IB · Mathematics Applications & Interpretation (AI)
AISL 3.6.1 Voronoi diagrams intro
Learn AISL 3.6.1 Voronoi diagrams intro in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 3.6—Voronoi diagrams.
Learn AISL 3.6.1 Voronoi diagrams intro in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 3.6—Voronoi diagrams.
Learn AISL 3.6.1 Voronoi diagrams intro in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 3.6—Voronoi diagrams.
The video provides an introduction to Voronoi diagrams, explaining their significance and applications in real life, such as in the patterns of a giraffe and the design of the Water Cube in Beijing.
Key concepts covered include:
The video also emphasizes the relevance of understanding these diagrams for exam preparation, particularly in recognizing the relationships between the different components of Voronoi diagrams.
00:00Hi everybody. So I'm going
00:02to do two lessons on
00:03Vorni diagrams. This first lesson
00:05we will look at what
00:07they are, what they mean,
00:10and where they come from,
00:11and we're going to create
00:12a Vorni diagram. And then
00:14in the second lesson, I'm
00:15going to go through the
00:16types of questions you could
00:17be asked in an exam.
00:20Okay, so firstly, let me
00:21bring your attention to these
00:22two photos. I had too
00:26many examples to choose from.
00:28where Varni diagrams appear in
00:30real life if you just
00:33go to Wikipedia and type
00:34in Varni diagrams and look
00:36at the applications you'll see
00:38that they appear and present
00:39themselves all over the place
00:42in the real world. I
00:44have chosen to show you
00:45the giraffe whose pattern is
00:48a Varni diagram and this
00:50is the water cube it's
00:53called, it's the aquatic center
00:55in Beijing where they
00:56held the swimming in the
00:572008 Olympics and what the
01:00design is, is, well, it
01:03represents bubbles and when when
01:06bubbles come together, they form
01:08a Voronai diagram like this.
01:11Okay, now the example I'm
01:13going to use to explain
01:14what a Voronai diagram is
01:15is a different example again.
01:17It's actually schools. So this
01:21is Voronai diagram, ignoring the
01:22arrows and the writing.
01:24This is Varnadar. Imagine each
01:27of these points, this is
01:29my example, imagine each of
01:30these points is a school.
01:32Now some states or cities
01:35or towns or countries even
01:37have rules where you have
01:39to go to the school
01:40that is closest to where
01:42you live. Now this can
01:43be quite a controversial policy
01:46and quite an annoying policy
01:48if you live in the
01:48wrong place. So for example,
01:51if you imagine you lived
01:52here
01:52here. And let's say these
01:55three schools here are all
01:58brilliant schools and this is
02:00a terrible school. Well, you
02:02have to go to this
02:03school because this is the
02:04school that's closest to you.
02:05In fact, this school might
02:06be easier to get to,
02:08but this is still the
02:09one that's closest to you
02:11so you have to go
02:12to it. But that is
02:14what Averna diagram is. It
02:16divides this whole area into
02:19regions
02:20Which kind of show you
02:22all the points that are
02:23closest to a given school
02:25in my example But site
02:27is the more technical name
02:29in Varni diagrams. So anywhere
02:31in here if you if
02:32you live here, that's your
02:33closest school if you live
02:34here That's your closest school
02:36if you live here That's
02:38your closest school anywhere in
02:42here that's your closest school
02:45etc. So that's basically what
02:47a Varni diagram
02:48What I want to now
02:51show you, I just want
02:53to go through these kind
02:55of technical terms. So edges
02:56are outside the region. So
02:58this is an edge, this
03:00is an edge, this is
03:00an edge, this is an
03:01edge, this is an edge.
03:02Sites are what I've called
03:05schools. So these are the
03:06sites, site, site, site, site,
03:09cells are the region. So
03:11this is a cell, this
03:12is a cell, this is
03:14a cell, each cell has
03:15one site.
03:16And then vertices are where
03:18the edges meet. So the
03:20edges meet at a vertex.
03:22So that's a vertex. That's
03:24a vertex. That's a vertex.
03:25Now note as well that
03:26it's always three vertices that
03:29meet. Sorry, three edges that
03:32meet at a vertex. Look,
03:35three edges, three edges, three
03:38edges, three edges, three edges.
03:41Okay, next thing I want
03:42to show you is
03:44that these edges are actually
03:47perpendicular bisectors. And that's the
03:49whole point of this thing.
03:51So look, the perpendicular bisector
03:54of these two points is
03:56this edge. So if I
03:58draw this, this bisects these
04:01two points. So clearly anything
04:02below this line is closest
04:04to this point and anything
04:05above this line, well, it's
04:08nearest school is this school.
04:11If I draw the perpendicular
04:12a regular bicepter here. Well,
04:16these two points, this is
04:19the perpendicular bicepter of these
04:20two points. Or anything to
04:21the left of it, if
04:23you live anywhere to the
04:24left of this line, that's
04:25your nearest school and you're
04:26at the right, this is
04:27your nearest school. And they're
04:28all perpendicular bicepters. Look, that's
04:30the perpendicular bicepter of these
04:31two. That is the perpendicular
04:33bicepter of these two. And
04:37it kind of shows you
04:38why this school is closest
04:40to this
04:40point and this school is
04:42close to this point because
04:43that's the perpendicular bisector. It
04:45bisects, it cuts in half
04:48the area between these two
04:51points. It's going to obviously
04:53anywhere to the left of
04:54this is closer to this
04:56point. The final thing I
04:58want to show you is
05:01the, let me get rid
05:03of these. So these circles
05:08So each vertex is actually
05:13the center of a circle
05:15that goes through the three
05:19sides closest to it. So
05:21here I've just picked a
05:22circle. This is my center
05:25and it goes through this
05:27point, this point, this point.
05:29If I pick another circle
05:31that I've already made, it
05:33goes through these three points.
05:36to 3 and this is
05:38its center. And you could
05:39do that for any, if
05:40you get any vertex, if
05:42you actually had a set
05:43of compasses and you had
05:45this on paper, you could
05:46do it with set of
05:47compasses, put the pointy bit
05:49right on the vertex and
05:50draw a circle and you'll
05:51see that it goes through
05:53the three closest points. Okay,
05:57that's your kind of introduction
05:58to Varnad diagrams. What I
06:01want to do now is
06:03draw one. So I want
06:04want to create a Vona
06:06diagram, you don't need to
06:08know how to create a
06:09full one from scratch in
06:11your exam. Normally they'll just
06:14ask you to complete one,
06:15but certainly it's going to
06:18help with your understanding if
06:21you know how it's done.
06:22So let's start with two,
06:24let's just start with two
06:25points. Imagine I have here's
06:26a point and here's a
06:28point and let's stick with
06:29our school example. Let's say
06:32I wanted to decide
06:33it. How do I split
06:34up this region, this whole
06:37area, split it up into
06:38two regions where one region
06:41is everywhere as closest to
06:43this point and the other
06:44region everywhere in it is
06:45closest to this point. What
06:46I do is I bisect
06:49the two points like this.
06:54Obviously this isn't perfect, but
06:57there you go. There's a
06:59straight line everywhere to
07:01the left of this line,
07:02that's your closest school. If
07:04you live anywhere to the
07:05right of this line, that's
07:06your closest school. So that
07:07is actually technically a VORNAI
07:09diagram with two points, or
07:11two sites. But obviously I'm
07:13going to increase the number
07:14of sites and the whole
07:15thing gets a little bit
07:17more complicated. So let's put
07:19another site here. Now, let
07:24me go out there, let's
07:25say here. Okay, now I
07:28need
07:29to draw perpendicular by sectors
07:32of the perpendicular by sector
07:34for these two points, which
07:35would be let's say something
07:37like that. And then the
07:41perpendicular by sector for these
07:43two points, which would be
07:44something like that. Now note,
07:49they all meet here in
07:51the middle, and that would
07:52be the center of this
07:54circle. But I want to
07:57remove some of these lines
07:59because this line here, it's
08:05only important for these two
08:07points. It's not really, it
08:08doesn't really affect this point.
08:11So I'm actually going to
08:11remove that there. And similarly,
08:18this line is only for
08:20these two points. So I'm
08:21going to remove this here.
08:25then this point is enough
08:26for these ones. So I'm
08:27going to remove this down
08:29here. So now there we
08:31have our var and i
08:32diagram for three points. Anywhere
08:34here is closest to this
08:36school and you were here
08:37is closest to this school
08:38and you were here is
08:38closest to this school. Fine.
08:41And I can keep going.
08:42I'll do one more. Obviously
08:44you can try and keep
08:47going as far as you
08:48want. If you were doing
08:50an IA and this is
08:51a very good topic for
08:52an IA, you
08:53could you might actually want
08:56to create the whole for
08:58an i diagram. So let's
08:59say I have another school
09:01down here. Right. I need
09:04to get the perpendicular bisector
09:07of these two, which is
09:10going to be there, but
09:13let's say here. And I
09:19need the perpendicular bisector of
09:20these two.
09:21which is here. Now again,
09:24it's not perfectly accurate, but
09:28note they meet here. That
09:30will be the center of
09:31the circle going through these
09:32three points. Now again, I'm
09:34going to get out my
09:34eraser and I'm going to
09:36remove this line because this
09:39line isn't to do with
09:41this point. It's for these
09:42two points and this line
09:45here is for these two
09:48points. So that's I don't
09:49need that there and yeah
09:54there we're actually sorry this
09:56one I don't need this
09:57line here because that is
09:59only for these two points
10:00so I'm gonna remove that
10:02there and that's my Vorta
10:05for an iDIGRA for four
10:07points and I can keep
10:08going on and on like
10:09that and eventually I'd end
10:10up with well depending where
10:12the points are you end
10:13up with a kind of
10:14more complete for an iDIGRA
10:16which looks something
10:17thing like this. Okay. So
10:20as I say, in the
10:21next lesson, we will in
10:24the next lesson, we're going
10:25to go through the different
10:26examples of the types of
10:27questions you could be asked
10:28in your exam. I do
10:30want you to know all
10:31the names of these different
10:32things, the edges, sites, cells
10:34and vertices. I think the
10:36school example is a good,
10:38that's a good kind of
10:40example that you can relate
10:42to possibly. And certainly it
10:45makes
10:45sense and it's easy to
10:46explain what a Varni diagram
10:48is using that example. I
10:50do want you to know,
10:52well I certainly want you
10:53to realize that it's perpendicular
10:54by sectors that is creating
10:58the Varni diagram and as
11:00I say in the next
11:00lesson we will go through
11:03those specific exam type questions.
11:06See you in that lesson.