>> This is the lecture for Week 2.
We're going to talk about engineering geometry,
which is very important in graphics.
It involves geometric elements and forms used
in engineering design.
In order to be able to represent things accurately in graphics,
we need to be able to describe locations.
Shapes. Size.
And operation of parts and products.
Very important in graphics obviously in order
to clearly communicate our ideas and design intent.
Let's start with the basic, the coordinate system, the 2-D space
that you learned in Algebra 1.
The Cartesian coordinate system.
And also the polar coordinate system that you learned in trig,
if you have taken trig.
Okay, so in two-dimensional space the location of a point
such as this point here, okay, can be expressed
by you saying the Cartesian coordinate system,
which consists of two mutually perpendicular lines, okay.
Horizontal is X. Vertical is Y. To the right is positive X,
upwards positive Y. Okay.
And intersection at coordinate [zero,
zero] that's called the origin.
And anywhere here with positive X
and Y coordinates is called quadrant one.
Quadrant two is when X is negative and Y is positive.
When both are negative, it's quadrant three.
And this is quadrant four.
So to locate the position of a point such as this guy here,
we specify the X coordinate.
In this case it's 3.
And the Y coordinate in this case is 5.
So to locate the point, you measure 3,
which is positive, to the right.
And 5 upwards, okay.
On the other hand, in the polar coordinate system,
here is still your X and your Y, okay.
But to locate the point, say this point here.
Instead of giving the X and Y coordinates,
we give one distance "r" and the angle theta measured
from the positive X. So the polar coordinates
of this point are distance "r" and angle theta.
Theta is measured clockwise, counterclockwise
from the positive X. A positive angle is counterclockwise.
Extending it to the 3-D coordinate space,
we have also the three Cartesian coordinate system.
Instead of having just X and Y, we have X, Y,
Z. And they still intersect
at that single point called the origin.
And the three axes are truly perpendicular.
And the orientations are defined by the right-hand rule.
I will illustrate that later.
And the location of points can be specified by using
or creating rectangular prisms, as I will illustrate later.
In addition to Cartesian coordinate system,
3-D also has the cylindrical coordinate system, okay.
Which is the 3-D counterpart
of polar wherein you specify one distance,
the radius of the cylinder.
And another, and angle in the X-Y plane.
And another distance in the Z direction.
So Cartesian coordinates here, three distances.
X, Y, Z. Cylindrical, you have two distances.
R and Z. And one angle.
On the other hand, the spherical coordinate system is one
in which you have only one distance, okay, and two angles.
So three distances here for Cartesian.
Two distances and one angle for cylindrical.
And two angles and one distance for spherical.
We also need to distinguish
between absolute coordinate system.
Absolute coordinates are always measured relative
to your origin zero or zero.
Or zero, zero, zero in 3-D.
Whereas, relative coordinates are coordinates measured
to some other point, temporary point.
May be the previous one rather than the origin.
World coordinate system is attached to the world, okay,
as opposed to a local coordinate system, which you can move
to ease your construction of your drawing.
A local coordinate system, for instance,
is also called user-coordinate system.
It's defined by the user.
It's orientation and location can be defined
at any point in time.
Here's an illustration, the Cartesian coordinate system.
Here is the positive X horizontal.
Positive Y vertical.
Positive Z is also horizontal.
The three are mutually perpendicular.
90 degrees, 90 degrees.
And the location of any points, which,
and axis, this point here.
X is zero.
Y is 4. And Z is zero.
Now, the location of any point in 3-D space is obtained by,
established by creating a rectangular parallel.
In fact, like our prism, okay.
So, for instance, here this point here
with coordinates X is 4.
Y is 3. And Z is 2 is obtained as follows.
You start from the origin, and then you move
in the X direction by four.
Because the X coordinate's 4.
So we need to move one, two, three, four.
And then the Y coordinate's 3.
So we move upwards, the positive Y direction, by three.
One, two, three.
Takes us to this point [4,3].
Still, Z is zero.
And then to account for a Z coordinate of 2,
we move in the positive Z direction of one, two.
Taking it to this corner of this rectangular prism.
Here's an illustration of the right-hand rule, okay.
To verify whether a given coordinate system is
right-handed, you should be able to align your thumb
to the positive X. Your index to the positive Y.
And your middle finger to the positive Z. If you can do that,
then it's a right-handed coordinate.
Of course, using your right hand.
Another way, by using your right hand.
It's a right-handed coordinate system if you align your thumb
in the positive Z direction.
And you're able to curl your fingers from the X to the Y.
So positive Z for your thumb, you should be able
to curl your fingers from X to Y. Or, if you align your thumb
in the positive X, you should be able to curl from Y to Z.
On the other hand, if you align your finger,
your thumb in the positive Y, the curling will be from Z
to X. Cylindrical coordinate system, okay, as I said,
gives you two distances.
In this case, 4.5 and 7.
And one angle.
In this case, 60 degrees, okay.
Now, focusing on this circle here that is
in the X-Y plane, okay.
It's this point here, okay, with 4.5 and 60.
It's just like giving you the polar coordinates.
One distance, the radius of the circle.
And one angle from the positive X axis, 60.
Then we turn the polar coordinates into 3-D
by simply giving you a Z component
or a Z distance parallel to the Z axis.
So two distances and one angle.
As opposed to spherical coordinate system,
you only have one distance.
The radius of the sphere.
And two angles, okay.
One angle.
In this case, 60 is measured from the X-Y plane
to the vertical plane containing your point right here.
And then the third coordinate is another angle, 20 degrees.
Which is the angle from the X axis
to where you are in the vertical.
So this is commonly called also the latitude.
Longitude.
And this is the radius of your globe.
You can look at this as the Earth or the globe.
And just to summarize, one distance and two angles.
So Cartesian has three distances.
X, Y, Z. Cylindrical has two distances.
Radius of the cylinder and the Z. And one angle.
Whereas, spherical has two angles.
Latitude, longitude.
And the radius, only one distance.
Basic geometric elements that we need
to understand before we do further construction.
We all know what a point is.
Specified in 3D space by its location or coordinates.
Line is the shortest distance between two points.
And we understand lines that are parallel, they never intersect.
They are the same direction or slope orientation.
Lines that are perpendicular make an angle 90 degrees
relative to each other.
Intersecting lines are lines that have a common point.
Tangent line.
A line is tangent to a circle
if it touches the circle at only one point.
Curved lines, okay, single versus double.
A single curve is one in which the entire curve can be drawn
on two-dimensional plane of your paper.
And then a double curve is one
that cannot be drawn in the paper.
An example of a double curve would be a helix.
Because, as it curves, it's also moving up in the Z direction.
Circle is a set of points of the same distance
from a given point called the center.
Some elements of the circle.
The radius is the distance from the center to any point
in the circle, on the circle.
Diameter is twice the circle.
Circumference is 2 pi R or diameter times pi.
This is called a chord, okay.
This is an arc.
Semicircle is 1/2 of a circle.
Quadrant is 1/4.
Here's a sector, a slice of pizza.
Here's a segment, a piece of the pizza.
Tangent touches the circle at only one point.
A tangent line is always going
to be perpendicular to a radial line.
Secant line, on the other hand,
intersects the circle at two points.
When two circles have the same circle, the same center,
they are called concentric.
If they have different center, they are called excentric.
A circle circumscribes a polygon if it's,
if the polygon is inside.
So it's outside, the circle's outside the polygon.
Inside, if the circle is inside the polygon is called
inscribed circle.
More geometric elements.
These conic sections you learned in algebra, algebra two maybe.
Formed by an intersection of a plane
with a right circular cone.
And you understand how these are formed with a parabola.
We have the hyperbola.
Ellipse. And, of course,
a circle is just a special type of ellipse.
Okay, here's an illustration
of why they're called conic sections.
Here's your cone, okay.
It's a right circular cone
because the axis is perfectly vertical, okay.
And we have a cutting plane.
Here's our cutting plane.
Depending on the orientation of the cutting plane, in this case,
the intersection of the cutting plane
with a cone is an ellipse, okay.
Further illustration here.
It's the same cone.
The cone actually has two sides, okay.
And here's the equation of the cone.
I actually plotted this
in 3-D space using a program called graphing calculator.
Now, if the orientation of the cutting plane is as shown here,
okay, this cutting plane intersects the upper part
of the cone along the curve, okay.
And the lower part of the cone along another curve.
So there's actually two parts of the curve, okay,
corresponding to here is one part.
And here is the other part.
And this gets us what we call the hyperbola, okay.
If I rotate the cutting plane now,
such as the cutting plane is parallel
to now this inclined surface of the cone, okay.
What you see now is that the cutting plane only cuts
or intersects the cone along a single curve.
And that curve, single curve here is your parabola.
So hyperbola here becomes a parabola.
If I rotate it even further, it intersects along this curve
that is formed that is now an ellipse.
If I rotate it further so that the cutting plane is now
perpendicular to the axis of the cone, okay,
the curve of intersection is now a circle.
So just by changing the orientation
of the cutting plane you can go from hyperbola to a parabola
to an ellipse to a circle.
Those are called conic sections.
Some quadrilaterals.
Quadrilaterals are polygons of four sides.
You, of course, know the square and the rectangle.
The rhombus is also.
Parallelogram.
The trapezoid and so on.
Polygons are named according to the number
of sides or corners they have.
Triangle. Square.
Penta is five.
Hex is six.
And so on and so forth.
Polyhedra.
Polyhedra are named according to the number of faces they have.
This is called tetrahedron.
Tetra is four.
So you have four sides, four triangles.
One, two, three, four.
So there's four faces not sides.
This is hexahedron.
Hex means six.
Because the cube actually has six faces.
Front. Back.
Right. Left.
Top and bottom.
Here's an octahedron because it has eight faces.
Prism, on the other hand, is a 3-D object generated
by taking a cross-sectional area and sweeping
or giving it a height, okay.
It's called a right prism if the direction
of sweep is perpendicular to the original cross-sectional area.
It's oblique if it's not, if the direction
of sweep is not perpendicular
to the original cross-sectional area.
Pyramid, okay.
Instead of sweeping the cross-sectional area,
you actually draw this line from a central point here
into the cross-sectional area.
It's right pyramid if this axis is perpendicular
to the original cross-sectional area.
If it's along, if it's not perpendicular,
the axis is not perpendicular, it's called oblique.
And, if you cut off part of, it's called truncated.
Very important, this class, one of the main objectives,
goals of this class is to help develop your visualization
skills, okay.
And visualization is an interactive dynamic process
that involves the mind, eyes and some physical stimulus,
either the object you're looking at or the image you're drawing.
This is similar to the sketching that we talked about.
There's an interactive process
between representing with your hand.
Imaging when looking, and imaging using your mind.
So eye-mind coordination.
Seeing, imaging and representing.
Some more terminologies
so we will understand each other better
as we talk about graphics.
Edges. Intersections between two faces of an object.
Faces. Areas of uniform gradual changing,
gradually changing lightness and are always bounded by edges.
Limiting element, okay, is for curved surfaces, okay.
And they represent the farthest outside feature.
I will illustrate that in the next slide.
Vertex is intersection of two or more edges.
Here's an illustration.
Here is a vertex where these three edges --
one, two, three -- intersect.
Here's an edge because it's an intersection
between these two faces, okay.
Here's also an edge here.
This circle here represents an edge
because it's an intersection between this right face
and the cylindrical surface.
These two lines here, one over here and one over here,
are called the limiting elements.
These lines here do not actually exist in the physical world.
If I rotate the cylinder you would not see a line here.
The, what this line's representing are the farthermost
farthest extent of the curved surface.
Those are called limiting elements.
In order to help you visualize,
develop your visualization skills,
there's some techniques you can use.
By looking at objects, complicated objects,
that's nothing but combinations of simpler objects.
By looking at cutting planes, similar to what we did
for the conic sections.
And changing the orientation, it's called normal plane
if it's parallel to one of the principal directions.
You can rotate it about a single axis
to give inclined plane.
Or rotate it again gives you an oblique cutting plane.
You can also look for planes of symmetry.
And also look at what's called a development.
Meaning try to slice the skin of the object very, very thin.
And then try to flatten it
on two-dimensional plane of the paper.
Here's an illustration of visualization
by combining solid objects.
So this object here, you can look at as a combination
of a box and a tiny little cylinder, okay.
And, if you do this combination in your visualization,
it's very important to keep in mind the relative location
and orientation of the two boxes.
So these two are correct representations of this.
Because the location of the cylinder relative
to box are correct, orientation location.
This is not correct because the orientation location relative
to each other is not correct.
Here's another way of combining the two basic solids, the box
and the cylinder, by removing or subtracting.
If you look at this object here, these three objects here.
At first glance you might think
that they are very, very different, okay.
But one way to visualize them is that they're fairly similar.
That involves the subtraction or removing
of a smaller box from the bigger box.
The only difference is that, okay,
for the first one you have a tiny little skinny box.
And then for the second one a slightly bigger box.
And for the third one an even bigger box.
So you can look at it, these three as being similar
in the way that they are being constructed
by removing one solid from another.
Same thing here.
These three objects here can be viewed
as removing these progressively getting larger wedges
from the original box.
Same thing here.
These three objects here are obtained
by removing a pyramid from the box, okay.
It's just a matter of increasing the size
of the pyramid that you're removing.
Okay, another illustration of combining simpler objects
into a more complex object is looking at this compound object
as the combination of a big box, about this size here.
And then from that big box you subtract a cube, the pink one.
And then subtract this tiny little cylinder
to give you that hole.
And then subtract from the corner this wedge here, okay.
Giving you this.
And then adding the yellow rectangle to account
for the protruding part of this compound object.
Okay, just like we did for conic sections.
One way to visualize what the object looks like in 3-D is
by looking at cutting planes.
A cutting plane is said to be normal if it's parallel to one
of the three principal planes.
So look at this rectangular box.
The three principal planes are front, frontal plane.
Top, horizontal plane.
And right, profile plane, okay.
A cutting plane that is parallel to one
of those three principal planes is called a normal plane.
This case it's normal because the cutting plane is parallel
to the right profile plane, okay.
If I rotate that, say, relative to the X axis
and the Z axis here, okay.
So it was originally parallel to the right profile plane.
I rotate it slightly relative to the Z axis.
The plane becomes an incline plane, okay.
So you can use, vary, you can vary the orientation
of your incline cutting plane and visualize the curves
of intersection like we did for the conic sections in order
to develop your visualization skills.
If I rotated the plane again,
it becomes about another axis, about Y axis here.
It becomes an oblique plane, okay.
So an oblique plane is not perpendicular to any
of the three principal plains.
Here the incline plane, here,
it's still perpendicular to one of the planes.
It's still perpendicular to the frontal plane.
So just by changing the angle of rotation, okay,
and trying to visualize what the intersection would be can
develop your visualization.
Here it, from a circle, when the plane is horizontal,
to an ellipse to an even bigger ellipse of higher eccentricity.
Another way of developing your visualization skills is
by looking for planes of symmetry.
So, for instance, this cylinder here, okay,
is perfectly symmetrical with respect
to this vertical plane here that passes through its center.
Because whatever you see on the front,
in front of the plane is exactly a mirror reflection
of what's behind it.
It's also, that same cylinder also has this horizontal plane
of symmetry.
Because whatever you find above it is a mirror image
of what's under it or below it.
Now, development.
So imagine moving your cutting plane so that it's as close
as possible, your normal cutting plane so it's as close
as possible to the surface.
So you're slicing only a very,
very thin skin of your 3-D object.
You do that for the right profile plane, the top
and the horizontal, okay.
So slice, slice, slice very thinly.
And then imagine flattening the skin of your 3-D object.
And that's what you call development.
How you can get from 2-D -- like this cardboard box,
it's flat initially -- into a 3-D object.
In one of your homework exercises you'll actually be
doing this, cutting paper in order
to generate 3-D boxes and prisms, okay.
You can also try to visualize the resulting image.
If you take a projection plane or an image plane, okay,
and you project the features you have onto this image plane.
So this image plane we have is perfectly parallel
to the frontal plane of the 3-D object, okay.
And the projection lines are perfectly perpendicular
to the image.
As a result, okay, as it says here,
we only capture 2 dimensions.
The height of the object and the width of the object.
But you lose the depth.
Which makes sense because an object
in 3-D space actually has three dimensions.
The height the width and the depth.
And you're trying to represent an image in the plane,
only has two dimensions in this case.
The dimensions that you keep are the height and the width, okay.
So in our coordinate system, Z corresponds to depth.
X can correspond to the width.
And Y to the height.
If you do that for the three principal projection planes,
so I have one that's parallel to the front.
One that's top, horizontal.
One that's right profile plane.
Do the same thing.
Project all the features onto the frontal plane.
You'll get the front view.
Do the same thing for the horizontal plane,
you get the top view.
This one here will get the right side view.
It's like having a camera, okay, and taking three pictures.
One wherein your camera perfectly placed on,
perpendicular to the frontal plane
to give you the front view.
Another one perfectly, your axis
of your camera is perfectly vertical.
So it's, actually, it's perpendicular to the horizontal.
Gives you the top view.
Here will give you the right side view.
Now, when you have a normal plane like this plane,
it's normal because it's parallel
to the frontal plane, okay.
As you project it to the frontal plane, it gives you an image
that is representing the precise size
and shape of that normal plane.
However, the same normal plane, when projected to the top,
will become just an edge or a line.
And also the same for the right side view, okay.
So to repeat, a given normal plane, in this case parallel
to the front, becomes two sides and shape in the front.
And becomes just edges or lines in the top
and right profile plane.
On the other hand, an incline plane like this, okay,
when you project it to the top, it becomes foreshortened.
So looking at this plane here.
The projection at the top is still a rectangle,
but it's foreshortened.
Meaning the size is smaller than what it it's real size is.
If I project the right side too, it's also foreshortened.
However, if I project that incline plane
into the front view, since the plane is actually perpendicular
to the frontal plane, it would still appear as an edge.
So for an incline plane it appears
as two foreshortened planes in two of the views.
And an edge in one of the views, okay.
So this incline plane here, plane here foreshortened.
And edge here.
On the other hand, an oblique face like this,
or an oblique plane, when projected to any of the three,
okay, will actually appear as planes.
But these planes, the one seen on the views
and the front, top, right.