensions
in this case.

00:25:31.136 --> 00:25:36.526 A:middle
The dimensions that you keep are
the height and the width, okay.

00:25:36.666 --> 00:25:40.586 A:middle
So in our coordinate system,
Z corresponds to depth.

00:25:40.896 --> 00:25:43.156 A:middle
X can correspond to the width.

00:25:43.866 --> 00:25:45.256 A:middle
And Y to the height.

00:25:46.056 --> 00:25:51.536 A:middle
If you do that for the three
principal projection planes,

00:25:51.536 --> 00:25:53.516 A:middle
so I have one that's
parallel to the front.

00:25:53.836 --> 00:25:55.296 A:middle
One that's top, horizontal.

00:25:55.296 --> 00:25:56.646 A:middle
One that's right profile plane.

00:25:57.106 --> 00:25:57.966 A:middle
Do the same thing.

00:25:57.966 --> 00:26:01.206 A:middle
Project all the features
onto the frontal plane.

00:26:01.206 --> 00:26:02.456 A:middle
You'll get the front view.

00:26:02.906 --> 00:26:06.236 A:middle
Do the same thing for
the horizontal plane,

00:26:06.236 --> 00:26:07.376 A:middle
you get the top view.

00:26:07.506 --> 00:26:09.226 A:middle
This one here will get
the right side view.

00:26:10.066 --> 00:26:14.616 A:middle
It's like having a camera,
okay, and taking three pictures.

00:26:15.106 --> 00:26:18.776 A:middle
One wherein your camera
perfectly placed on,

00:26:18.996 --> 00:26:22.156 A:middle
perpendicular to
the frontal plane

00:26:22.156 --> 00:26:23.146 A:middle
to give you the front view.

00:26:23.446 --> 00:26:26.856 A:middle
Another one perfectly, your axis

00:26:26.856 --> 00:26:28.256 A:middle
of your camera is
perfectly vertical.

00:26:28.866 --> 00:26:31.576 A:middle
So it's, actually, it's
perpendicular to the horizontal.

00:26:32.036 --> 00:26:33.026 A:middle
Gives you the top view.

00:26:33.416 --> 00:26:35.916 A:middle
Here will give you
the right side view.

00:26:36.956 --> 00:26:42.466 A:middle
Now, when you have a normal
plane like this plane,

00:26:43.596 --> 00:26:45.416 A:middle
it's normal because
it's parallel

00:26:45.416 --> 00:26:47.816 A:middle
to the frontal plane, okay.

00:26:47.816 --> 00:26:51.046 A:middle
As you project it to the frontal
plane, it gives you an image

00:26:51.136 --> 00:26:53.586 A:middle
that is representing
the precise size

00:26:53.586 --> 00:26:56.526 A:middle
and shape of that normal plane.

00:26:57.216 --> 00:26:59.796 A:middle
However, the same normal plane,
when projected to the top,

00:27:00.186 --> 00:27:02.146 A:middle
will become just
an edge or a line.

00:27:02.646 --> 00:27:06.446 A:middle
And also the same for the
right side view, okay.

00:27:06.826 --> 00:27:12.406 A:middle
So to repeat, a given normal
plane, in this case parallel

00:27:12.406 --> 00:27:15.716 A:middle
to the front, becomes two
sides and shape in the front.

00:27:16.246 --> 00:27:19.986 A:middle
And becomes just edges
or lines in the top

00:27:20.206 --> 00:27:21.656 A:middle
and right profile plane.

00:27:22.576 --> 00:27:26.956 A:middle
On the other hand, an
incline plane like this, okay,

00:27:26.956 --> 00:27:29.866 A:middle
when you project it to the
top, it becomes foreshortened.

00:27:30.266 --> 00:27:32.416 A:middle
So looking at this plane here.

00:27:33.236 --> 00:27:35.896 A:middle
The projection at the
top is still a rectangle,

00:27:35.946 --> 00:27:37.066 A:middle
but it's foreshortened.

00:27:37.066 --> 00:27:40.296 A:middle
Meaning the size is smaller
than what it it's real size is.

00:27:40.296 --> 00:27:44.236 A:middle
If I project the right side
too, it's also foreshortened.

00:27:44.866 --> 00:27:48.556 A:middle
However, if I project
that incline plane

00:27:48.556 --> 00:27:51.836 A:middle
into the front view, since the
plane is actually perpendicular

00:27:51.836 --> 00:27:56.036 A:middle
to the frontal plane, it
would still appear as an edge.

00:27:56.136 --> 00:27:58.896 A:middle
So for an incline
plane it appears

00:27:58.896 --> 00:28:03.496 A:middle
as two foreshortened
planes in two of the views.

00:28:04.316 --> 00:28:09.576 A:middle
And an edge in one
of the views, okay.

00:28:09.686 --> 00:28:13.546 A:middle
So this incline plane here,
plane here foreshortened.

00:28:13.726 --> 00:28:14.986 A:middle
And edge here.

00:28:16.006 --> 00:28:20.406 A:middle
On the other hand, an
oblique face like this,

00:28:20.406 --> 00:28:23.666 A:middle
or an oblique plane, when
projected to any of the three,

00:28:24.476 --> 00:28:27.366 A:middle
okay, will
actually appear as planes.

00:28:27.976 --> 00:28:30.056 A:middle
But these planes, the one
seen on the views

00:28:30.056 --> 00:28:32.456 A:middle
and the front, top, right.