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Differential Geometry of Curves
Cartesian coordinate system
René Descartes (1596-1650) (lat. Renatus Cartesius)
French philosopher, mathematician, and scientist.
Rationalism
„Ego cogito, ergo sum (I think, therefore I am)
Father of analytic geometry
long correspondence with Princess Elisabeth of Bohemia
(1618-1680) devoted mainly to moral and psychological
subjects.
1649 Queen Christina of Sweden invited Descartes to her
court to tutor her in his ideas about love (1626-1689)
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M
x
y
2
3
Polar coordinate system
Cartesian coordinates Polar coordinates.
M
x
y
f
r M
x
y
cos
sin , 0
x
y
r f
r f r
2 2
arctan
x y
y
x
r
f
4
Spirals
Archimedova spirála fr a
Logaritmická spirála fr ba e
3
5
Spira mirabilis
Jacob Bernoulli 1654-1705) requested that the curve be engraved upon his tomb with the phrase "Eadem
mutata resurgo".
I shall arise the same, though changed
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Representation of a curve
Parametric (vector form)
Implicit equation
Graph of a function
,X t x t y t
2 2
, 0, . . 14 9
x yF x y e g
2, e. g. ( 3)x
y f x y x e
Example:
1 3 , 2
3 5 0
1 5
3 3
X t t t
x y
y x
funkce_sing.ggb
4
Equation of Motion, Position Vector
r(t)
x(t)
z(t)
y(t) x
y
z
Position vector defines the motion of a
particle (i.e. a point mass) – its location
relative to a given coordinate system at some
time t
r(t) = [x(t), y(t), z(t)]
Kinematic quantities of a
classical particle: mass m,
position r, velocity v,
acceleration a
Velocity
Average velocity.
2 1 2 1; ; ( ) ( )s
v t t t s s t s tt
The velocity of an object is the rate of change of its position
with respect to a frame of reference, and is a function of time.
Velocity is equivalent to a specification of its speed and
direction of motion (e.g. 60 km/h to the north).
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Instantaneous velocity
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the rate of change of position with respect to time
Speed – the magnitude of the velocity
If we consider v as velocity and r as the position vector, then we can
express the (instantaneous) velocity of a particle or object, at any
particular time t, as the derivative of the position with respect to time.
0 0
( ) ( )lim limt t
s s t t s t dsv
t t dt
d d
, ,d d
x yt v t v t x t y tt t
v
Free fall
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the rate of change of position with respect to time
0 0
( ) ( )lim limt t
s s t t s t dsv
t t dt
Example: Free fall
The absence of an air resistance 21
2s gt
dsv gt
dt
Heavy objects fall faster than lighter ones, in direct proportion to weight.
Aristotélēs (384 – 322)
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Inclined plane
t s s
1 1
2 4 3
3 9 5
4 16 7
5 25 9
6 36 11
Acceleration the rate of change of velocity of an object
with respect to time
0lim
( ) ( )
td v v
d v v t dt v t
Acceleration is NOT necessary
on a normal line
0limt
d v va
dt t
0
dlim
d
d d, ,
d d
t
x y x y
t t tt t
t t
t a t a t v t v tt t
v va v
a
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Tangential and normal acceleration
tr aaa
The tangential component at is due to the change in speed of
traversal, and points along the curve in the direction of the
velocity vector (or in the opposite direction).
The normal component ar is due to the change in direction of
the velocity vector and is normal to the trajectory, pointing
toward the center of curvature of the path.
Uniform circular motion
Instantaneous velocity
Angular rate
.cos ( ), .sin ( )s r t r t
sin ( ); cos ( ) ;ds d d d
v r t r t v rdt dt dt dt
vd
dt r
( )v v
t dt tr r
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Circular motion
Pohyb_kruznice.ggb
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The period of swing of a simple gravity
pendulum depends on its length and the
local strength of gravity.
0( ) cos
sin ( ); cos ( )
gt t
l
X t l t l l t
( ) ( )
sin ( ) ( )
gt t
l
assume t t
2l
Tg
Simple gravity pendulum
Oscilator.ggb
The period is independent of
amplitude
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Vector function as parametrization of smooth curve Cn
The set kE3 of all points in space is called a space curve Cn if
point coordinates could be expressed by mapping IR3, t X(t),
where parameter t varies throughout the interval I and
X(t) is continuous on an interval I
X(t) is injective (one to one)
X(t) has continuous n-th derivatives on a interval I
X(t) is regular - derivative vector X´(t) o.
Curve in plane
Curve in space
;
; ;
X t x t y t
X t x t y t z t
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Cycloid
A cycloid is the curve traced by a point on the perimeter of a circular
wheel as the wheel rolls along a straight line without slipping.
trrtrrttX cos;sin)(
x(t)
y(t)
Cykloida.ggb
10
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Cycloid trrtrrttX cos;sin)(
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Reparametrization
Let X(t) tI and Y(u), u J denote parametrizations for a curve. They
have the same image (and run through it in the same direction) if there
exist a strictly increasing differentiable function t = f(u) such that
1. f(u) is one to one I J
2. Y(u) = X(f(u))
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21
Tangent line
Curve X(t) has a tangent line at regular point X(t0):
0
0 0( ) ( )
x t r
y f t r f t
0 0( ) ,T r X t r X t r R
X(t0)
X(t)
X´(t0) t
X(t0) X(t)
0 0
00
lim
h
X t h X tX t
h
X(t0+h)
0 0 0
0 0
( )
( ) [ , ( )] ( ) , ( )
( ) [1, ( )] ( ) 1, ( )
y f x
X t t f t X t t f t
X t f t X t f t
Tangent line is the limiting position of the secants connecting two
points ont the curve close to the given one.
Ex: Tangent line to the curve y = f(x) at point f(x0):
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Tangent line
0 0( ) ,T r X t r X t r R
Reparametrization give the derivative vector with the same
direction.
( ) [cos( ),sin( )]; [ sin( ),cos( )],
tangent vector at point K(0): ( 0)
( ) [cos(2 ),sin(2 )]; [ 2sin(2 ),2cos(2 )],
tangent vector at point L(0): ( 0)
[0,1]
[0,2]
dKK t t t t t
dt
dKt
dt
dLL u u u u u
du
dLu
du
Curve X(t) has a tangent line at regular point X(t0):
12
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Frenet-Serret moving frame
Vectors t, n, b form an orthonormal basis spanning ℝ3 and are defined
as follows:
Osculating plane –plane spanned by X’ and X’’, best approximating
plane.
0 0 0 ; ,X t r X t s X t r s R
nt
b
T
k
t – tangent
• n – normal
• b – binormal
• = (t,n) – osculating plane
• = (b,n) – normal plane
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Unit vector tangent to the curve pointing in the direction
of motion.
Binormal unit vector
Normal unit vector
nt
b
T
k
X
btn
XX
XXb
X
Xt
Frenet-Serret moving frame
13
25
tangent
normal
binormal
Frenet-Serret moving frame of the helix
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Point of inflection
The Frenet–Serret formulas apply to curves which are non-degenerate, which
roughly means that they have nonzero curvature. More formally, in this
situation the velocity vector X′(t) and the acceleration vector X′′(t) are required
not to be proportional.
00 tXtX
14
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Arc length – rectification o a curve
Length of the curve X(t) between points X(a) a X(b)
b b
a a
l X t dt X t X t dt
X(t)
X(t0)
b=X(tn)
X(t1)
X(t2)
X(t3)
1
1
0
n
i i
i
l X t X t
We can then approximate the curve by a series of
straight lines connecting the points.
Polygonal path
Integral
A definite integral of a function can be represented as the signed
area of the region bounded by its graph.
15
Approximating circles and osculating circle
29
oskulačni_parabola.ggb
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Osculating circle of the ellipse
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32
Osculating circle of the
cycloid
Cykloida.ggb
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Curvature
0
lims
k X ss
Near a point P on the curve, that curve may be
approximated by osculating circle.
The curvature is in inverse proportion to the
radius of this osculating circle.
17
34
Geometric meaning of the curvature
Circles and lines have constant curvature.
Vertices of the curve have extremal curvature.
Point of inflection has zero curvature.
Difgeo24_krivost.ggb
36
Curvature Calculator
1. X(l) parametrized by arc length
2. X(t) parametrized by general parameter
3. Curve as a graph of a function y = f(x)
k X l
3
X Xk
X X
3
21
yk
y
Př: Vypočítejte funkci křivosti paraboly y = x2
2
2
2
y x
y x
y
3
2
2
1 4
k
x
32
2
1 4
k x
x
2y x x
18
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Order of continuity
Quantity of touching, smooth connection between two
curves
0 0 0
0 0
2 2
0 02 2
0 0
1 1
0 01 1
n n
n n
n n
n n
X l Y s P
dX dYl s
dl ds
d X d Yl s
dl ds
d X d Yl s
dl ds
d X d Yl s
dl ds
t
n
O
k
k
38
Taylor approximation y=sin(x)
20 0 0
0 0 0 01! 2! !
nnf x f x f x
T x f x x x x x x xn
Taylor_sin.ggb TaylorovPolynomial[sin(x), 0, n]
19
39
Cubic Taylor approximation of circle c(t) = (sin t, cos t)
The components of the vector function are just the k-th degree
Taylor polynomials of the components
40
Helix
20
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Rotation about axis simultaneously with
translation along the same axis.
0( ) cos ; sin ; X r r v
Circular Helix – trajectory in the screw motion
Helix is determined by r, rate v0 and axis o = z .
42
Tangent line to the helix
0cos , sin ,X r r v
0 0
1
tan v v
t r
0
1
sin , cos ,
sin , cos ,0
t r r v
t r r
Helix:
Top view of t:
Gradient (slope):
Constant slope
tangent vektor:
It has the property that the tangent
line at any point makes a constant
angle with a fixed line called the
axis.
21
43
Arc length reparametrization
0
2 2
0
sin , cos ,
t r r v
t r v0
cos
sin
;
x r
y r
z v R
2 2 2 2
0 02 2
0 0 0
ll t d r v d r v
r v
2 2
0
2 2
0
02 2
0
cos
sin
;
lx r
r v
ly r
r v
lz v s R
r v
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Curvature of the helix
0
2 2 2 2 2 2
0 0 0
0
2 2 2 2 2 2 2 2 2 2
0 0 0 0 0
2 2 2 22 2 2 20 00 0
( ) cos ; sin ;
( ) sin ; cos ;
( ) cos ; sin
v ll lX l r r
r v r v r v
vr l r lX l
r v r v r v r v r v
r l r lX l
r v r vr v r v
; 0
2 2
0
( )r
k X lr v
2 2 2 2
0 0
( ); 1
( )cos ; sin ; 0
t X l t
X l l ln
k r v r v
Helix has a constant curvature
22
45
Track transition curve
Track transition curve (spiral easement) is designed to
prevent sudden changes in lateral (or centripetal)
acceleration.
The start of the transition of the horizontal curve is at infinite
radius and at the end of the transition it has the same
radius as the curve itself, thus forming a very broad spiral.
s
an
s
an
Continuous curvature
46
Clothoid – Euler spiral
Curvature is proportional tu the arc length k(l) = a.l
2
0
2
0
cos2
sin2
l
l
atx l dt
aty l dt
cos ,sin
sin , cos
X l l l
X l l l l l
2
2
a l k X l l
a ll
l
23
48
DifGeo25_klotoida.ggb
Determine the length of transition curve k = l / 225
used between the straight section and circular arc
with radius r = 15 m
49
Clothoid and cubic approximation
Determine Taylor cubic approximation at point X(0) = [0,0].
2 2
0 0
2 2
2 2
2 2 2 22 2 2 2
cos ; sin 0 0,02 2
cos ; sin 0 1,02 2
sin ; cos 0 0,02 2
sin cos ; cos sin ; 0 0,2 2 2 2
l lat at
X l dt dt X
al alX l X
al alX l al al X
al al al alX l a a l a a l X a
20 0 0
0 0 0 01! 2! !
nnX t X t X t
T t X t t t t t t tn
2 30,[1,0] [0,0]
[0,0] 0 0 01! 2! 3!
aT t t t t
3
,3!
atT t t
