AISL 3.6.1 Voronoi diagrams intro | Free Mathematics Applications & Interpretation (AI) Video | RevisionDojo
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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.
About this video 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:
Definition : Voronoi diagrams partition a space into regions based on the proximity to a set of points, known as sites .Technical Terms : Important terms include edges (boundaries of regions), cells (the regions themselves), and vertices (points where edges meet).Construction : The process of creating a Voronoi diagram involves drawing perpendicular bisectors between points to determine the closest regions.
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.
Video transcript 00:00 Hi everybody. So I'm going
00:02 to do two lessons on
00:03 Vorni diagrams. This first lesson
00:05 we will look at what
00:07 they are, what they mean,
00:10
and where they come from,
00:11 and we're going to create
00:12 a Vorni diagram. And then
00:14 in the second lesson, I'm
00:15 going to go through the
00:16 types of questions you could
00:17 be asked in an exam.
00:20 Okay, so firstly, let me
00:21 bring your attention to these
00:22 two photos. I had too
00:26 many examples to choose from.
00:28 where Varni diagrams appear in
00:30 real life if you just
00:33 go to Wikipedia and type
00:34 in Varni diagrams and look
00:36 at the applications you'll see
00:38 that they appear and present
00:39 themselves all over the place
00:42 in the real world. I
00:44 have chosen to show you
00:45 the giraffe whose pattern is
00:48 a Varni diagram and this
00:50 is the water cube it's
00:53 called, it's the aquatic center
00:55 in Beijing where they
00:56 held the swimming in the
00:57 2008 Olympics and what the
01:00 design is, is, well, it
01:03 represents bubbles and when when
01:06 bubbles come together, they form
01:08 a Voronai diagram like this.
01:11 Okay, now the example I'm
01:13 going to use to explain
01:14 what a Voronai diagram is
01:15 is a different example again.
01:17 It's actually schools. So this
01:21 is Voronai diagram, ignoring the
01:22 arrows and the writing.
01:24 This is Varnadar. Imagine each
01:27 of these points, this is
01:29 my example, imagine each of
01:30 these points is a school.
01:32 Now some states or cities
01:35 or towns or countries even
01:37 have rules where you have
01:40 that is closest to where
01:42 you live. Now this can
01:43 be quite a controversial policy
01:46 and quite an annoying policy
01:48 wrong place. So for example,
01:51 if you imagine you lived
01:52 here. And let's say these
01:55 three schools here are all
01:58 brilliant schools and this is
02:00 a terrible school. Well, you
02:03 school because this is the
02:04 school that's closest to you.
02:05 In fact, this school might
02:06 be easier to get to,
02:08 but this is still the
02:09 one that's closest to you
02:14 what Averna diagram is. It
02:16 divides this whole area into
02:20 Which kind of show you
02:22 all the points that are
02:23 closest to a given school
02:25 in my example But site
02:27 is the more technical name
02:29 in Varni diagrams. So anywhere
02:32 you live here, that's your
02:33 closest school if you live
02:34 here That's your closest school
02:36 if you live here That's
02:38 your closest school anywhere in
02:42 here that's your closest school
02:45 etc. So that's basically what
02:51 show you, I just want
02:53 to go through these kind
02:55 of technical terms. So edges
02:56 are outside the region. So
02:58 this is an edge, this
03:01 edge, this is an edge.
03:02 Sites are what I've called
03:05 schools. So these are the
03:06 sites, site, site, site, site,
03:09 cells are the region. So
03:11 this is a cell, this
03:14 a cell, each cell has
03:16 And then vertices are where
03:18 the edges meet. So the
03:20 edges meet at a vertex.
03:22 So that's a vertex. That's
03:24 a vertex. That's a vertex.
03:25 Now note as well that
03:26 it's always three vertices that
03:29 meet. Sorry, three edges that
03:32 meet at a vertex. Look,
03:35 three edges, three edges, three
03:38 edges, three edges, three edges.
03:41 Okay, next thing I want
03:44 that these edges are actually
03:47 perpendicular bisectors. And that's the
03:49 whole point of this thing.
03:51 So look, the perpendicular bisector
03:54 of these two points is
03:58 draw this, this bisects these
04:01 two points. So clearly anything
04:02 below this line is closest
04:04 to this point and anything
04:05 above this line, well, it's
04:08 nearest school is this school.
04:11 If I draw the perpendicular
04:12 a regular bicepter here. Well,
04:16 these two points, this is
04:19 the perpendicular bicepter of these
04:20 two points. Or anything to
04:23 you live anywhere to the
04:24 left of this line, that's
04:25 your nearest school and you're
04:26 at the right, this is
04:27 your nearest school. And they're
04:28 all perpendicular bicepters. Look, that's
04:30 the perpendicular bicepter of these
04:31 two. That is the perpendicular
04:33 bicepter of these two. And
04:37 it kind of shows you
04:38 why this school is closest
04:40 point and this school is
04:42 close to this point because
04:43 that's the perpendicular bisector. It
04:45 bisects, it cuts in half
04:48 the area between these two
04:51 points. It's going to obviously
04:53 anywhere to the left of
04:54 this is closer to this
04:56 point. The final thing I
05:03 of these. So these circles
05:08 So each vertex is actually
05:13 the center of a circle
05:15 that goes through the three
05:19 sides closest to it. So
05:21 here I've just picked a
05:22 circle. This is my center
05:25 and it goes through this
05:27 point, this point, this point.
05:29 If I pick another circle
05:31 that I've already made, it
05:33 goes through these three points.
05:38 its center. And you could
05:40 you get any vertex, if
05:42 you actually had a set
05:43 of compasses and you had
05:45 this on paper, you could
05:47 compasses, put the pointy bit
05:49 right on the vertex and
05:50 draw a circle and you'll
05:51 see that it goes through
05:53 the three closest points. Okay,
05:57 that's your kind of introduction
05:58 to Varnad diagrams. What I
06:04 want to create a Vona
06:06 diagram, you don't need to
06:08 know how to create a
06:09 full one from scratch in
06:11 your exam. Normally they'll just
06:14 ask you to complete one,
06:15 but certainly it's going to
06:18 help with your understanding if
06:21 you know how it's done.
06:22 So let's start with two,
06:24 let's just start with two
06:25 points. Imagine I have here's
06:26 a point and here's a
06:28 point and let's stick with
06:29 our school example. Let's say
06:34 up this region, this whole
06:37 area, split it up into
06:38 two regions where one region
06:41 is everywhere as closest to
06:43 this point and the other
06:44 region everywhere in it is
06:45 closest to this point. What
06:49 the two points like this.
06:54 Obviously this isn't perfect, but
06:57 there you go. There's a
06:59 straight line everywhere to
07:01 the left of this line,
07:02 that's your closest school. If
07:04 you live anywhere to the
07:05 right of this line, that's
07:06 your closest school. So that
07:07 is actually technically a VORNAI
07:09 diagram with two points, or
07:11 two sites. But obviously I'm
07:13 going to increase the number
07:14 of sites and the whole
07:15 thing gets a little bit
07:17 more complicated. So let's put
07:19 another site here. Now, let
07:24 me go out there, let's
07:25 say here. Okay, now I
07:29 to draw perpendicular by sectors
07:32 of the perpendicular by sector
07:34 for these two points, which
07:35 would be let's say something
07:37 like that. And then the
07:41 perpendicular by sector for these
07:43 two points, which would be
07:44 something like that. Now note,
07:49 they all meet here in
07:51 the middle, and that would
07:52 be the center of this
07:54 circle. But I want to
07:57 remove some of these lines
07:59 because this line here, it's
08:05 only important for these two
08:07 points. It's not really, it
08:08 doesn't really affect this point.
08:11 So I'm actually going to
08:11 remove that there. And similarly,
08:18 this line is only for
08:20 these two points. So I'm
08:21 going to remove this here.
08:25 then this point is enough
08:26 for these ones. So I'm
08:27 going to remove this down
08:29 here. So now there we
08:32 diagram for three points. Anywhere
08:34 here is closest to this
08:36 school and you were here
08:37 is closest to this school
08:38 and you were here is
08:38 closest to this school. Fine.
08:41 And I can keep going.
08:42 I'll do one more. Obviously
08:44 you can try and keep
08:48 want. If you were doing
08:51 a very good topic for
08:53 could you might actually want
08:56 to create the whole for
08:58 an i diagram. So let's
08:59 say I have another school
09:01 down here. Right. I need
09:04 to get the perpendicular bisector
09:07 of these two, which is
09:10 going to be there, but
09:13 let's say here. And I
09:19 need the perpendicular bisector of
09:21 which is here. Now again,
09:24 it's not perfectly accurate, but
09:28 note they meet here. That
09:30 will be the center of
09:31 the circle going through these
09:32 three points. Now again, I'm
09:34 eraser and I'm going to
09:36 remove this line because this
09:39 line isn't to do with
09:41 this point. It's for these
09:42 two points and this line
09:45 here is for these two
09:48 points. So that's I don't
09:49 need that there and yeah
09:54 there we're actually sorry this
09:56 one I don't need this
09:57 line here because that is
09:59 only for these two points
10:00 so I'm gonna remove that
10:02 there and that's my Vorta
10:05 for an iDIGRA for four
10:07 points and I can keep
10:08 going on and on like
10:09 that and eventually I'd end
10:10 up with well depending where
10:12 the points are you end
10:14 more complete for an iDIGRA
10:16 which looks something
10:17 thing like this. Okay. So
10:21 next lesson, we will in
10:24 the next lesson, we're going
10:25 to go through the different
10:26 examples of the types of
10:27 questions you could be asked
10:30 want you to know all
10:31 the names of these different
10:32 things, the edges, sites, cells
10:34 and vertices. I think the
10:36 school example is a good,
10:38 that's a good kind of
10:40 example that you can relate
10:42 to possibly. And certainly it
10:45 sense and it's easy to
10:46 explain what a Varni diagram
10:48 is using that example. I
10:50 do want you to know,
10:52 well I certainly want you
10:53 to realize that it's perpendicular
10:54 by sectors that is creating
10:58 the Varni diagram and as
11:00 lesson we will go through
11:03 those specific exam type questions.
11:06 See you in that lesson.