T h e
M a t h e m a t i c s
o f
H o r s e s
B y
K a t r i n a
A
T h e
T h e s i s
F a c u l t y
C a l i f o r n i a
T o p o l i n s k i
P r e s e n t e d
to
o f the M a t h e m a t i c s
State U n i v e r s i t y
P r o g r a m
C h a n n e l
Islands
2 0 1 3
In Partial
F u l f i l l m e n t
O f the R e q u i r e m e n t s
M a s t e r s
o f
for the
S c i e n c e
D e g r e e
M S THESIS BY K A T R I N A TOPOLINSKI
A P P R O V E D FOR THE M A T H E M A T I C S P R O G R A M
Doctor I v o n a G r z e g o r c z y k
DoctorGregoryWood
D a t e 1/7/2013
Date 1/7/2013
A P P R O V E D FOR THE UNIVERSITY
Doctor
Gary A. Berg
Date 1/7/2013
copyright
2013
Katrina Topolinski
ALL RIGHTS
RESERVED
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Title ofItem:TheMathematicsofHorses
3 to 5 keywords or p h r a s e s to d e s c r i b e t h eitem:ThesisforMastersofMathematics
Authors
N a m e(Print):KatrinaTopolinksi
Authors
Signature
Date: 1/11/2013
This is a permitted, modified version of the Non- exclusive Distribution
License from M I T Libraries and the University of Kansas
A c k n o w l e d g e m e n t s
I w o u l d l i k e t o t h a n k m y a d v i s o r ,DoctorI v o n a G r z e g o r c z y k , f o r e n c o u r a g i n g m e t o
study something I w a s passionate about, spending m a n y hours helping m e study, and
editing m y paper.
I could not have done this without her help and support.
I would also
l i k e t o t h a n kDoctorG r e g o r y W o o d f o r t a k i n g t h e t i m e t o r e a d a n d e d i t m y t h e s i s .
I a m so
appreciative to T a m m y Terzian for spending countless hours being m y thesis buddy,
helping m e edit curves in M a p l e , editing sections of m y thesis, and reassuring m e w h e n I
doubted things.
I w o u l d also like to thank m y husband, Jacob Topolinski, m y parents,
Terry and Will H a m m e r , and m y in- laws, Sheri and R o n Topolinski for encouraging,
supporting, and helping m e along the way.
letting m e u s e her horses in this study.
I owe M o r g a n Gillaspy a big thank you for
Also, t h a n k y o u to e v e r y o n e in the M a t h e m a t i c s
department for everything they have done for m e throughout m y journey to acquire m y
degree.
A b s t r a c t
T h e r e are m a n y c o m p o n e n t s in the m o v e m e n t of horses that can b e mathematically
modeled. In this research w e h a v e classified footprint patterns of various strides and f o u n d
an analytical f a m i l y of c u r v e s d e s c r i b i n g s i m p l e j u m p s . T h i s a p p r o a c h is d i f f e r e n t f r o m t h e
standard
center
of mass
method
typically
used
in
physics,
as
it is
more
visual
practically useful. W e will s h o w curve fitting for four different horses and the
and
equations
for the curves describing the j u m p s . W e standardize the curves as quartics with prescribed
inflection points that contain basic information about each horse. W e
will
discuss
the
properties of this family of functions and various mathematical m e t h o d s of c o m p a r i n g the
horses.
T a b l e
o f
C o n t e n t s
C h a p t e r 1- H o r s e s
1
C h a p t e r 2- Strip Patterns
5
Chapter 3- Physics Model
13
Chapter 4- J u m p
17
Curves
Chapter 5- C o m p a r i s o n of Trajectories
24
Chapter 6- Analyzing Space of Trajectories of Horse Jumps
28
Chapter 7- Family of Curves and Future Research
33
Appendix
A
36
Appendix
B
38
Appendix C
41
Appendix D
44
Bibliography
47
C h a p t e r
1 -
H o r s e s
In this study w e used seven horses to e x a m i n e their m o v e m e n t s both o n the ground
and over obstacles.
W e m e a s u r e d the horses' heights, lengths of front and hind legs,
g i r t h s , h o r i z o n t a l l e n g t h s , a n d l e n g t h b e t w e e n t h e i r l e g s . F i g u r e 1point1 s h o w s h o w t h e s e
measurements were taken.
F i g u r e 1point1 H o r s e m e a s u r e m e n t s
Carmelita
C a r m e l i t a is a t e n - y e a r - o l d C a m a r i l l o W h i t e H o r s e .
Height:
5 6 inches
Front leg:
2 9 inches
H i n d leg:
3 7 inches
Girth:
6 6 inches
Length:
6 0 inches
L e n g t h b e t w e e n legs:
3 1 inches
1
Coyote
C o y o t e is a n e l e v e n - y e a r - old A m e r i c a n Q u a r t e r H o r s e .
Height:
5 8 inches
Front leg:
3 1 inches
H i n d leg:
3 7 inches
Girth:
7 1 inches
Length:
6 0 inches
L e n g t h b e t w e e n legs:
3 3 inches
Hollywood
H o l l y w o o d is a t h r e e - y e a r - o l d A m e r i c a n Q u a r t e r H o r s e .
Height:
5 6 inches
Front leg:
2 9 inches
H i n d leg:
3 9 inches
Girth:
6 9 inches
Length:
5 9 inches
L e n g t h b e t w e e n legs:
3 2 inches
2
Honey
H o n e y is a n e i g h t - y e a r - o l d A m e r i c a n P a i n t H o r s e .
Height:
5 9 inches
Front leg:
3 0 inches
H i n d leg:
4 0 inches
Girth:
7 1 inches
Length:
6 6 inches
L e n g t h b e t w e e n legs:
3 4 inches
Patriot
Patriot is a s i x - m o n t h - o l d C a m a r i l l o W h i t e h o r s e .
Height:
5 3 inches
Front leg:
3 2 inches
H i n d leg:
3 8 inches
Girth:
5 3 inches
Length:
5 3 inches
L e n g t h b e t w e e n legs:
2 7 inches
3
Poker
P o k e r is a t h r e e - y e a r - o l d A m e r i c a n M i n i a t u r e H o r s e .
Height:
3 2 inches
Front leg:
1 6 inches
H i n d leg:
1 6 inches
Girth:
4 0 inches
Length:
3 1 inches
L e n g t h b e t w e e n legs:
1 7 inches
Roxie
R o x i e is a p r e g n a n t s e v e n - y e a r - o l d A m e r i c a n Q u a r t e r H o r s e .
Height:
6 0 inches
Front leg:
3 3 inches
H i n d leg:
4 1 inches
Girth:
7 2 inches
Length:
6 4 inches
L e n g t h b e t w e e n legs:
3 6 inches
4
C h a p t e r
2point1 - S t r i p
2 - S t r i p
P a t t e r n s
Patterns
T h e t e r m strip patterns refers to patters that exhibit s y m m e t r i e s a n d are b o u n d e d b y
t w o parallel lines.
The possible symmetries of these patterns consist of translations (Figure
2point1 ) , r o t a t i o n s ( F i g u r e 2point2 ) , r e f l e c t i o n s ( F i g u r e 2point3 ) , a n d g l i d e r e f l e c t i o n s ( F i g u r e 2point4 ) .
E a c h pattern h a s a " p r e - i m a g e " w h i c h is translated, rotated, reflected, a n d / or g l i d e r e f l e c t e d
o n t o itself.
T h e patterns continue infinitely and w e usually display t h e m in a horizontal
direction.
F i g u r e 2point1 Strip pattern obtained by translation of the L s h a p e d motif.
T o g e n e r a t e a strip p a t t e r n it is e n o u g h t o d e f i n e a s y m m e t r y ( r u l e ) t h a t t r a n s f o r m s a
given motif to the next one.
F i g u r e 2point2 s h o w s g e n e r a t i n g p a t t e r n s b y r o t a t i o n s .
Rotated 90 degrees
clockwise
Rotated 90 degrees
counterclockwise
Rotated 180 degrees
F i g u r e 2point2 T h e three d i f f e r e n t types of rotations that can b e u s e d to generate strip patterns.
5
F i g u r e 2point3 s h o w s h o w r e f l e c t i o n s c a n g e n e r a t e p a t t e r n s .
and 1m for examples;
S e e p a g e 7 a n d 8, m l , m 2,
[G].
F i g u r e 2point3.H o r i z o n t a l and vertical r e f l e c t i o n s can b e u s e d to m a k e strip patterns.
A g l i d e r e f l e c t i o n t r a n s l a t e s a n d r e f l e c t s a m o t i f t o g e n e r a t e p a t t e r n a s i n F i g u r e 2point4 .
S e e p a g e s 7 a n d 8, g 1 a n d m g f o r e x a m p l e s , [ G ] .
F i g u r e 2point4.A glide reflection is m a d e w i t h a translation f o l l o w e d b y a reflection.
6
E a c h strip pattern h a s a specific classification b a s e d o n their s y m m e t r i e s .
u s e t h e f o l l o w i n g chart t o classify a n y strip p a t t e r n b y its s y m m e t r y g r o u p .
We
can
For a
c o m p r e h e n s i v e study of strip patterns, see [G].
A s w e see, there are only seven t y p e s of patterns as classified b y the s y m m e t r y
groups.
T h e following e x a m p l e s are visual representations of these patterns generalized by
footsteps.
7
2point2 - H o r s e
Gaits
M o s t horses h a v e four gaits that they travel.
have m o r e gaits and these are called "gaited horses".
the horse and each has a different rhythm.
There are several horse breeds that
These four gaits c o m e naturally to
Every time one or m o r e of a horse's feet
8
t o u c h e s t h e g r o u n d it is c o n s i d e r e d a b e a t .
These beats create the different rhythms of each
gait. W h e t h e r they are m o v i n g quickly in the gait or slowly in the gait, their feet f o l l o w
the s a m e pattern a n d r h y t h m . W e a n a l y z e d patterns m a d e b y h o r s e s in the typical gaits.
T h e first a n d s l o w e s t g a i t is t h e w a l k .
T h e w a l k is a f o u r b e a t gait. A h o r s e w i l l
a l w a y s h a v e t w o or three feet o n the g r o u n d at a n y g i v e n time.
F i g u r e 2point5 s h o w s t h e o r d e r
that the legs take a step and h o w the four beats of the w a l k are created.
(first
beat) right hind
leg
(second b e a t ) r i g h t f o r e
(third b e a t ) l e f t h i n d
(fourth b e a t ) l e f t f o r e
leg
leg
leg
Figure 2point5.Steps of a w a l k i n g horse
W e photographed walks of several horses to analyze their patterns.
This picture s h o w s the h o o f prints of Patriot at the w a l k .
This picture has a sketch edited in over w h e r e e a c h print is in the sand.
This picture s h o w s the h o o f prints of R o x i e at the walk.
9
This picture has a sketch edited in over w h e r e e a c h print is in the sand.
Figure 2point6.H o o f prints m a d e of t w o h o r s e s w a l k i n g
T h i s strip pattern h a s n o vertical reflection, n o horizontal reflection, a n d n o
d e g r e e r o t a t i o n , b u t h a s a g l i d e r e f l e c t i o n s o t h a t m a k e s it a g 1 s t r i p p a t t e r n .
turns out to be typical for walks.
180
This pattern
Please see A p p e n d i x A for m o r e pictures of walking hoof
prints.
T h e s e c o n d g a i t t h a t w e s t u d i e d is t h e t r o t . T h e trot is a t w o b e a t gait, t h e d i a g o n a l
l e g s m o v e t o g e t h e r a n d t h e r e is a m o m e n t o f s u s p e n s i o n w h e n t h e h o r s e is s w i t c h i n g
diagonals. First the front left and hind right m o v e f o r w a r d and then the front right and
hind left m o v e forward.
F i g u r e 2point7 s h o w s h o w t h e l e g s m o v e t o g e t h e r t o c r e a t e t w o b e a t s .
(first b e a t ) l e f t f o r e / r i g h t
(second b e a t ) r i g h t f o r e / l e f t
F i g u r e 2point7.Steps of a trotting horse
This picture s h o w s the h o o f prints of C a r m e l i t a at the trot.
10
hind
hind
T h i s picture h a s a sketch edited in over the h o o f prints in the sand.
F i g u r e 2point8.H o o f prints of a horse trotting
This strip pattern h a s n o vertical reflection, n o horizontal reflection, a n d n o
d e g r e e r o t a t i o n , b u t h a s a g l i d e r e f l e c t i o n s o t h a t m a k e s it a g 1 s t r i p p a t t e r n .
analyzing different trots of various horses w e got the s a m e pattern.
180
Again, when
Please see A p p e n d i x
A
f o r m o r e p i c t u r e s of trot h o o f p r i n t s .
T h e last g a i t w e s t u d i e d w a s t h e c a n t e r , w h i c h is a t h r e e b e a t gait.
O n e hind leg
c o n t a c t s t h e g r o u n d first, f o l l o w e d b y t h e o t h e r h i n d leg a n d t h e d i a g o n a l f r o n t leg l a n d i n g
at t h e s a m e t i m e , a n d last t h e o t h e r f r o n t l e g l a n d s . T h e last f r o n t l e g t o l a n d is r e f e r r e d t o
as the " l e a d " leg.
T h e r e is a m o m e n t o f s u s p e n s i o n b e t w e e n e a c h stride o f t h e c a n t e r .
F i g u r e 2point9 s h o w s h o w t h e l e g s m o v e a t a c a n t e r o n t h e l e f t l e a d .
The horses were on the
left lead for our strip patterns.
(first b e a t ) r i g h t h i n d
leg
(second b e a t ) l e f t h i n d / r i g h t
(third b e a t ) l e f t f o r e
F i g u r e 2point9.Steps of a cantering horse
This picture s h o w s the hoof prints of C a r m e l i t a at the canter.
11
leg
fore
This picture h a s a sketch edited over the prints in the sand.
F i g u r e 2point10.H o o f prints of a h o r s e cantering
T h i s p a t t e r n c a n b e s k e t c h e d as:
Picture shows a pattern represented by two pairs of rightward pointing arrows.
T h e r e is n o v e r t i c a l r e f l e c t i o n , n o h o r i z o n t a l r e f l e c t i o n , n o g l i d e r e f l e c t i o n , a n d n o
1 8 0 - d e g r e e r o t a t i o n s o t h a t m a k e s it a 1 1 t y p e t h a t a l l o w s t r a n s l a t i o n s o n l y .
includes m o r e canter strip patterns.
12
Appendix
A
C h a p t e r
3 -
P h y s i c s
m o d e l
In this c h a p t e r w e e x p l a i n t h a t h o w o u r s t u d y is d i f f e r e n t f r o m t h e u s u a l a p p r o a c h
of analyzing the actual center of m a s s m o v e m e n t during horse jumping.
W h e n a h o r s e is s t a n d i n g still its c e n t e r o f m a s s is l o c a t e d t o w a r d t h e f r o n t o f t h e
body, slightly above and behind their elbow.
T o find the center of m a s s w e n e e d to d r a w
t w o lines on the horse and their intersection gives u s the center of mass.
T h e first line is
d r a w n f r o m t h e withers, w h i c h is t h e h i g h e s t p o i n t of t h e h o r s e at t h e b a s e of their n e c k ,
straight d o w n to the ground.
T h e s e c o n d line is d r a w n f r o m farthest p o i n t f o r w a r d , w h i c h
is t h e p o i n t of their shoulder, t o t h e farthest p o i n t b a c k , w h i c h is t h e p o i n t of t h e b u t t o c k .
[W]
T h e s t a t i o n a r y c e n t e r o f m a s s i s l a b e l e d C i n F i g u r e 3point1.
F i g u r e 3point1 F i n d i n g t h e center of m a s s
In our study, w e m a r k e d the point representing the stationary center of m a s s of a
h o r s e and tried to describe the m o v e m e n t of the m a r k in h o p e that this w o u l d give u s a
g o o d u n d e r s t a n d i n g of the m o v e m e n t .
For practical purposes the usual physical model for
j u m p i n g that r e d u c e s t h e o b j e c t to a center of m a s s is n o t v e r y u s e f u l f o r p r e d i c t i n g a n
animal's position and behavior.
A s t h e h o r s e j u m p s , its c e n t e r o f m a s s m o v e s a r o u n d
13
w i t h i n t h e h o r s e ' s b o d y d e p e n d i n g o n t h e p o s i t i o n o f its extremities.
During a j u m p to
c l e a r a n o b s t a c l e t h e h o r s e m o v e s its n e c k a n d h e a d f o r w a r d , w h i c h c a u s e s t h e c e n t e r of
m a s s t o m o v e t o t h e f r o n t , a n d it b e n d s its l e g s u p , w h i c h c a u s e s t h e c e n t e r of m a s s t o
m o v e u p as well.
body.
Theses adjustments change the position of the center of mass within the
H o w e v e r , w h i l e t h e h o r s e is in t h e air, t h e r e a r e n o a d d i t i o n a l f o r c e s a c t i n g o n t h e
system besides gravity.
Therefore, according to laws of physics, the center of m a s s will
m o v e along a parabola.
T h e ideal case of m o t i o n of a small projectile (or a center of m a s s
of a larger o b j e c t ) in a u n i f o r m gravitational field is v e r y w e l l u n d e r s t o o d a n d w a s
d e s c r i b e d b y Galileo, Torricelli, a n d N e w t o n a n d its s i m p l i f i e d p a r a b o l i c t r a j e c t o r y p r o v e s
essentially correct.
S i n c e a h o r s e is quite h e a v y a n d n o t m o v i n g v e r y fast, the air resistance
can be neglected.
It is n o t h a r d t o d e r i v e t h e e q u a t i o n of t h e p a r a b o l a d e s c r i b i n g t h e m o t i o n of t h e
center of m a s s of a j u m p i n g horse.
T h e x- axis is parallel t o t h e g r o u n d a n d t h e y- axis
perpendicular to the g r o u n d and parallel to the gravitational field lines. W e n e e d to
m e a s u r e t h e i n i t i a l h o r i z o n t a l s p e e d b evsubhequalsvsub0cosparenthesisthetaparenthesisa n d t h e i n i t i a l v e r t i c a l s p e e d b e
vsubvequalsvsub0sinparenthesisthetaparenthesis,s e e f i g u r e 3point2
F i g u r e 3point2.P a r a b o l i c m o t i o n of an object.
U s i n g trigonometry and physics laws of m o t i o n in a u n i f o r m gravitational field and
combining horizontal and vertical m o v e m e n t s , w e can derive the algebraic equation for the
p a r a b o l i c t r a j e c t o r y i n t h e f o r m yequalsa xsquaredplusb xplusc .
14
W e c a n u s e t h e f o r c e o f g r a v i t yrightar owaboveFsubgequalsnegativem gcaratovery a n d N e w t o n ' s S e c o n d L a wrightarrowabove
F
equals
mrightaowbvea t o
rightarrowaboveaequals0cartbovexminusgcartbovey .
W e c a n a s s u m e t h e acceleration is c o n s t a n t so w e get t h e f u n c t i o n
vparenthesistparenthesisequalsvsub0plusa t ,
t a k i n g t h e i n t e g r a l of it w e f i n d
t above integral symbol above 0 v parenthesis t prime parenthesis d t prime
equals t above integral symbol above 0 parenthesis v sub 0 plus a t prime
parenthesis
d
t
prime
which
xparenthesistparenthesisminusxsub0equalsvsubhtplusone-halfa t squared.
T h e r e f o r e , o u r c o n s t a n t acceleration f u n c t i o n is
x parenthesis t parenthesis equals x sub 0 plus v sub h t plus one- half a sub x t squared.
W e k n o w asubxequals0 s o o u r f u n c t i o n w i l l b e c o m e
xparenthesistparenthesisequalsxsub0plusvsubht.
Similarly, w e f i n d t h a t o u r y f u n c t i o n is
y parenthesis t parenthesis equals y sub 0 plus v sub
v t minus one- half g t squared.
L e t ' s s e txsub0equals0 a n d ysub0equals0 t o p u t t h e o r i g i n a t t h e b e g i n n i n g o f o u r p a r a b o l a .
If w e solve
f o r t i n xparentheistparenthesisw e o b t a i n
t equals x parenthesis t parenthesis divided by v sub h
S u b s t i t u t i n g t h a t i n t o yparenthesi tparenthesisw e f i n d
y parenthesis t parenthesis equals 0 plus v sub v parenthesis x parenthesis t parenthesis divided by v sub h parenthesis minus one- half
g parenthesis x parenthesis t parenthesis divided by v sub h parenthesis squared.
C l e a n i n g i t u p b y d r o p p i n g t h eparenthesistparenthesisw e f i n d t h a t t h e e q u a t i o n o f o u r p a r a b o l a i s
y equals v sub v divided by v sub h times x minus g divided by 2 v sub h squared times x squared.
H e n c e , y i s i n t h e f o r m yequalsb xminusa xsquaredw h e r e agreaterthan0 a n d
greaterthan0 .
15
b
F i g u r e 3point3.T h e s c h e m a t i c s h o w i n g a h o r s e j u m p i n g
If a h o r s e j u m p s optimally a n d w i t h g o o d t i m i n g t h e n the vertex of the p a r a b o l a
should b e directly over the obstacle.
S i n c e it is h a r d t o c o n t r o l t h e c e n t e r o f m a s s w i t h r e s p e c t t o t h e p o s i t i o n o f t h e
horse's body, the parabolic function does not provide a practical description of j u m p s that
could s o m e h o w help to predict the actual position of the horse.
the c e n t e r of m a s s is n o t h e l p f u l f o r a rider.
Therefore, trying to track
In our study w e m a r k a specific, easily
i d e n t i f i a b l e p o i n t o n t h e h o r s e a n d study its trajectories.
16
C h a p t e r
4
J u m p
c u r v e s
In o u r study, w e m a r k e d t h e stationary c e n t e r o f m a s s f o r e a c h h o r s e t o o b s e r v e its
trajectory w h e n a horse was jumping.
W e marked the point representing the center of
mass of a standing horse with colored tape and photographed each j u m p to create accurate
graphs. The horses were videotaped from the center of the arena with a camera on a tripod
to keep consistency.
W e h a d f o u r h o r s e s j u m p a 9 - i n c h j u m p a n d a n 18 i n c h j u m p ; w e h a d
t w o of those horses also j u m p a 24 inch jump.
O n c e t h e v i d e o s w e r e c o m p l e t e d , still i m a g e s w e r e t a k e n f r o m t h e v i d e o a n d p i e c e d
t o g e t h e r t o s e e t h e h o r s e ' s m o v e m e n t f l u i d l y a s r e p r e s e n t e d b y F i g u r e 4point1.
F i g u r e 4point1.Still i m a g e s f r o m v i d e o p u t t o g e t h e r
W e a d d e d a n x- a n d y- axis a n d a grid t o h e l p f i n d t h e c o o r d i n a t e s o f t h e c e n t e r o f
mass.
E a c h d a r k e r line o n the axis is c o n s i d e r e d t o b e t w o units, so t h e m e s h h a s
increments of 1 unit.
A pink point w a s put on the intersection of the taped X on the horse
t o b e t t e r s e e w h e r e it l a n d e d o n t h e g r a p h .
17
F i g u r e 4point2.H o r s e j u m p i n g w i t h point and grids a d d e d
W e m a d e several initial o b s e r v a t i o n s a b o u t t h e trajectories to f i g u r e out w h a t w o u l d
b e the best m e t h o d for curve fitting. W e noticed that the sections of the curves w h i l e the
horses w e r e over the j u m p are c o n c a v e down.
W e also observed that the curves w e r e
concave u p after the horse landed, so there w e r e inflection points b e t w e e n the peaks of the
j u m p s and landings.
W e also s a w that the slope is steeper w h e n t h e h o r s e is l a n d i n g t h a n
w h e n t h e h o r s e is t a k i n g off.
F i g u r e 4point3.Z o o m e d in i m a g e of f i g u r e 4point2 to better see the c u r v e
18
O u r trajectories of the m a r k e d point could not be represented by lines because the
curves h a d various slopes in our graphs; and horses cannot travel forever u p or d o w n .
Our
g r a p h s c o u l d also n o t b e a q u a d r a t i c c u r v e b e c a u s e o u r g r a p h is n o t s y m m e t r i c a n d t h e r e
are t w o d i f f e r e n t c o n c a v i t i e s in o u r g r a p h w h i l e t h e r e is o n l y o n e in a q u a d r a t i c curve.
U s i n g s o f t w a r e w e t h e n t r i e d f i t t i n g o u r t r a j e c t o r i e s i n t o a c u b i c c u r v e t o s e e i f it
w o u l d b e a g o o d fit a n d m a d e the f o l l o w i n g observation.
T h e o r e m : A cubic f u n c t i o n is n o t a g o o df i t curve f o r h o r s e j u m p i n g .
P r o o f : C o n s i d e r g e n e r a l c u b i c w i t h t h e e q u a t i o n yequalsa xcubedplusb xsquaredpluscxplusd , d i v i d i n g
e v e r y t h i n g b y a w e c a n s i m p l i f y t h e e q u a t i o n yequalsxcubedplusb xsquaredplusc xplusd .
xequalsxminusbdiv dedby3 w e g e t a c u b i c e q u a t i o n w i t h o u t a q u a d r a t i c t e r m .
yequalsxcubedpluscxplusd , w h e r e c a n d d a r e n e w c o n s t a n t s .
Substituting
W e d e n o t e it a g a i n b y
S h i f t i n g o u r c u r v e v e r t i c a l l y b ynegatived
we
o b t a i n t h e e q u a t i o n yequalsxcubedplusc xequalsxparenthesi xsquaredpluscparenthesisw h i c h h a s e x a c t l y t h e s a m e s h a p e a s o u r
o r i g i n a l c u r v e . N o w w e s t u d y it i n m o r e detail.
O u r r o o t s a r enegativesquarerootofnegativeccomma0commasquarerootofnegativec
Notice
c
F i g u r e 4point4.A c u b i c w i t h rootsnegativesquarerootofnegativeccomma0commasquarerootofnegativec
N o w b y taking the first derivative w e find the m a x i m a and m i n i m a .
y prime equals 3 x squared plus c
0 equals 3 x squared plus c
19
W e f i n d t h a t o u r m a x i m u m i sxequals quarerootofnegativecdividedby3a n d o u r m i n i m u m i s xequalsnegativesquarerootof
negative
c
divided
by
3.
Rewriting this
0point57squarerootofnegativec.C o m b i n i n g t h i s i n f o r m a t i o n , w e c a n s e e t h a t o u r m a x i m u m
and m i n i m u m are closer to our non- zero roots than to our zero root.
This means the curve
i s s t e e p e r o n t h e i n t e r v a lsquarebracketnegativesquarerootofnegativeccommanegative0point57squarerootofnegativec
square
bracket
t h a n t h e i n t e r v a lsquarebracketnegative0point5 7squarerootofnegativeccomma0squarebracket.
Recall that our horse j u m p i n g trajectories showed the slope to be steeper on the
landing side (the side to the right of the m a x i m u m ) .
Therefore, a cubic function would not
a p p r o x i m a t e o u r g r a p h s w e l l b e c a u s e it is s t e e p e r t o t h e l e f t o f t h e m a x i m u m .
T o c o n f i r m this, w e tried experimentally fitting our trajectories u s i n g cubic curves
a n d the fit p a r a m e t e r s w e r e not g o o d e n o u g h .
For example, Excel gives the equation of the
b e s t f i t c u r v e a n d t h e Rsquaredv a l u e , w h i c h i s t h e c o r r e l a t i o n c o e f f i c i e n t a n d i t t e l l s u s h o w g o o d
o u r f i t i s . W h e n Rsquaredi s c l o s e t o 1 w e h a v e a v e r y g o o d f i t , Rsquaredequals1 w o u l d b e a p e r f e c t f i t .
F i g u r e 4point5 a n d 4point6 s h o w s h o w o u r c u r v e s l o o k i n a c u b i c c u r v e a n d t h e Rsquaredv a l u e s a r e n o t
close e n o u g h to 1 to b e an accurate fit for our trajectories.
F i g u r e 4point5.C o y o t e 2 4 inches as a cubic
T h e c o r r e l a t i o n e r r o r is g r e a t e r
F i g u r e 4point6.H o l l y w o o d 2 4 inches as a c u b i c
Therefore, w e e x a m i n e d quartic curves, did experimental fitting, and noticed that
o u r a p p r o x i m a t i o n w o u l d b e m u c h b e t t e r w i t h a q u a r t i c c u r v e w i t h Rsquaredv e r y c l o s e t o 1.
u s e d t h e p o i n t s i n t h e h o r s e -j u m p i n g p i c t u r e a n d p l o t t e d t h e m o n a g r a p h i n E x c e l .
see A p p e n d i x B to see m o r e pictures and graphs.
20
We
Please
F i g u r e 4point7.C o y o t e 2 4 inches as a quartic c u r v e
N o t i c e that there are t w o inflection points w h e r e the graph changes; first f r o m
c o n c a v e u p t o c o n c a v e d o w n , a n d s e c o n d w h e r e it c h a n g e s f r o m c o n c a v e d o w n t o c o n c a v e
up.
These curves are similar to our trajectories.
W e also plotted t h e points o n t h e c u r v e fitting website, [C].
t h e r e d u c e d c h i s q u a r e d s t a t i s t i c w h i c h t e l l s h o w g o o d t h e f i t is.
This website calculates
T h e e q u a t i o n is:
chi squared r equals 1 divided by N minus f sigma above i square bracket y parenthesis x sub i parenthesis minus y sub i square
bracketsquareddividedbysigmasquaredsubiw h e r e N i s t h e n u m b e r o f d a t a p o i n t s a n d
f is t h e n u m b e r of
p a r a m e t e r s i n t h e f i t . I fchisquaredrgreaterthan1 t h e n t h e c u r v e i s a p o o r f i t , i fchisquaredrap roximatelyequalto1 t h e n t h e c u r v e i s a g o o d
f i t , a n d i fchisquaredrlessthan1 t h e n t h e f i t i s v e r y g o o d .
21!
F i g u r e 4point8.C o y o t e 2 4 inches as a quartic c u r v e
F i g u r e 4point8 s h o w s t h e c u r v e o f C o y o t e j u m p i n g a 2 4 - i n c h j u m p .
W e find that the
c u r v e f i t t i n g w e b s i t e g i v e s u s c l o s e t o t h e s a m e c u r v e a s E x c e l a n d t h a tchisquaredequals0point2 9 s o t h i s
c u r v e is a g o o d fit.
W e fit t h e c u r v e s f o r e v e r y h o r s e at e a c h size j u m p a n d f o u n d an e q u a t i o n that
described each curve.
T a b l e 4point9 s h o w s t h e c o e f f i c i e n t s f o r t h e e q u a t i o n s i n t h e f o r m :
yequalsa xsuperscript4plusb xcubedpluscxsquaredplusd xpluse .
A n a l y z i n g the table, w e w e r e able to m a k e
simplifications described in the f o l l o w i n g chapters.
close to zero.
22
some
N o t e that b is a l w a y s n e g a t i v e a n d c is
Horse
Jump
(in)
size
a equals
sign
b equals
sign
c equals
sign
d equals
sign
e equals
sign
Coyote
9
0point0 0 2 4
negative 0point0 6 8 3
0point6 2 1 2
negative 1point9 4 4 1
6point4 1 0 9
Coyote
18
0point0 0 0 4
negative 0point0 1 0 5
0point0 6 0 9
0point1 3 7 8
4point4 8 3 5
Coyote
24
0point0 0 0 6
negative 0point0 1 7
0point1 2 6 1
negative 0point0 9 7
4point7 1 6 8
Hollywood
9
0point0 0 0 3
negative 0point0 0 5 7
negative 0point0 0 3 9
0point2 7 0 1
4point5 3 8 1
Hollywood
18
0point0 0 0 5
negative 0point0 1 3 2
0point0 9 1 7
0point0 0 3 2
4point4 0 5 8
Hollywood
24
0point0 0 0 2
negative 0point0 0 7 7
0point0 5 1
0point2 2 4 9
4point4 0 3 7
Honey
9
0point0 0 0 2
negative 0point0 0 5 9
0point0 2 1 7
0point1 8 7 9
4point7 2 7 1
Honey
18
0point0 0 0 4
negative 0point0 1 0 6
0point0 6 5 9
0point0 4 5 3
4point9 9 8 2
Poker
9
0point0 0 4 8
negative 0point0 9 1 4
0point4 8 7 6
negative 0point5 4 5 1
3point3 4 2 6
Poker
18
0point0 0 7 1
negative 0point1 7 5 1
1point4 1 5 6
negative 4point2 1 3 7
7point0 7 4 2
F i g u r e 4point9.C o e f f i c i e n t s f o r t h e e q u a t i o n s of o u r curves
23
C h a p t e r
5
C o m p a r i s o n
o f
T r a j e c t o r i e s
In M a p l e , w e rescaled a n d shifted our curves so that the inflection points w e r e at
xequals0 a n d xequals1. T o d o t h i s , w e f i r s t f o u n d o u r i n f l e c t i o n p o i n t s a n d s h i f t e d t h e g r a p h s o t h e
f i r s t i n f l e c t i o n p o i n t i s a t xequals0 . T h e p o i n t sparenthesis0commafparenthsi0parenthesisparenthesisr e p r e s e n t t h e m o m e n t s t h e
horses take
points.
T h i s m a d e o u r s e c o n d i n f l e c t i o n p o i n t a t xequals1. T h e p o i n t sparenthesis1commafparenthesi 1parenthesisparenthesisr e p r e s e n t w h e n
the horses are landing.
This change helped us to compare the curves of different horses
and of different sized j u m p s .
F i g u r e 5point1 s h o w s a l l t h e c u r v e s o f t h e j u m p s t o g e t h e r a f t e r
being shifted.
F r o m a P h y s i c s p o i n t o f v i e w , it is i n t e r e s t i n g t o n o t i c e t h a t t h e a c c e l e r a t i o n is
d e c r e a s i n g f r o mparenthesis0commafparenthesi 0parenthesisparenthesist o o u r l o c a l m a x i m u m .
T h e a c c e l e r a t i o n is i n c r e a s i n g f r o m o u r
l o c a l m a x i m u m t oparenthesis1commafparenthesi 1parenthesisparenthesis.
F i g u r e 5point1.R e s c a l e d curves of all h o r s e s j u m p i n g all h e i g h t s
Notice that curve hw 3 and h 1 are not like the other curves.
Curve hw 3 was a curve
of a horse that j u m p e d further than the rest of the horses, therefore the last data points w e r e
24
w h e n h e w a s still l a n d i n g a n d h a d n ' t r e c o v e r e d h i s n o r m a l s t r i d e y e t .
The data points that
w e r e outside the c a m e r a f r a m e w o u l d h a v e b e e n higher than our last data point and w o u l d
have m a d e the curve increase.
than the others.
C u r v e h 1 is also d i f f e r e n t b e c a u s e its m i n i m u m is l o w e r
This curve w a s for H o n e y j u m p i n g a 9-inch j u m p and she did not j u m p
v e r y h i g h o v e r the j u m p , w h i c h m a k e s o u r c u r v e s h a l l o w e r t h a n t h e rest.
F i g u r e 5point2 s h o w s t h e c u r v e s m a d e b y C o y o t e j u m p i n g .
It is i n t e r e s t i n g t o n o t i c e t h a t t h e
l o c a l m a x i m a a r e all at a b o u t t h e s a m e x t e r m w h i c h is w h a t w e w o u l d e x p e c t a s t h i s w o u l d
occur over the jump.
T h e y values vary because she w a s j u m p i n g different sized jumps.
F i g u r e 5point2.R e s c a l e d c u r v e s of C o y o t e j u m p i n g three d i f f e r e n t h e i g h t s
F i g u r e 5point3 s h o w s a l l f o u r h o r s e s ' c u r v e s o v e r t h e 1 8 - i n c h j u m p .
i n F i g u r e 5point3 i s P o k e r w h o w a s t h e s h o r t e s t h o r s e .
All of the curves have about the s a m e
s h a p e a n d their m a x i m a are at a b o u t t h e s a m e x value.
25
The lowest curve
F i g u r e 5point3.R e s c a l e d c u r v e s of f o u r h o r s e s j u m p i n g an 18 inch j u m p
F r o m these observations w e can see that the curves share several properties.
In the
n e x t c h a p t e r w e w i l l p r o v e t h a t t h e r e is a g e n e r a l " h o r s e " c u r v e t h a t s u m s u p all t h e
properties they share.
After rescaling and shifting each curve our original equations changed.
T a b l e 5point4
s h o w s t h e c o e f f i c i e n t s o f o u r n e w e q u a t i o n s o f t h e f o r m : yequalsa xsuperscript4plusb xcubedpluscxsquaredplusd x
plus
e.
26
This
Horse
J u m p
size
(in)
a equals
sign
b equals
sign
c equals
sign
d equals
sign
e equals
sign
Coyote
9
2point1 4 7 6
negative 4point2 9 5 2
0
2point0 4 2
2
4point9 5 7
3
Coyote
18
2point0 0 3 1
negative 4point0 0 6 2
0
2point2 7 8
3
5point0 2 1
3
Coyote
24
2point2 0 2 2
negative 4point4 0 4 4
0
2point0 7 3
9
5point2 0 1
3
Hollywood
9
2point9 3 5 4
negative 5point8 7 0 7
0
2point6 9 5
0
4point4 7 7
8
Hollywood
18
1point3 5 0 6
negative 2point7 0 1 2
0
1point8 0 9
4point9 2 3
6
Hollywood
24
8point0 4 5 1
negative 1 6point0 9 0 1
0
4point9 2 9
2
5point1 8 7
5
Honey
9
4point2 1 8 3
negative 8point4 3 6 6
0
2point6 0 5
4
5point0 0 6
3
Honey
18
1point7 2 8 1
negative 3point4 5 6 3
0
1point6 3 0
5point3 8 7
7
Poker
9
2point5 2 2 5
negative 5point0 4 5 0
0
2point3 0 5
7
3point7 2 2
5
Poker
18
2point5 9 9 1
negative 5point1 9 8 2
0
2point2 9 2
0
3point4 6 8
9
5
8
F i g u r e 5point4.C o e f f i c i e n t s f o r t h e e q u a t i o n s of o u r curves after shifting
F o r e x a m p l e , w e c a n s e e t h a t cequals0 a n d w e c a n c o n j e c t u r e t h a t bequalsnegative2 a .
o n t o p r o v e it i n t h e n e x t c h a p t e r .
27
W e will g o
C h a p t e r
6 - A n a l y z i n g
S p a c e
o f T r a j e c t o r i e s
o f H o r s e
J u m p s
In this chapter w e will study the family of quartic curves describing the trajectories
of j u m p i n g horses f r o m an abstract point of view.
In order to c o m p a r e theses trajectories
w e will try to standardize t h e m and describe their interesting properties b y f o r m u l a t i n g
several theorems.
T h e o r e m 6point1 : E v e r y t r a j e c t o r y o f a h o r s e ' s j u m p c a n b e c h a r a c t e r i z e d b y t h e q u a r t i c
f u n c t i o n given by a n equation o f the f o r m
fparenthesisxparenthesisequalsa xsuperscript4minus2 a xcubedplusd xplush
w h e r e a,
d, a n d h a r e p o s i t i v e a n d h is the h e i g h t o f the h o r s e .
P r o o f : In chapter 3 w e showed that polynomials of degree four give a very close
approximation of the trajectories.
Consider the general equation of a quartic curve given
by
fparenthesisxparenthesisequalsa xsuperscript4plusb xcubedplusc xsquaredplusd xpluse
w h e r e a , b , c , d , a n d e a r e r e a l n u m b e r s a n d agreaterthan0 .
T o b e a b l e t o c o m p a r e o u r h o r s e c u r v e s , w e r e s c a l e all o f t h e m s o t h a t t h e t w o
i n f l e c t i o n p o i n t s a r e a t xequals0 a n d xequals1 a s i n c h a p t e r 5 .
This w a y each horse begins his
j u m p w h e n xequals0 h e n c e e r e p r e s e n t s t h e h e i g h t o f t h e m a r k o n e a c h h o r s e , w h i c h c a n b e
e a s i l y m e a s u r e d a s i n c h a p t e r 1. T h e r e f o r e w e s u b s t i t u t e eequalsh a n d w e g e t
fparenthesisxparenthesisequalsa xsuperscript4plusb xcubedplusc xsquaredplusd xplush .
N o w observe that
fprimeparenthesi xparenthesisequals4 a xcubedplus3 b xsquaredplus2 cxplusd ,
and
fdoubleprimeparenthesi xparenthesisequals12a xsquaredplus6 b xplus2 c .
S i n c e w e h a v e r e s c a l e d , w e c a n s e tfdoubleprimeparenthesi 0parenthesisequals0
so,
fdoubleprimeparenthesi 0parenthesisequals1 2 aparenthesi 0parenthesissquaredplus6 bparenthesi 0parenthesisplus2 c
0equals1 2 aparenthesi 0parenthesissquaredplus6 bparenthesi 0parenthesisplus2 c
28
T h e r e f o r e , w e f i n d t h a t cequals0 ( c o m p a r e w i t h t h e t a b l e 5point4 ) . H e n c e , w e c a n w r i t e t h e
second derivative as follows:
fdoubleprimeparenthesi xparenthesisequals12a xsquaredplus6 b x equals
6 xparenthesi 2 a xplusb parenthesis.
N o t e t h a t s i n c e fdoubleprimeparenthesi 1parenthesisequals0 w e c a n c a l c u l a t e
fdoubleprimeequals6parenthesi 2 aplusb parenthesis
0equals6parenthesi 2 aplusb parenthesis
h e n c e bequalsnegative2 aparenthesisc o m p a r e w i t h o u r p r e d i c t i o n s f r o m t a b l e 5point4parenthesis.A f t e r w e s u b s t i t u t e b a c k
into the original function w e get the required f o r m
fparenthesisxparenthesisequalsa xsuperscript4minus2 a xcubedplusd xplush
L e m m a 6point2 : C o n s i d e r t h e e q u a t i o n f r o m T h e o r e m 6point1 o n t h e i n t e r v a lsquarebracket0comma1squarebracketd e s c r i b i n g t h e
horse curve.
T h e d e c r e a s i n g p a r t i s s t e e p e r t h a n t h e i n c r e a s i n g p a r t (i.e. t h e h o r s e i s
P r o o f : C o n s i d e r the equation describing the curve in question
fparenthesisxparenthesisequalsa xsuperscript4minus2 a xcubedplusd xplush
w i t h agreaterthan0 a n d t w o i n f l e c t i o n p o i n t s a t xequals0 a n d xequals1. W e
have
fparenthesi 0parenthesisequalsh
and
fparenthesi 1parenthesisequalsnegative2 aplusd xplush .
S i n c e fparenthsixparenthesisi s p o s i t i v e a t b o t h i n f l e c t i o n p o i n t s a n d hgreaterthan0 w e h a v enegative2 aplusdplushgreaterthan0
so
dplushgreaterthan2 a ( c o m p a r e w i t h t a b l e 5point4 a n d s e e i t w o r k s w i t h a l l e x c e p t o u r d i f f e r e n t c u r v e ,
H o l l y w o o d 24 inch).
Now
observe
fprimeparenthesi xparenthesisequals4 a xcubedminus6 aplusd
S i n c e t h e s l o p e a t xequals1 i s n e g a t i v e , w e o b t a i n t h e i n e q u a l i t y
fprimeparenthesi xparenthesisequals4 aminus6 aplusd
negative
2 aplusdlessthan0
T h e r e f o r e ,negative2 aplusdplushlessthanh w h i c h m e a n s o u r y c o o r d i n a t e a t xequals1 i s l o w e r t h a n o u r y
29
N o w w e c h e c k t o s e e if o u r c u r v e is a s c e n d i n g o r d e s c e n d i n g h a l f w a y b e t w e e n o u r
i n f l e c t i o n p o i n t s a t xequalsone-half.fprimeparenthesisone-halfparenthesisequals4 aparenthesi one-halfparenthesi cubedminus6 a
parenthesisone-halfparenthesissquaredplusd
plus
h.
fprimeparenthesi one-halfparenthesi equalsnegativeaplusdplush
R e c a l l dplushgreaterthan2 a a n d agreaterthan0 t h e n dplushgreaterthana .
Therefore
negative
aplusdplushgreaterthan0 .
O u r c u r v e i s s t i l l a s c e n d i n g w h i c h m e a n s o u r l o c a l m a x i m u m i s c l o s e r t o xequals1. S i n c e o u r
c u r v e i s l o w e r a t xequals1 a n d t h e l o c a l m a x i m u m i s c l o s e r t o
xequals1, t h e r i g h t s i d e o f t h e h o r s e
T h e o r e m 6point3 : T h e f a m i l y o f c u r v e s o f d e g r e e l e s s t h a n 5 w i t h fdoubleprimeparenthesi 0parenthesisequals0 a n d fdoubleprime
parenthesis
1
parenthesis
P r o o f : N o t e that our horse curves belong to
equals
Hsuperscriopt0comma1.H e n c e i t i s a g e n e r a l i z a t i o n o f t h e
T o s h o w Hsuperscipt0com a1i s a v e c t o r s p a c e w e n e e d t o p r o v e t h e f o l l o w i n g :
roman numeral 1: f plus g set membership symbol H superscript 0 comma 1
roman numeral 2: f minus g set membership symbol H superscript 0 comma 1
roman numeral 3: alpha times f set membership symbol H superscript 0 comma 1
f o r a n y t w ofunctionsfcommagsetmembershipHsuperscript0comma1anda n y r e a l n u m b e ralpha.
L e t fcommagsetmembershipsymbolHsuperscript0comma1t h e n w e h a v e :
romannumeral1:f plus gequalsparenthesisfplusg parenthesis
fdoubleprimeparenthesi 0parenthesisplusgdoubleprimeparenthsi0 parenthesis
equals
parenthesis
parenthesis
fdoubleprimeplusgdoubleprime
parenthesis
0parenthesisequals0
fdoubleprimeparenthesi 1parenthesisplusgdoubleprimeparenthesi 1parenthesisequalsparenthesisfdoubleprimeplusg
double
prime
parenthesis
parenthesis
H e n c e , f plus gsetmembershipsymbolHsuperscript0comma1
S i m i l a r l y ,roman umeral2:fminusgequalsparenthesisfminusg parenthesis
1parenthesisequals0
30
0
fdoubleprimeparenthesi 0parenthesisminusgdoubleprimeparenthesi 0parenthesisequalsparenthesisf
double
prime
minus
gdoubleprimeparenthesi parenthesi 0parenthesisequals0
fdoubleprimeparenthesi 1parenthesisminusgdoubleprimeparenthesis1parenthesisequalsparenthesisf
double
prime
minus
symbol
gdoubleprimeparenthesi parenthesi 1
parenthesis
Hsuperscript0comma1.A n d
times
equals
0.
H e n c e ,
f minus g
set
membership
o b v i o u s l yroman umeral3:parenthesi alphatimesfparenthesisdoubleprimeequalsparenthesisalpha
fprimeparenthesisprimeequalsalphatimesfdoubleprime.H e n c e ,alphatimesfsetmembershipsymbolHsuperscript0comma1 w i t halphaarealnumber.
C o n s i d e r
parenthesis
the B a n a c h
x
parenthesis
s p a c e
Csquarebracket0comma1squarebracketw i t h
vertical
bar
x
t h e n o r mdoubleverticalbarfdoubleverticalbarequalss u pcurlybraceverticalbarf
set
s u b s p a c e
membership
o f
symbol
Csquarebracket0comma1squarebracketb e c a u s e
square
bracket
0comma1
o u r p o l y n o m i a l
square
f u n c t i o n s are
differences inside
xparenthesisminusgparenthesisxparenthesisverticalbarf o r a l l
T h i s will
T h e o r e m
Hsuperscipt0com a1squarebracket0comma1squarebracketw i t h m e t r i c
o u r t w o
curves.
xparenthesi verticalbarf o r a l l
bar
fparenthesisxparenthesi verticalbarf o r a l l
xsetmembershipsymbolsquarebracket0comma1squarebracketi s a
s h o w
that
c o m p l e t e .
C o n s i d e r
right
c o m p l e t e
n o r m e d B a n a c h
Hsuperscript0comma1i s c l o s e d
a c o n v e r g i n g
arrow
in
s e q u e n c e
f
Csquarebracket0comma1squarebracket,a s b y
parenthesis
fdoubleprimec a n n o t b e
fdoubleprimew i t h
set
fdoubleprimeparenthesi 0parenthesisequals0
w h e r e f
membership
equals
0
a linear function.
a n d
fdoubleprimeparenthesis1parenthesisequals0
symbol
w e
c a n
c o m p a r e
H e n c e ,
the
(or the z e r o
H
case.
superscript
3 1
T h e o r e m
as
less t h a n
5 in
Csquarebracket0comma1squarebracketi s c o m p l e t e .
W e
Hsuperscript0comma1i s a c l o s e d
N o t e
that
T h e r e f o r e ,
0
vertical
2point3hyphen1 i nsquarebracketKsquarebracketi t w i l l
o f d e g r e e
subspace.
F o r
all
fdoubleprimesubna r e p o l y n o m i a l s
s e q u e n c e fdoubleprimesubnrightarrowfdoubleprimea l s o
function).
double
space.
is c o n t i n u o u s ,
in o u r
the
xsetmembershipsymbolsquarebracket0comma1squarebracketa n d n o r m
fsubno f p o l y n o m i a l s
i m p l i e s that
1
H e n c e
s u p fparenthesisxparenthesisminusg
parenthesis
vertical
Hsuperscript0comma1u s i n g
xsetmembershipsymbolsquarebracket0comma1squarebracket.
g i v e u s the largest distance b e t w e e n
6point4 :
bracket
c o n v e r g e s
to a
o f
p a r a b o l a
f
comma
1,
w h i c h
s h o w s
bar
f
do
C o n s i d e r Lsuperscipt2squarebracket0comma1squarebracketw i t h t h e n o r mdoubleverticalbarfdoubleverticalbarequalssquareroot
1 above integral symbol above 0 f squared d x. T h i s n o r m i n d u c e s t h e f o l l o w i n g m e t r i c
measuring distance between two curves
dparenthesi fcommagparenthesisequalsintegralsymbolsuperscript1sub0
parenthesisfminusgparenthesissquareddxa n d i s a s s o c i a t e d w i t h t h e i n n e r p r o d u c t1aboveintegralsymbolabove0ftimesgdx.
In this c a s e t h e d i s t a n c e is r e p r e s e n t e d b y t h e area b e t w e e n t h e t w o curves.
N o t e that by
s i m i l a r a r g u m e n t a s i n T h e o r e m 6point4 o u r v e c t o r s p a c e Hsuperscript0comma1i s a s u b s p a c e o f t h e H i l b e r t s p a c e
Lsquaredsquarebracket0comma1squarebracketw i t h t h e i n d u c e d i n n e r p r o d u c t a n d t h e m e t r i c .
Hence w e can compare our jumping
trajectories in t e r m s of differences of the areas b e t w e e n the t w o
curves.
T h e o r e m 6point5 : Hsuperscipt0com a1squarebracket0comma1squarebracketw i t h t h e m e t r i c dparenthesi f
comma
parenthesis equals 1 above integral symbol above 0 parenthesis fminusgparenthesi squaredd x i s a c o m p l e t e H i l b e r t s p a c e .
P r o o f : S i n c e all o u r c u r v e s a r e p o l y n o m i a l s o f d e g r e e l e s s t h a n 5, s i m i l a r l y t o t h e
superscript
3
0
comma
point
1
is c l o s e d v e c t o r s u b s p a c e of t h e
2hyphen4 i nsquarebracketKsquarebracketi t i s a c o m p l e t e s u b s p a c e , h e n c e
squarebracketi s c o m p l e t e a n d a H i l b e r t s p a c e .
32
g
C h a p t e r
7point1 - H e a d
7 - F a m i l y
o f
C u r v e s
a n d
F u t u r e
R e s e a r c h
curves
The study of the curves given by the m a r k of the center of mass of the horse gave
i n t e r e s t i n g r e s u l t s . N o w , it w o u l d b e i n t e r e s t i n g t o s e e if t h e r e a r e a n y o t h e r b o d y p a r t s t h a t
will give us interesting curves as well.
W e used the same video and edited pictures f r o m
Chapter 4 to e x a m i n e w h a t curves w e r e best fits w h e n w e f o c u s e d on other parts of the
horse's body.
First w e p u t a p o i n t o n t h e h o r s e s ' h e a d s at t h e b a s e s o f their ears.
W e then
added our axes and grid and f o u n d the coordinates of each point using the s a m e scale as
before. Putting these points in Excel, w e h a v e curves for each horse.
F i g u r e 7point1.P o k e r ' s h e a d c u r v e o v e r an 18- inch j u m p
33
F i g u r e 7point2.H o l l y w o o d ' s h e a d c u r v e o v e r a 24- inch j u m p
The curves of the head differ substantially between horses unlike our previous
c u r v e s , w h i c h a r e all p o s i t i v e q u a r t i c s .
F i g u r e 7point1 i s a q u a r t i c s i m i l a r t o o u r c e n t e r o f m a s s
c u r v e s b u t f i g u r e 7point2 o f a d i f f e r e n t h o r s e i s a c u b i c .
H o r s e s h a v e the ability to m o v e their
head vertically to shift their weight to balance while j u m p i n g obstacles.
Since each horse
is built differently t h e direction a n d a m o u n t t h e y n e e d t o m o v e their h e a d is different.
This
explains w h y our curves w o u l d b e so different. Please see A p p e n d i x D for m o r e graphs.
7point2.T a i l c u r v e s
Using the same video of our horses j u m p i n g w e reversed the w a y the pictures were
overlaid to see the horses' tails as they j u m p e d .
W e p l a c e d p o i n t s at t h e t o p o f t h e h o r s e s '
tails a n d plotted the points in Excel to find the curves m a d e .
34
F i g u r e 7point3.H o l l y w o o d ' s tail c u r v e o v e r an 18- inch j u m p
T h e f a m i l y o f tail c u r v e s is m o r e d i v e r s e t h a n o u r c e n t e r o f m a s s c u r v e s b u t n o t as
m u c h as the head curves.
B y f i t t i n g o u r d a t a , w e c o n c l u d e t h a t all t h e tail c u r v e s h a v e a
g o o d f i t i n t o a c u b i c c u r v e w i t h Rsquaredgreaterthan0point8 2 .
7point3.F u t u r e
Please see A p p e n d i x D for m o r e graphs.
Research
O u r research can be taken further by studying the front and hind legs w h e n the
horses are jumping.
T h e c u r v e s o f t h e h e a d , tail, a n d c e n t e r o f m a s s c a n also b e c o m p a r e d
to see any relationships that m a y exist.
Another interesting thing would be studying the
c u r v e s m a d e w h e n t h e r e is a rider o n t h e h o r s e s a n d w h e t h e r t h e r e is a d i f f e r e n c e m a d e in
the curves w h e n the horses jump.
S i n c e t h e r e is a b i g d i v e r s i t y o f h e a d p o s i t i o n s , it w o u l d b e i n t e r e s t i n g t o f i n d t h e
o p t i m a l position d u r i n g t h e j u m p as w e l l as correlations w i t h o u r initial curves.
Research
c o u l d b e d o n e to see if t h e r e are a n y correlations b e t w e e n t h e c u r v e s of h o r s e s of t h e s a m e
breeds or of the s a m e heights.
35
A p p e n d i x
Carmelita
A
walking:
R a w image
Edited image:
Roxie trotting:
R a w image
Edited image
36
Patriot trotting:
Raw
image
Edited image
Patriot cantering:
Raw
image
Edited image
37
A p p e n d i x
Hollywood 24 inches:
38
B
H o n e y 18 inches:
39
P o k e r 18 i n c h e s :
40
A p p e n d i x
All horses j u m p i n g the 9-inch j u m p :
T w o horses j u m p i n g 24- inch j u m p :
41
C
H o l l y w o o d j u m p i n g 9-inch, 18- inch, a n d 24- inch j u m p s :
H o n e y j u m p i n g 9- inch a n d 18- inch j u m p s :
42
P o k e r j u m p i n g 9- inch a n d 18- inch j u m p s :
43
A p p e n d i x
Head
curves:
44
D
Tail
curves:
45
46
B i b l i o g r a p h y
[ C ] " C u r v e F i t t i n g 2point0 2 . " P h E T : F r e e o n l i n e p h y s i c s , c h e m i s t r y , b i o l o g y , e a r t h
Web. 22 May
2012.
< h t t pcol nforwardslashforwardslashp h e tdotc o l o r a d odote d uforwardslashs i m sforwardslashc u r v e hyphen
f i t t i n gforwardslashc u r v ehyphenf i t t i n gunderscoree ndoth t m l > .
[ D ] D a u s e n d , T o n j a . " T h e H o r s e ' s B a l a n c e . " r i d i n g a r tdotc o m H o m e . W e b . 6 J a n .
2 0 1 3 . < h t t pcol nforwardslashforwardslashw w wdotr i d i n g a r tdotc o mforwardslashb a l a n c edoth t m > .
[G] G r z e g o r c z y k ,
Ivona. M a t h e m a t i c s a n d fine arts. D u b u q u e ,
Pub. Co., 2000.
I o w a : K e n d a l lforwardslashH u n l
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Wiley, 1978.
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2012.
< m a t h e m a t i c sdotg u l f c o a s tdote d uforwardslashm g f 1 1 0 7 l lforwardslashC h 5 S e c 2 L e s s o n 1doth t m > .
[St] " S t r i p P a t t e r n s . " W e b .
1 7 S e p t . 2 0 1 2 . < w w wdoti m adotu m ndote d uforwardslashtildes h a k i b a n forward
slash
strip
[ S ] S u t o r , C h e r y l . " E q u u s i t edotc o m - H o r s e G a i t s : w a l k , t r o t , c a n t e r a n d
E q u u s i t edotc o m - T h e U l t i m a t e H o r s e R e s o u r c e . W e b . 2 0 M a y
gallop.."
2012.
< h t t pcol nforwardslashforwardslashw w wdote q u u s i t edotc o mforwardslasha r t i c l e sforwardslashb a s i c sforwardslash
basics
G a i t sdots h t m l > .
< h t t pcol nforwardslashforwardslashw w wdotw i n p o s s i b l edotc o mforwardslashl e s s o n sforwardslashG e o m e t r y
underscore
T r a n s l a t i o n ,underscoreR e f l e c t i o n ,
[ W i] W i l s o n , J e r r y D . , A n t h o n y J . B u f f a , a n d B o . L o u . C o l l e g e p h y s i c s J e r r y D .
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un
Wilson, A n t h o n y J. Buffa, B o Lou. 6th ed. U p p e r S a d d l e River, NJ:
P r e n t i c e Hall, 2 0 0 7 .
Pearson
Print.
[W] W o o d f o r d , C h r i s t i n e . " E q u i n e B i o m e c h a n i c s a n d G a i t A n a l y s i s . "
Vipsvet.
2 6 S e p t . 2 0 1 2 . < w w wdotv i p s v e tdotn e tforwardslasha r t i c l e sforwardslashb i o m e c h a n i c sdotp d f > .
[Z] Z a f r a n , L a r r y . " M a t h w i t h L a r r y - I n t r o d u c t i o n t o R o t a t i o n . " M a t h w i t h L a r r y
(Zafran). W e b . 6 May
2012.
< h t t pcol nforwardslashforwardslashm a t h w i t h l a r r ydotc o mforwardslashl e s s o n sforwardslashl e s s o n 0 2 8doth t m > .
48
Web.