Alg Calc dCB Curves 11

Wild Egg Maths

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Harriot's difference calculus is an important part of the Algebraic Calculus online course, and leads to the remarkable interpolation/extrapolation formula of Gregory Newton, which gives us a direct connection between discrete calculus and continuous calculus.

The Gregory Newton formula relates directly to the familiar Pascal array and really ought to be a standard highlight of any calculus course. From our point of view, it is essential to shift from the rows to the columns of the Pascal / Harriot array, and in terms of these "Harriot sequences" the Gregory Newton formula has both a simple statement and the proof is relatively simple. This can be encoded in what we call the Difference Transform on sequences, and allows an interpretation of the terms of the de Casteljau Bezier curves when written as a polynumber in terms of the control points.

00:02

hello everyone i'm norman wahlberger

00:03

thanks for joining me on this adventure

00:05

towards understanding the castle albizia

00:07

curves from an algebraic calculus point

00:09

of view

00:11

today we're going to bridge

00:13

somewhat the gap between the discrete

00:16

and the continuous which is already sort

00:18

of inherent in our de castelia bezier

00:21

story where we start with this polygon

00:23

p0 through p4 and then construct from

00:27

that this curve you see

00:29

so

00:30

this is a really interesting

00:33

aspect of calculus that there's actually

00:35

a discrete side to most of the

00:38

continuous

00:40

theory that you usually treat but

00:42

unfortunately the discrete version

00:44

doesn't get as much airplay as it ought

00:45

to so that's one of the things that

00:47

we're going to be uh touching upon today

00:50

so we're going to go right back to

00:51

thomas harriet who introduced the basic

00:53

idea of the discrete or difference

00:55

calculus and would aim towards a

00:57

remarkable

00:58

result of gregory and newton

01:01

um that

01:02

really connects

01:03

very dramatically to the story of the

01:05

castle bezier curves

01:08

so i think we'll see what we've been

01:09

talking about in the last video or two

01:11

in a new light

01:13

so the slides that i'm going to be

01:14

presenting to you here are taken

01:16

directly from my algebraic calculus one

01:18

online course which is sort of the

01:21

foundation for

01:22

the basic orientation that we have to

01:24

this entire subject and this is the sort

01:26

of welcome page here you can see some

01:28

the topics and the topic number eight

01:31

the discrete calculus is what we're

01:33

going to be focusing on today

01:35

so this is not a replacement for taking

01:38

this course and if you go to this uh

01:42

place here you're gonna see a dozens um

01:44

of work problems and homework exercises

01:47

and lots of information and variants and

01:50

and interesting challenges so i'm just

01:53

going to very briefly uh thread our way

01:55

through

01:56

this topic just giving you sort of the

01:58

essentials so that you can be in a

02:00

position to understand

02:02

um this

02:04

remarkable connection between the

02:05

discrete and the continuous that

02:08

ultimately goes back to harriet all

02:10

right

02:11

so

02:12

i'm gonna for purposes of brevity um

02:14

just take some screenshots of some of

02:17

the slides in my

02:20

my videos so these videos are actually

02:22

publicly available and they are on the

02:24

wild egg maths channel

02:26

which is also where you can become a

02:28

member and watch all of these great

02:30

explore research level mass

02:33

videos so here i want to juice going

02:35

right back to thomas harriet who lived

02:36

from 1560 to 1621. the idea of a

02:40

difference table so this was actually

02:42

pretty well found in harriet so he was

02:43

interested in say the pattern of fourth

02:46

powers here the fourth powers of the

02:47

first few natural numbers and let's say

02:50

you've just gone up to 1296 okay

02:54

which is the

02:56

fourth power of six

02:58

and

02:59

you might want what's the next one in

03:00

this series

03:01

okay it's the fourth power of seven but

03:03

suppose you didn't know what

03:05

the formula was you just wanted to sort

03:06

of extrapolate

03:08

this pattern can you tell what the next

03:11

entry here is going to be somehow

03:13

so this is a some kind of standard iq

03:16

kind of test kind of question so one

03:19

standard approach to this is to look at

03:20

successive differences between these

03:22

entries okay and over here we have the

03:24

difference operator delta

03:27

acting on

03:28

this uh sequence s okay so this is delta

03:31

s and the entries here as you can see

03:33

are just the differences of successive

03:35

entries in the original s

03:38

so 1 minus 0 is 1 16 minus 1 is 15 81

03:41

minus 16 is 65 etc so if we just have

03:45

the original sequence up to 1296 we can

03:47

generate this delta s up to 671

03:51

and then

03:52

the in some sense the the question

03:54

reduces of trying to find a pattern in

03:56

this sequence if we can understand the

03:59

pattern in this sequence then we can use

04:00

that to extend the original sequence

04:02

that's the basic idea

04:04

okay so then by extrapolation we carry

04:07

on and say okay we don't quite

04:09

understand this one let's look at the

04:10

differences between the entries in this

04:13

sequence so we take del squared of s we

04:16

take the difference of this sequence we

04:17

get 14 50 110 and so on down to 302

04:21

and then again iterating the differences

04:24

in this sequence give us this third

04:27

difference power

04:28

and then one more gives us the fourth

04:30

difference powers and at this stage here

04:33

the smoke clears and we start to see the

04:36

pattern and maybe we can already sort of

04:39

start seeing it here but it becomes very

04:40

visible at this stage here

04:42

and if we didn't see the pattern here

04:43

well if we want one more then we would

04:46

immediately see what it is it's just

04:48

successive differences of zero meaning

04:50

that these things are constant so that

04:51

means it's easy to predict what the next

04:53

one is it's going to be 24 and if we

04:55

know what this next one is here then we

04:56

can tell what the next one is over here

04:58

because

04:59

this 24 should be the difference between

05:01

this one and this one and so this one

05:03

has to be 108 plus 24.

05:05

so once we have this one then we can add

05:07

this to 302 to get 334 and then we can

05:10

use this one add that to this to get

05:12

1005 and finally we can add this

05:15

difference to the 1296 to get

05:18

2301 so this allows or allowed thomas

05:22

harriet to

05:23

extrapolate a given sequence by looking

05:27

at this table of differences okay until

05:31

eventually you sort of go over far

05:33

enough where the the pattern becomes

05:35

clear

05:36

okay so this is really

05:38

a crucial idea

05:39

and it's pretty interesting because

05:42

somehow this difference operator acting

05:44

on sequences say of natural numbers or

05:48

of integers

05:49

is very parallel to the idea of the

05:52

derivative of a function okay and that

05:55

correspondence is a

05:57

source of a lot of really interesting

05:58

ideas

06:01

okay so now let's move to this next

06:03

slide here where we're talking about the

06:05

inverse operation so on sequences

06:08

in the difference or discrete calculus

06:10

there are two basic operations there's

06:12

the

06:13

the difference operator that we've just

06:14

talked about represented by that delta

06:17

and there's a summation operation which

06:20

is

06:21

sort of the inverse of it okay it's just

06:23

like the discrete analog of integration

06:26

okay so here is an example of that and

06:30

here is what we might call harriet's

06:31

triangular summation table so this is a

06:34

pretty interesting table which harriet

06:36

thinks about in the following way so we

06:39

start with a very fundamental sequence

06:40

just the sequence of ones

06:42

and now to construct this summation

06:45

operator what we do is we

06:47

first of all always start a summation

06:49

with zero that's the convention okay and

06:52

then the next entry is the first entry

06:54

of whatever sequence we're examining

06:57

and then after that what we do is we

06:58

just take successive sums okay so like

07:01

the zeroth sum if you haven't gone

07:03

anywhere is zero and then the first sum

07:06

is is is this one and then the next sum

07:08

is one plus one which is two and the

07:10

next sum is one plus one plus one which

07:12

is three etc so we generate this

07:14

sequence as sums of successive elements

07:18

in this original sequence

07:20

now we can iterate that as well so if we

07:22

take the second

07:24

summation so the sigma operator on on on

07:27

this it's like taking the sigma operator

07:29

on this what do we get well we always

07:31

start with a zero and then the sum of

07:33

the the zeroth one is zero and then the

07:35

sum of these two is one and the subnext

07:37

sum is three and then the next sum is

07:39

six and so on and you can see we're

07:40

getting what's usually called the

07:41

triangular numbers

07:44

performing another summation we get

07:45

what's often called the pyramidal

07:47

numbers 0 0 0 1 4 10 20 et cetera and

07:51

then

07:52

these sort of five

07:55

dimensional analogs okay 0 0 0 0 1 5 15

08:00

etc

08:01

so each of these columns is generated

08:04

successively from the previous column by

08:06

summing up

08:08

successive entries in the previous

08:10

column

08:12

now we've seen this matrix before okay

08:14

this is a very familiar matrix

08:17

to mathematics students because it's

08:19

essentially the pascal matrix

08:21

or pascal triangle as it's sometimes

08:23

called sort of extended in into make it

08:25

a matrix

08:27

and usually when we look at it as a

08:29

pascal matrix it's the rows which are

08:31

interesting for us right so this this

08:34

row here is the expansion of x plus y to

08:36

the fourth it's giving us the

08:38

coefficients

08:40

so

08:41

what we're doing here is we're subtly

08:43

changing our point of view on on this

08:46

data structure and starting to think

08:48

about the columns as being at least as

08:50

interesting as the rows and from this

08:53

sequence point of view it's really the

08:55

columns which are of primary interest

08:57

here

08:59

and because harriet uh did this before

09:01

pascal

09:03

let's actually change notation okay so

09:05

instead of calling this p for a pascal

09:07

matrix let's call it h for harriet

09:10

matrix

09:12

okay but when we do that it's with the

09:14

understanding that we're really

09:16

interested in the columns of this thing

09:19

okay so to make this a little bit more

09:21

clear let's um let's go down here okay

09:24

so the columns of harriet's

09:26

triangular summation table are important

09:29

so the zeroth column just consists of

09:31

one so here's a sort of my notation for

09:33

a sequence which i developed in the

09:35

algebraic calculus one course okay

09:38

it's given by the function of one

09:40

function one and then the next column

09:43

the first column

09:44

entries are zero one two etcetera that's

09:46

given by the the function n the nth

09:49

entry is n

09:51

the next one which are the triangular

09:53

numbers is given by the function n

09:55

choose 2. it's a binomial coefficient if

09:57

we write it out it's n times n minus 1

10:00

over 2. that's these triangular numbers

10:02

and we're starting all our indices from

10:05

zero so this is the case n equals zero

10:07

this is the case n equals one and then

10:09

here's the case n equals two that's when

10:12

you get a one

10:14

and then the next column h3 is given by

10:18

n choose three there is the official

10:20

formula n times n minus one times n

10:22

minus two over three factorial and there

10:25

is the sequence that it generates for

10:28

natural number of values of n starting

10:30

with zero the pyramidal numbers

10:33

okay so we're giving the columns of this

10:36

harriet matrix h

10:38

names

10:39

h0

10:40

h1 h2 etc and here is the formula for

10:44

the kth one okay so if we go across to

10:46

the kth column where our

10:49

indexing starts at zero then this is the

10:52

column whose

10:53

entries as a sequence

10:55

is n choose k

10:58

okay and explicitly this is n times n

11:00

minus 1 all the way down to n minus k

11:02

plus 1 over k factorial it's another way

11:04

of writing the binomial coefficient

11:07

and

11:07

if we write it this way we can utilize

11:10

donald kluth's falling

11:13

factorial notation so n to the k falling

11:16

here so there's a k with a little bar

11:18

underneath it okay so that's not

11:20

n to the k not k copies of n but rather

11:23

k k numbers multiplied together starting

11:26

with n and going down by 1 each time n

11:29

times n minus 1 times n minus 2 down to

11:31

n minus k plus 1. so all together k

11:34

numbers being multiplied together

11:37

okay

11:38

so this is a succinct formula for the

11:43

the nth element in the k-th

11:46

column of this h matrix

11:49

okay so it's a very familiar thing this

11:52

is just the pascal triangle but we're

11:54

looking at it differently we're looking

11:55

at it from the point of view of these

11:57

columns and we're thinking of these

11:59

columns as being sequences

12:01

which are

12:03

in some sense a basis for sequences

12:07

okay

12:08

because of the triangular nature if we

12:10

go back up here you see

12:12

because these these columns have this

12:14

pleasant property that they sort of form

12:15

this triangular matrix it's not too hard

12:18

to see that if you have any sort of

12:20

initial sequence any finite sequence say

12:22

of you know m

12:24

elements that you can write it as a

12:26

linear combination of the first m

12:29

of these

12:32

harriet columns

12:36

okay because uh you know the top entry

12:38

will determine this one and so when

12:40

when you when you know that then the

12:42

next entry will determine that the

12:43

coefficient of this one and then the

12:44

next entry will determine the

12:45

coefficient of this one so so you can

12:47

recursively just calculate what the

12:49

various coefficients

12:51

of of these sort of basis columns are

12:54

going to be when you expand on any

12:56

sequence okay so these are our

12:59

particularly attractive

13:01

sequences from the point of view of

13:04

sequence

13:05

calculus

13:06

and the the fundamental reason for that

13:09

is because what of this thing that we

13:10

make called harriet's difference theorem

13:13

so i'm going very fast here right okay

13:15

so

13:16

i'm not expecting you to immediately

13:18

absorb this if you're really interested

13:19

in this you can watch this video several

13:21

times or you can go over to the wild egg

13:25

mass

13:27

channel and and watch the videos um in

13:29

in the algebraic calculus one

13:32

course so they're publicly available and

13:34

if you're really interested of course

13:35

you just

13:37

join up i've learned algebraic calculus

13:39

one properly and you'll learn all this

13:41

stuff and and much more

13:43

okay so what is the crucial

13:45

property of these harriet sequences well

13:48

it's contained in this little

13:50

formula here that

13:52

the delta of h sub k is h sub k minus

13:55

one

13:56

okay and this is a snappy little proof

13:58

okay so what we end up is we have this

14:01

sort of very pleasant

14:03

little

14:05

property here that if you think of these

14:07

these sequences h0 is a sequence h1 is a

14:10

sequence h2 is a sequence that the

14:13

summation operator takes h0 to h1 takes

14:16

h1 to h2 takes h2 to h3

14:19

that's sort of how we generated the

14:20

table okay and

14:23

conversely the

14:25

difference operator delta

14:27

takes h3 to h2 takes h2 to h1 takes h1

14:31

to h0

14:32

so this is a great basis in which to try

14:36

to express sequences if you're

14:38

interested in differences and summations

14:42

of

14:43

sequences because exactly of this

14:45

property that these particular sequences

14:48

are so pleasant with respect to these

14:50

crucial

14:52

difference and summation operators

14:56

okay so let us go down here and sort of

14:59

summarize what we have so here's another

15:00

look at our matrix h

15:02

and really of course this is uh an

15:05

ongoing matrix

15:07

okay so properly

15:08

one really wants to adopt a kind of

15:10

maxwell point of view where you're not

15:12

thinking about matrices as having a

15:14

sort of a bounded size you're thinking

15:16

of maxwell's sort of you know things

15:18

which kind of extend beyond your view

15:21

and you make statements about finite

15:24

portions of it but you allow the object

15:27

to be defined by

15:29

formulas or recursions which generate

15:32

more and more elements

15:34

okay so here's

15:36

just another copy of h here is the

15:38

indexing so we'll think of n as being

15:40

indexing the rows and k as indexing the

15:42

columns so here are the columns here's

15:44

h0 here's h1 here's h2 here's h3 here's

15:47

h4 here's h5 etc

15:50

okay and here are the actual formulas

15:52

for these columns in terms of the

15:56

the height variable n okay so n n choose

16:00

2 and choose 3 and choose 4 and choose 5

16:02

etc

16:04

ok now we come to one of the great

16:07

theorems in in mathematics which

16:10

is often not even taught to to first or

16:13

second year students in fact i think a

16:15

lot of undergraduates may go you know

16:17

maths undergraduates will go through

16:19

their career and not even see this just

16:21

somehow mind-boggling okay so this is

16:23

the gregory newton interpolation formula

16:25

it's absolutely a key

16:28

understanding and it's actually a really

16:30

useful

16:32

tool that every mathematics student

16:34

should have under their belt

16:36

it's actually a real practical utility

16:39

because

16:40

we are often in a position of wanting to

16:43

see what the pattern of something is or

16:46

to write down a formula

16:48

for

16:49

some pattern and

16:50

gregory newton tells us

16:52

what to do okay so here it is

16:56

if you have a poly number on sequences

16:58

that's a okay it's a

17:00

a sequence okay which is given by a

17:02

polynomial of our polynomial is what i

17:05

prefer to call them okay it's called s

17:07

okay so you've got some so we say

17:09

function of of n okay and let's say it's

17:13

a degree k

17:15

okay

17:16

that means that it can be written as

17:18

a linear combination of the hjs from h0

17:22

up to hk

17:24

okay that's the remark i made before

17:26

that the h

17:27

these columns in the

17:29

heria matrix they form a basis in some

17:32

sense of sequences okay

17:35

and this formula is telling us what the

17:38

coefficients are

17:40

okay

17:41

telling us that the coefficient

17:44

of h j in the expansion of this original

17:47

sequence s

17:48

is obtained by taking

17:50

the

17:52

jth difference delta to the j of s

17:55

which is another sequence and looking at

17:57

the zeroth or top element

18:03

now

18:04

this

18:05

h

18:06

sub j as a function of n so if we want

18:09

to state everything in terms of a

18:10

function of n it's that the nth element

18:12

in s okay is obtained here you the nth

18:15

element of h j is n choose j

18:18

okay and

18:19

these are these numbers delta to the j

18:21

of s at zero these are the coefficients

18:24

that you have to multiply these little

18:26

polynomials you think of this as a

18:27

polynomial in n by to get s n

18:31

and the proof is as you can see it's

18:34

pretty short and sweet

18:36

okay so let us have a look at an example

18:39

okay so here is some sequence that we

18:41

stumble across somewhere it starts out

18:44

two zero four twenty fifty four two zero

18:46

four twenty fifty four so we write it as

18:48

a vertical thing as we're thinking of it

18:50

like a poly number but you know it

18:52

potentially keeps going we don't know

18:54

what the additional terms are we want to

18:56

get some kind of control over this

18:59

sequence so this is such an important

19:00

thing to do

19:01

okay so following thomas harriet okay

19:04

what we are going to do is we are just

19:05

going to take differences of this thing

19:07

so we just take successive differences

19:09

zero minus two is minus two four minus

19:11

zero is four twenty minus four sixteen

19:13

fifty four minus twenty is thirty four

19:15

and that's as far as we can go because

19:17

we don't know any more

19:19

terms in s there may be some more terms

19:21

but we don't know what they are so this

19:22

is as much as we're legally allowed to

19:25

write down

19:26

okay

19:27

now we apply delta again so delta

19:30

squared 4 minus 2 is 6. 16 minus 4 is 12

19:34

34 minus 16 is 18. okay

19:37

then we apply delta again

19:39

now we only get 12 minus 6 which is 6

19:42

and 13 minus 12 which is 6.

19:45

we only get two entries and of course

19:46

the difference here is zero

19:49

now

19:50

there's all kinds of

19:52

sequences that could

19:54

could uh you know carry on here but

19:55

there's a kind of a simplest one there's

19:58

a simplest one which is

20:01

of

20:02

uh degree four okay so if we only go up

20:05

to degree four as represented by just

20:08

looking at these four

20:09

um

20:10

difference

20:12

operators or numbers that we've obtained

20:14

so across this first row here the minus

20:16

two the six the six and the zero

20:19

we can then recover this

20:22

this sequence by using the gregory

20:24

newton formula okay what is that okay so

20:27

we start with two okay that's the

20:30

initial element and then we have to

20:31

subtract minus two times the h1 the h1

20:34

is the sequence which is just n

20:36

so we multiply by n

20:38

and then plus 6 times n choose 2.

20:40

there's n choose 2 plus 6 times n choose

20:43

3 and actually plus 0 times n choose 4.

20:49

and then we just expand this thing out

20:50

so this is a polynomial n and if we

20:52

expand it out we get two minus three n

20:55

plus n cubed

20:57

and then you can check that this is then

20:59

a polynomial

21:00

function which agrees with the original

21:03

sequence

21:05

for these five terms

21:07

and this is a great thing to have for

21:09

reasons um that gregory and newton were

21:12

very interested in they were interested

21:13

in extrapolating this sequence so for

21:16

example if you want to know what the

21:17

next one might be well then you could

21:20

just take this formula now and plug in n

21:22

equals

21:23

five

21:24

but you could also use it to interpolate

21:27

let's say you wanted to to have some

21:29

understanding of this sequence not just

21:31

for natural number of values but for

21:33

intermediate values okay so for the

21:35

value at

21:36

at two is four the value at uh three is

21:39

twenty what is the value at two and a

21:41

half

21:43

well it's not so obvious at all right

21:45

but here this gives us a potential

21:47

answer you just plug two and a half into

21:49

this formula and you're going to get

21:50

some value intermediate in into in

21:53

between these two values

21:55

now note also that if

21:58

subsequently we um were given more terms

22:02

of the original

22:03

series okay some additional terms then

22:05

we would not have to rub off this table

22:07

this table would still be valid as it

22:09

stands it's just that we would be adding

22:11

more differences and we'd be able to

22:13

then extend our

22:16

our

22:17

top row of

22:19

of differences evaluate zero

22:22

further

22:23

okay so there's lots to think about here

22:25

if you're new to this spend a week

22:26

thinking about it okay so the beautiful

22:28

thing here is you can see this working

22:31

by just working out lots of examples and

22:33

it's fun okay so this leads us to what i

22:36

call the difference transform t of s

22:39

okay so what we're really doing here is

22:41

we're starting with a sequence this

22:42

original

22:44

sequence written vertically and we're

22:46

ending up with this new sequence written

22:48

horizontally and it's the new

22:51

sequence written horizontally that we're

22:52

using in order to extrapolate or to to

22:55

create this

22:57

formula for the original sequence

23:00

okay so we actually have a

23:01

transformation here going from this

23:04

sequence this vertical original sequence

23:06

to this horizontal original sequence so

23:09

let's formalize that okay so we call it

23:12

the difference transform so here it is

23:15

so uh tiva so we start say with a

23:18

sequence let's say a0 a1 a2 a3 a4

23:23

rather general okay so then what exactly

23:26

is the difference

23:28

gonna be well this

23:30

zeroth difference is gonna be a1 minus

23:32

a0 and then a2 minus a1 then a3 minus a2

23:35

etc

23:36

then the next difference is going to be

23:38

the difference between a2 minus a1 and

23:40

a1 minus a0 well that's a2 minus 2a1

23:44

plus a0

23:46

and then the next one will be the same

23:47

kind of thing but with just the indices

23:49

jacked up by one

23:52

and what about the next difference well

23:54

that'll be the difference of these two

23:55

so we have to take this element here a3

23:58

minus 2a 2 plus a1 and subtract a2 minus

24:01

2a1 plus a0

24:03

if we do that we get this

24:05

a3 minus three a two plus three a one

24:08

minus a zero

24:10

i hope this is starting to look familiar

24:13

okay and then the

24:14

subsequent ones in this third difference

24:17

power will just be obtained by

24:20

upping the indices throughout

24:23

and then the next one difference between

24:25

this element and this element we get the

24:28

degree four sort of binomial alternating

24:31

sign

24:33

entry

24:35

so what was involved in the gregory

24:39

newton formula

24:42

the coefficients were obtained by taking

24:44

these higher difference

24:47

operators acting on the original

24:49

sequence and evaluating them at zero in

24:52

other words taking the zero for the

24:54

first element in each of those sequences

24:56

so we have to read down here so it's

24:58

it's a 0 and then a1 minus a0 and then

25:02

a2 minus 2a1 plus a0 and then this one

25:04

and then this one

25:06

so the difference transform of the

25:09

sequence s

25:10

is given by this new sequence

25:13

a zero a one minus a zero a two minus

25:16

two a one plus a zero

25:18

a three minus three a two plus three a

25:21

one minus a zero

25:22

a four minus four a three plus six a two

25:25

minus four a one plus a zero

25:29

okay so now i hope it's coming into

25:32

focus for you

25:33

just why

25:35

this is so

25:36

intimately connected with the castell

25:39

bezier story if you've been following

25:42

the last few videos

25:44

so to make this a connection work very

25:46

well we can think about applying this

25:48

theory of difference operators

25:51

to not just sequences of numbers

25:54

but actually also to sequences of affine

25:57

points

26:00

so

26:01

instead of having a0 and a1 and a2 etc

26:04

suppose we had

26:06

capital p0 capital p1 capital p2

26:09

where now these things are happen to be

26:11

affined points

26:13

well we're taking differences are we

26:15

allowed to take differences of affine

26:17

points

26:18

well yes we are those are vectors

26:22

so that that's that's defined

26:24

if we were taking sums of affine points

26:26

that wouldn't make any sense but taking

26:28

differences

26:30

is legitimate because that's a vector

26:32

okay so then this t of s

26:35

operation then we just recast this

26:37

instead of a's we have piece we are

26:40

getting the sequence p zero p one minus

26:42

p zero p two uh minus two p one plus p

26:45

zero etc the next one will be i'll say

26:49

in this language p three minus three p

26:52

one

26:53

minus

26:55

p sub zero

26:58

and happily these are just the

27:00

coefficients up to some binomial

27:02

coefficients of the dicastlia bezier

27:05

curve gamma of t

27:07

that has control points p0 p1 p2 p3 etc

27:12

okay so

27:14

that was this thing here

27:16

so what are we seeing here we're seeing

27:18

that the difference transform of a

27:21

sequence of dcb control points so we're

27:24

thinking of the elements in our polygon

27:27

as a sequence of affine points p0 p1 p2

27:31

p3 up to whatever okay

27:34

if we take the difference transform of

27:36

this exactly in the spirit of harriet

27:39

here okay

27:41

then what we're getting is

27:43

um

27:44

well

27:46

from a physical point of view

27:48

you take

27:49

just the first difference that's

27:51

going from position to velocity and you

27:53

take another difference or a difference

27:55

of differences you're getting an

27:56

acceleration you take another difference

27:58

of differences you're getting a jerk etc

28:01

so you're getting that sort of hierarchy

28:03

position velocity acceleration jerk snap

28:05

etc okay

28:07

but now

28:09

in addition we see that these are just

28:10

the coefficients

28:12

so these objects that we're getting

28:14

are just the coefficients up to these

28:16

binomial coefficients of the associated

28:19

dcb curve

28:21

gamma of t written as a polynomial in t

28:27

written as a polynomial in t these

28:30

entries which are appearing here which

28:31

we were discussing at some length in our

28:34

last video we now have an interpretation

28:36

of them

28:37

as coming from

28:39

the

28:41

um

28:42

coming from this difference transform

28:45

applied to

28:47

the

28:48

control points of the castiglial bezier

28:53

polygon

28:55

so this is a really sort of

28:58

somehow elementary but it's sort of a

29:00

deep and important thing to understand

29:04

and to make

29:05

a lot of this um

29:07

even more workable and more pleasant

29:10

i'm going to show you next time how we

29:11

can

29:13

utilize ideas from linear algebra to

29:15

help us work with all of these things

29:18

much more smoothly and to to understand

29:20

more deeply also what's going on