Дзжйиз ¡ ¢ ; Parametric Form: ¦ ! " ¢$ #%¦ & ' ¡ ¦ ! )(. a BDC'e ¦0 1 4 ¢$ F

Home , Center of curvature, Evolute, Osculating circle

Alexander Belyaev ([email protected], http://www.mpi-sb.mpg.de/ belyaev/gm04) Plane and Space Curves. Curvature. Curvature-based Features.

Description of Plane Curves Orientation, Curvature, Curvature Vector

¦ m¢x ¨¦ ¡ ¦

e e e e Plane curves can be described mathematically in Consider a curve parameterized by arc length .

explicit form: ¡£¢¥¤§¦©¨ (as a function graph); The parameterization deﬁnes a prescribed direction along the curve at

¤§¦©¨§ ¡ ¢ ¦

y e

implicit form: ; which e grows. Let be the unit tangent vector associated with the

¦ ¢ ¨¦ ¡ ¦ ¦ ¦

z e y z parametric form: . direction. Let be the unit normal vector such that forms the

Here and everywhere below the vectors are denoted by bold letters. counter-clockwise oriented frame.

¦ ¨¦ ¡ ¦

e e e Denote by { the angle between the tangent at a point and the positive direction of the ¨ -axis. The curvature at a point mea-

Arc Length sures the rate of curving (bending) as the point moves along the curve

¨! " ¡# § ¨%$& ¡'$(

Let and be two points with coordinates and , with unit speed and can be deﬁned as

respectively. The distance between and is given by

{ e

¦ (¢ 9

)*,+-

¦ "¢. ¦/¨ 102¨ $3 456¦©¡# 708¡'$3 4:9

Thus the curvature of a straight line is zero. The curvature of a circle

¦ ;¢< ¨¦ ¡ ¦

Consider a curve . Let us approximate the curve oriented by its inner normal is equal to the reciprocal of the radius of the

@9A9@9B ¢< ¨§¦ ¡!¦

="? =C =D D D

by a polygonal line => such that and circle.

0F ¢HGI

D@E D ? . Now let us approximate the length of the curve y d k(s)= n (s) t(s) ds s P P (x(s),y(s)) k−1 k+1 Pk

x Fig. 1: Approximation of a smooth curve by a polygonal line.

Fig. 2: Deﬁnition of curvature.

@9B9@9@

=> ="? =§C by the length of the polygonal line . The length of the ¢ polygonal line is given by the sum Points where are called the points of inﬂection. Surprisingly, the curvature is all that is needed to deﬁne a curve (up

J to rigid motions).

. ¦©¨ 08¨ 45¥¦©¡ 0K¡ 4:9

D@E D D@E D ?

? An oriented curve is a curve such that at every point a unit normal

z e vector z is deﬁned, provided is continuous along the curve. The

curvature depends on the orientation. If orientation is changed, the sign

¦ ¦

Rewriting the sum in a more convenient form and passing to the limit as ¢

e z e

of curvature changes. The curvature vector } does not LM

yields depend on the orientation.

N O OTS O O!S

¨ ¨ 02¨ ¡ ¡ 02¡

J 4 4

4 4

D@E D D@E D

? ?

S S 5 GILRQ 5

P GI P !P !P

GI The Formulas of Frenet

¦ ~¢M ¨¦ ¡ ¦

e e e

Let a curve be parameterized by arc length. The

¦ ¦ I¢ S

Thus the length of the curve segment between two points

e e y e

derivative of with respect to arc length , is a unit

$ $

¨¦ ¡ ¦ ¢U ¨¦ ¡ ¦ e

and is given by S

vector tangent to the S curve (prove it). Differentiation of the scalar prod-

y y

O!S O!S

S S

¢us ¢

y y y y

¨ ¡

4 uct gives and, therefore, the vectors and are

v v

)*,+- V:W,X:Y[Z \@]^_'`

e e

S 5 S ¦ (¢baQ

\@X:+ + *,Y

TP P

S ¢o

y { {

mutually perpendicular: y . Since , see Fig. 2,

then S

S S S

+ *,Y \@X+

S ¢u0 %¢

¨ ¡ ¨

{ { z

¨ S ¡ S ¨ S

{

c d

We will use Newton’s dot notations c for , for , for , etc.

4 ¦

Let e be the length of the curve segment between the point on the is a unit vector normal to the curve. Differentiating with respect to arc

£¢f ¨¦ ¡ ¦

> >

curve corresponding to the value > , , and a general length gives

S S S S S S ¨¦ ¡ ¦

point on the curve . Then

| V:Y[) |

y { z z { y

S S S S S S

¢ ¢ ¢ ¢t0

z y

v v

S S

e { e e { e

h h

S S

Q S S

¦ (¢ ¨ 5 ¡ ¢ ¦ (¢ ¨ 5 ¡ ¢ji i

4 4 4 4

e c c e c c c

i i

| |

y z

S ¢ S ¢t0

i i y

We arrive at the famous formulas of Frenet: z

e e

¦ ¨¦ ¡ ¦

e >

Note that measures the distance between points > and Curvature Computation

¨¦ k¡!¦ ¦ mln mlo ¦ mpn

e e

along the curve and if > and if

¦ ¢

1pq ¦

S e

> . The value is called arc length of the curve. The value Parameterized curves. Consider a parameterized curve

¦©¨¦ ¡ ¦

©y z

e and assume that forms a right-hand basis. The curva-

S ¢ ¦ ;¢ ¨ 5 ¡

4 4

e c c c indicates the rate of change of arc length

ture is given by

¦ e

with respect to the curve parameter . In other words, c gives the

S S

¨ ¡0 ¨ ¡

c d d c

r ¢tsBr ¨ 5 ¡

¦ (¢

4 4

c c

speed of the curve parameterization. Note that e . (1)

¦ ¨ 5 ¡ 4

4 4 c Often it is very convenient to parameterize the curve with respect to c

its arc length:

(¢u ¨§¦ ¡!¦ :9

¦ Let us derive this formula:

e e e

S S

S S S S S S

¡r ¡%¦ ¡%¦

-V:Y -VYT

c c

¢ S ¢ S S ¢ ¢

S S S S S S

¢ ¢ i i i i ¢ts

{ {

¨ r ¨¦ ¨¦ :

Note that and, hence, ¨

2v 3w i i i i

c c

e e e

i i i i

S S S S

s s ¡

{ { c

¢ ¢ ¢

S S S S

h Curve Quality Evaluation via Curvature Proﬁle

v v v

s"5¦ ¡ r ¨

¨ 5 ¡

4 4

e e c

c c c

c The shape of the curvature proﬁle is an important tool for the curve qual-

¢ ¦

¨ ¨ ¡I0 ¨ ¡ s ¨ ¡0 ¨ ¡

ity evaluation. Fig. 4 displays an original curve (the left image)

c c d d c c d d c

¢ ¢

¦ ¦

¨ 5 ¡ v ¨ v

e z e

4 4 4

¨ 5 ¡ 4

¦ and the curve together with its curvature vectors attached to

¨ 5 ¡

4 4

c c c

c c c c curve points (the right image). Curves in explicit form. Consider a plane curve given as the graph of

a function: ¡m¢¤§¦©¨ . Assume that the curve is oriented by its upward

¢ ,0 ¤ ¦©¨ As:r s"5 ¤ ¦©¨% 4 normal z (it is equivalent that we move along the curve from left to right). The curvature of the curve is given

¡ ¡

¤ ¦©¨

| ¢

¢ (2)

4¤£

s"5 ¤ ¦©¨%

This formula for the curvature is easily derived from the previous one if Fig. 4: Curve quality evaluation via curvature proﬁle.

we represent the curve in the following parametric form

¨ ¢ A ¡ ¢¥¤§¦

Circle of Curvature

Curves in implicit form. Consider a plane curve given by the equation The circle of curvature or osculating circle at a non-inﬂection point = ¥

¦©¨§ ¡ (¢ . The curvature vector of the curve is given by (where the curvature is not zero) on a plane curve is the circle that

¥ ¥ ¥§¦¤¦¨¥ ¥§¦ ¥ ¥§¦

¥§¦ ¥

¦ 1. is tangent to the curve at ;

5 0

4 4

¢ ¢t0 ¢

z z z

} 2. has the same curvature as the curve has at ;

¥ ¥ ¥ ¥

¦ ¦

4 4

¦ 5 ¦ 5

4 4 4 4 © © 3. lies toward the concave or inner side of the curve.

(3)

¥ ¢

To derive this formula let us consider a point on the curve where ©

¥ ¥

¦ R

¦ k =

means © , etc.). At some vicinity of the point the curve can be repre- R

¢¥¤§¦©¨

sented as the graph of a function ¡ . Thus

¥ 9 ¦©¨§ ¤§¦©¨ (¢ Fig. 5: The circle of curvature.

The center of the circle of curvature is called the center of curvature.

¨ ¨¦ ¡ ¦

Differentiating this equation with respect to one and two times gives Consider a parameterized curve . At each point of the

¢ ¨¦ ¡ ¦

> > >

¡ ¡ ¡ ¡ ¡ ¡

)

VY curve let us measure how closely the curve can

¥§¦ ¥ ¥§¦¤¦ ¥§¦ ¥§¦ ¥ ¥

5 ¤ ¢ 5 ¤ 5 ¤ 5 ¤ 5 ¤ ¢

¡ ¡ ¡

4 4 4

¦©¨ 0[ 5¦©¡0B ¢ ¤

respectively. Now expressing ¤ and through partial derivatives of (4) ¥ ¦©¨§ ¡ and substituting them in (2) we arrive at (3).

through => and the function

4 4 4

¦ (¢u¦¨¦ 0 56¦©¡ ¦ §0B 0 9

Reconstruction of Curve from its Curvature

S S S S

\AX+ + *,Y | ¦ "¢t

¦ r ¢ ¦ ¢f ¦ ¦ ¦ ¢ ¦ r

Since the circle goes through => , the equation has an obvious

e e y e { e { e e { e e

Since and ,

the curvature of a curve determines the curve up to rigid transformations. solution > . In order to ﬁnd the circle providing with the best ap-

One can reconstruct the curve from its curvature by integration proximation of the curve at > we require that itself and as many as

¦ ¢

possible derivatives of vanish at > :

S S S

| \@X+ + *,Y

D$#

%¦ (¢¥ %¦ ¢ 9@9@9@ "! ¦ ¢¥ 9@9@9A9

¦ '¢ Q ¦ 5 ¦ '¢©Q ¦ 5% Q ¦ 5

c > d > >

{ e e e { e { e e { e e > (5)

Actually, since in general a circle depends on three parameters, two co-

{ where > , , are constants of integration (a rigid transformation is ordinates of the center and the radius, at a generic curve point we are

deﬁned by three parameters). able to satisfy only three conditions

¦ ¢¥ %¦ "¢ %¦ "¢ 9

c d

> > >

¨¦ ¡ ¦ ¨T¦ ¡ ¦ ¨¦ ¡!¦ ¨ ¡ ¨ ¡ ¨ ¡

> c > c > d > d > > > c > c > d > d >

Denote > , , , , , by , , , , ,

¦ (¢¥ %¦ (¢¥

c d >

respectively. The equations > and give

¨ ¦©¨ 0[ !5 ¡ ¦©¡ 0 @ (¢¥

c c

> > > >

¨ ¦©¨ 0[ !5 ¡ ¦©¡ 0 @ (¢u0 ¨ 0 ¡

4 4

d d c c

> > > >

> >

¦& 'B

The ﬁrst equation says that lies on the normal to the curve at => .

Solving the system with respect to and we arrive at

s s ¡ ¨

c c

> >

( ¢¡ 5 £¢6¨ 0

h h

| |

> >

v v

¦ ¦

¨ 5 ¡ ¨ 5 ¡

4 4 4 4

> >

c c c c

> > > >

¨ ¡ 0 ¨ ¡

c d d c

where ¦ (¢ is the curvature of the curve. Note that

¦ ¨ 5 ¡

4 4

c c

)

0 ¡ ¨

c c

> >

h h

¨ 5 ¡ ¨ 5 ¡

4 4 4 4

c c c c

> > > >+*

Fig. 3: Curves reconstructed from prescribed curvature functions e . is a unit vector normal to the curve. Thus we have proved

¦ 5¨¢( (¢ ¦ T5

e z e

Theorem 1 The best approximation of the curve by a circle is given by z

¢ ¢ 4

the circle of curvature (osculating circle).

¡ ¡ ¡

| | | |

5 0¢ 0 5 0 £ 5 ¦¢ £5

y ©

A vertex of a smooth curve is a point of the curve where the curvature

¦ ¢

e > ¢

takes an extremum: . ¢

| | |

5¨ ¦¢ 0 5 0

z Theorem 2 (Four-Vertex Theorem) A simple closed curve has at least

four vertices.

¡ ¡ ¡

| | | |

¦ 5 ¢ ¢ ¦ T5¨¢ 5 5 ¦¢ e A sketch of a simple proof of the theorem will be given in the section e

devoted to the medial axis.

¡ ¡ ¡

We will say that the circle (4) has contact of order or higher with ¡

| | |

s s

¨¦ k¡ ¦ ¢M ¨¦ ¡ ¦ ¦

=> > > >

0¢ 5 ¢ 0 5 ¦¢ 9

curve at point if and the ﬁrst ¢

| | | | |

¦ 5 ¢ ¦

4 4

derivatives in (5) vanish. Thus the osculating circle has contact of order e

or higher. It is not difﬁcult to show that if > is a vertex then the

¡ ¦ 5§¢( osculating circle at = has contact of order or higher.

For the evolute point e we have

¦ 5 ¢

z e

¢¡ ¦ !5 ¦ 5 ¢( (¢ ¦ 5¨¢( T5

Evolute ¡

e e

e (7)

¦ 5¨¢(

¡ ¡ ¡

The locus of the centers of curvature of a given curve is called the evo- ¡

| | |

5 ¢ 5¨ ¦¢ 5 ¢ 0

lute of the curve. The evolute of a smooth curve has singularities (so- 5

| | |

called cusps). The cups of the evolute correspond to the points where

¡ ¡ ¡ ¡

| |

the curvature takes extremal values. |

5 0¢ 5¨¢ 0 5 ¦¢

| | |

4 4

¦ ¢

Thus, if e , in a small vicinity of the center of curvature

¦ 5 ¦ r ¦

e z e e

the evolute is approximated by a parabola tangent

¦ ¦ ¢

e e to the normal z (see the left image of Fig. 7). If , in a

small vicinity of the center of curvature the evolute is approximated by

¡ ¢ ¨ ¦

z e a semicubic parabola tangent to the normal (see the middle and right images of Fig. 7).

Fig. 6: Left: an ellipse and its evolute. Middle: a circle of curvature is evolute added. Right: a parabola and its evolute. center of center of curvature

curvature

¢ ¦

Given an oriented curve , its evolute is described by evolute center of evolute

¦ curvature

¦ ¢ ¦ !5

¡ n | n

¦ n

t t t

¦ ¦ where is the curvature of the curve at .

Let us consider a curve parameterized by arc length e . To study the

¦ 5£¢(

¦ Fig. 7: The evolute is tangent to the curve normals at the centers of

e e

shape of the evolute at a small vicinity of a point we expand

¢¥¤Rs ¢ curvature. The critical points of the curvature correspond to singularities into Taylor series with respect to , . Let

of the evolute.

V:Y[)

¢ ¢

y z y

We have proven the following theorem.

¢ ¦

e compose the Frenet frame at . The evolute is given by

Theorem 3 The evolute is tangent to the curve normals at the centers

z e

¦ ¢ ¦ !5

¡ of curvature.

e e

¦ e

| The middle image of Fig. 6 demonstrates this property of the evolute.

¦ ¦

e where e is the curvature of the curve at . According to the Frenet A more detail analysis of expansion (7) leads us to the following re-

formulas

¡ ¡

| |

¢t0

¢ sult.

y z z y

we have Theorem 4 The evolute of a curve has a cusp at > if and only if

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

| | |

| is a vertex of the curve. The cusp on the evolute is pointing towards

9 0 ¢ ¢ ¢ ¢u¦ ¢

y z y y z z

or away from the vertex depending on whether the absolute value of the

curvature has a local minimum or maximum at > .

It yields

¢ ¢

y z

¡ ¡ ¡ ¡ ¡

¡ Let and be the Frenet basic vectors of a given curve . If

¦ 5§¢( ¢ 5¨¢ 5 5 5 ¢ £¢

lq ¢ ¢ 0

z z y

, the vectors y and form the Frenet basis of

the evolute. Let deliver arc length parameterization of the evolute.

5 ¢ £¤ 5 ¦ T5 ¢10

¢ Reparameterization of (7) gives

e y

¢ ¢

¡ ¦ 5 (¢¡ ¦ 5¢( (¢¡ ¦ 5 5 ¦ :5 5 ¦

| |

5 5 5¨ ¢ £ 9

e e y z!

| |

¢"¢ r

Similarly, ¢0 where . Comparing the above expansion with (6) we con-

z y

| |

¡ ¡ ¡ ¡

| | |

¢0 0

¢u¦0 clude that the curvature of the evolute is is equal to . It follows that

| |

z y y z

r ¢U

the evolute has no inﬂections. Indeed, if then the curve has

¡ ¡ ¡ ¡ ¡ ¡

| | | |

¢u¦0 5 0

z z y an inﬂection and the corresponding evolute point is located at inﬁnity. 4

of curves

of family

Envelope Fig. 9: The envelope of all the normals to a given curve is the evolute of the curve.

Fig. 8: The envelope of the family of curves is tangent to every curve from the family

Families and Envelopes

3¢t

Consider a family of plane curves ¦©¨T ¡% parameterized by the

¦©¨§ k¡% ¢ parameter . The envelope of the family of curves is a Fig. 10: Coffeecup caustic.

curve which is tangent to every curve from the family, as seen in Fig.8

&¢u Theorem 5 The envelope of the family of curves ¦©¨T ¡% is the Caustics by Reﬂection solution to the system The envelope of the family of rays is called a caustic (i.e. ‘burning’,

¥ since light is concentrated at it). A caustic is clearly visible on the inner

¦©¨T ¡% ¢

¡ surface of a cup when the sun shines on it. See a coffeecup caustic in

¢ 9

¦©¨T ¡% Fig. 10: drinking from a cylindrical cup in the sunshine one can see a ¡ crescent of light as the sunshine reﬂects from the inside of the cup onto

the surface of the drink.

¢ ¦

Proof. Let be a parameterization of the envelope. Then for Consider a point and a curve not passing through . The locus of

¦ ¦©¨§ ¡% !¦

S S

each , belongs to some curve from the family. the reﬂections of with respect to all the tangent lines to the curve is

r ¢

Thus can be considered as a non-constant function of , , called the orthotomic of the curve relative to , see the left image of

since otherwise it would imply the envelope was a member of the family. Fig. 11.

Differentiating with respect to yields Let be a point source of light. The light from is reﬂected from the

curve according to the usual rule that the reﬂected ray and the incident

¥§¦ ¥ ¥£¢

¨I5 ¡5 ¢ 9

c c c ray make equal angles on opposite sides of the normal of the curve.

Since the curve and its envelope have the common tangent at Theorem 7 The caustic generated by the reﬂected rays is the evolute of

¦©¨¦ k¡ ¦ , we have the orthotomic.

¥ ¥

¡£¢¥ 9 ¨I5

c c

¥ ¥ ¢

¢ Proof. Suppose that is the origin of coordinates. Let the curve be

£¢ ¢¥ ¢¥

c Thus c . Since we get .

parameterized by arc length e :

¢ ¦ ¨§¦©¨§¦ k¡ ¦

e e e Theorem 6 The envelope of all the normals to a given curve is the locus

of centers of curvature – the evolute – of the curve.

¦ ¦ ¦

y e z e

and be the unit tangent and normal vectors respectively, e

~¢H ¨¦ ¡ ¦

¦ be the curvature of the curve. Then the orthotomic is given by

e e e

Proof. Let us consider a curve parameterized by

¦ ¢ ¦ ¦ ¦ ¦ 9

arc length . The equation of the straight line normal to the curve and ©

e e z e z e

¦ "¢u ¨¦ ¡ ¦

e e e

passing through a curve point is given by

¥¤10 ¦ ¦ ¢

¦ Differentiating with respect to yields

e e

¢ ¢

£ £

¤K¢ ¨§ ¡ e e e y e z e e z e y e v where is a point on the line. Thus the family of all the normals v is described by the equation

and the expression in square brackets is a vector tangent to the ortho-

¦¤3 ¨§u¦¦¤70 ¦ ¦ (¢ 9

¦ tomic. Therefore

e e

e (8) v

¦ ¦ ¦ ¦ 0 ¦ ¦ ¦ ¦

e y e y e e z e z e v

We have: v

¡ ¡ ¡ ¡

¡ ¡

§u¦¦¤10 ¦ ¦ 0 ¦ ¦ (¢

r is a vector normal of the orthotomic. Simultaneously this vector coin-

e e e e e

v v e

cides with the direction the ray reﬂected at . It remains to recall that

¢u¦¦¤70 ¦ ¦ ¦ 0 s ¢ 9

e z e e

(9)

v the evolute can be described as the envelope of the normals.

¤ ¢ ¦ T5 ¢ ¦ T5 ¦

e y e z e

Suppose . From (8) and (9) it follows that Conic Sections and Their Optical Properties

¢K¢ 1¢us r ¦

and e . It gives the evolute:

Ellipse. An ellipse given by the equation

¤2¢ ¦ T5 ¦ r ¦ 9

e z e e

¨ ¡

4 4

5 ¢ts l lF

4 4 Optically the evolute of a given curve can be described as the locus of points where the light rays emitted by the curve in the normal is represented in parametric form by

directions are concentrated. At the singularities of the evolute the

\@X:+ + *,Y

¨ ¢ ¡£¢ p 9

{ { concentration is even greater. { 5

ellipsoid hyperboloid

orthotomic paraboloid paraboloid paraboloid

tangent line Fig. 12: Telescope conﬁgurations: the Newtonian telescope (left) the normal to orthotomic mirror curvilinear Cassegrainian telescope (middle), and the Gregorian telescope (right). incident ray

ray source of light reflected Involute

Fig. 11: Left: The orthotomic is formed by the reflections of a given If a line rolls without slipping as a tangent along a curve, then the path source of light with respect to all the tangent lines. Top-right: A curve of a fixed point on the line forms a new curve called the involute. An (reflector), a source of light, and a caustic generated by the rays reflected involute of the curve may be thought as any curve which is always per- by the curve. Bottom-right: The same curve and source of light and the pendicular to the tangent lines of the original curve. orthotomic curve associated with them. The involute was introduced by Huygens who used it to improve a pendulum clock mechanism. For a simple pendulum clock, the time of

swing of the pendulum is not perfectly constant. It changes slightly if

¢¢¡ 0 ¢f¦§¦ 4

4 the angle of swing changes. Huygens proposed to change the length of a

S S

£¥¤ Let . The points are called the foci of the

pendulum in order to make the time of swing of the pendulum the same E ellipse. Let and be the distances between a point on the ellipse

for all angles. In 1673, Huygens built a clock with a modiﬁed pendulum.

E £ and the foci £ and , respectively. Then

The top of the pendulum was made of ﬂexible wire which would swing

S S S S

\@X:+ \@X+

¢ I0 ¢ 5 5 ¢

against curved metal cheeks, altering the pendulum’s length as it swings.

E { { E See ? for Huygens’ clock inventions. Therefore an ellipse can be also deﬁned as the locus of all points such that the sum of the distances from each point to two ﬁxed points (the foci) is constant.

The tangent and normal of the ellipse at { have the equations

\@X+ + *,Y

{ {

¨I5 ¡ ¢ts ¨I5 ¡£¢

\@X:+ + *,Y

and

{ {

\@X+ + * Y

¢¦&

{ { respectively. It is easy to verify that the tangent at

makes equal angles with the line segments connecting with the foci

¢u¦ ' ¢¦0 '

£ £ and . Fig. 13: Left: a circle and its involute. Right: a scheme of Huygens’s Thus rays emitted from a focus of an ellipse after being reﬂected by pendulum with ﬂexible wire and curved metal cheeks.

the ellipse will pass through the other focus of the ellipse.

¨ ¡ 4

4 Dual Curves ¢ts

Hyperbola. A hyperbola given by the equation 0 , where

4 4

l lF I hope to write here about the dual curves in the future.

, is represented in parametric form by

-V:Y ¨

¨ ¢ ¡ ¢ 9 ©¨ \@X:+ Offset Curves

There is a set of points, each of which has a constant distance from

¢ ¦ m¢b ¨¦ ¡ ¦

¢ 5 ¢f¦§¦

4 a corresponding point of a smooth curve in

S S

£¥¤

Let . The points are called the foci of the

^ S

the direction of its normal vector at that point. The set forms a curve

hyperbola. Let E and be the distances between a point on the

¦ ¢ ¦ T5 ¦ ¦

z z S

, where is the unit normal vector at the point

S S

E £

hyperbola and the foci £ and , respectively. It is easy to demonstrate

0 ¢

. The set is called an offset curve (or -offset) of the initial curve

that . ¦ Therefore a hyperbola can be deﬁned as a set of points the difference . of whose distances from two given points, the foci, is a given positive Offset curves and surfaces arise in many industrial applications, in constant. particular in mathematical modeling of cutting paths for milling ma-

The hyperbola has the important property that a ray originating at one chines. For example, let us consider a ball-end milling machine cutting

focus reﬂects in such a way that the outgoing path lies along the line a surface. The trajectory of the center of the machine is -offset of the

from the other focus through the point of intersection. surface where is the radius of the cutting ball.

¨8¢ ¡ lU 4

Parabola. A parabola given by the equation , , can

¨K¢ # ¡ ¢ #

be represented in parametric form by . The point

¢¦&r

£ mashine

is called the focus of the parabola. Calculations show that milling

¦©¨T ¡

S £

the distance between a point on the parabola and is given by

r ¨ ¢u0 r

¢¨5 R . The straight line is called the focal line of the desired parabola. shape Thus a parabola can be deﬁned as the locus of all points equidistant Fig. 14: A ball-end milling machine cutting a desired shape.

from a given line (the focal line) and a given point not on the line (the

| |

ptsBr ¦ ¦

focus). ¦

S S

Let be the curvature of the curve . As long as > ,

¢UsBr ¦ ¦ 3¢

It can be shown that rays parallel to the axis of a parabola after being ¦ §5

> z > the offset curve is smooth at . If then

reﬂected by the parabola will pass through the focus of the parabola. S

the offset curvS e is not smooth: a singularity (so called cusp) appears at

¢n l s r ¦ > > . For -offset is not smooth: it has singularities Ellipse, parabola, and hyperbola for telescope design. (cusps).

Fig. 12 demonstrates how optical properties of the ellipse, hyperbola, ? and parabola are used in telescope design. http://www.sciencemuseum.org.uk/collections/exhiblets/huygens/1673.asp 6

Theorem 8 For a given curve, the locus of the singularities of the offset

curves is the evolute of the curve.

¢ ¦

Proof. Let the curve be parameterized by arc length e : . Let

¦ ¢ ¦ §5 ¦

e e z e

be the arc length parameter of the offset curve .

¦ ¦ ¢ S

e y e Denote by y the unit tangent vector of the curve: . Using

the Frenet formulas we get: e S

S Fig. 16: Left: medial axis is formed by ﬁrst self-intersections of off-

S S

¦ (¢ ¦ ¦ T5 ¦ (¢u¦s 0 ¦ ¦ 9 S

S set curves. Right: medial axis is formed by centers of inner bitangent

e e z e e y e e

e circles.

S S S

r ¢us&0 ¦

e e

Thus ,

S S

¦ ¦ ¦ ¦

e z e y e y e

S ¢ S S ¢ S

s 0 ¦

e e

and the curvature of the offset curve equals

S 9

0 ¦ ¦s Fig. 17: Left: closed curve, its evolute, and medial axis. Right: curvature

e proﬁle is added.

¢ts r ¦

The denominator vanishes when e .

In particular we showed that it is impossible to produce a curve which For a ﬁgure without holes, the skeleton is a tree (in the graph the-

r curvature attains values greater then s by a ball-end milling machine oretic sense). with the cutting ball of radius . Fig. 15 illustrates properties of the evolute and offset curves. Fig. 18 shows a closed curve and its skeleton. Medial axis transform (MAT) consists of transforming a given figure to its skeleton. Features of MAT can be used for pattern recognition, e.g. number of nodes, branches etc. Notably, it is applied in the medical field for chromosome identification, see the left image of Fig. 19. Unfortunately the skeleton of a figure is very sensitive to small per- turbations (noise) of the boundary of the figure, as demonstrated in the left image of Fig. 19. Consider a point on the skeleton and let us move it toward an endpoint of the skeleton. When that moving point approaches the endpoint, the to points of tangency merge and the bitangent circle becomes osculating.

Fig. 15: Left: A curve, its evolute, and an osculating circle; note that the That osculating circle has contact of order or higher. Thus the end- segment connecting the circle center and the tangency point is tangent points of the skeleton correspond to the vertices of the curve. Since the to the evolute. Right: The same curve, its evolute, and an offset curve; skeleton of a ﬁgure bounded by a simple closed curve has at least two note that the cusps of the offset curve are situated on the evolute. endpoints, the curve has at least four vertices. This completes the proof of the four-vertex theorem.

Practical Extraction of Medial Axis

Medial Axis ¥ ¥ Voronoi Diagram. Given a set of points on the plane, the Voronoi di-

The medial axis or skeleton of a ﬁgure is the set of points inside ¡ ¥ agram of the set is a partition of the plane into regions (Voronoi regions), that have more than one closest point among the points of .

¥ each of which consists of the points closer to one particular point of the

We can deﬁne the medial axis of as the locus of centers of circles

¡ ¥ ¥ set than to any others. Each Voronoi cell is a convex polygon, and its inside that are bitangent to .

¥ vertices are called Voronoi vertices. Each Voronoi vertex is equidistant

The medial axis of can be also deﬁned as the locus of centers of

¥ from or more original points. maximal circles: circles inside that are not themselves enclosed in any

¥ There are several efﬁcient algorithms to compute the Voronoi diagram other circle inside . 4

¥ of a set of points.

Equivalently, the medial axis of can be defined via the set of points ¡ where the offset curves of ¥ meet themselves. Perhaps the most intu- itive is the prairie fire analogy. Imagine that the interior of the figure is Practical Extraction of Medial Axis via Voronoi Diagram. composed of dry grass and that the exterior, or background, of the figure Since the medial axis of a closed curve and the Voronoi diagram of a has no grass. Suppose a fire is set simultaneously at all points along the polygon have much in common, it is natural to approximate the medial figure boundary. The fire will propagate, at uniform speed, toward the axis by Voronoi vertices of a proper set of points. Given a closed curve,

middle of the ﬁgure. At some points, however, the advancing line of the 4 Joseph O'Rourke. Computational Geometry in C. Cambridge Univ. Press.

fire from one region of the boundary will intersect the fire front from Mark de Berg, Marc H. Overmars, Otfried Schwarzkopf, Marc Van Kreveld. some other region and the two fronts will extinguish each other. These Computational Geometry: Algorithms and Applications. Springer-Verlag. points are called quench points of the fire; the set of quench points de- fines the skeleton of the figure. The above description of the medial axis via first self intersections of the offset curves can be used to demonstrate that the endpoints of the medial axis are located at cusps of the evolute and correspond to curvature extrema. It is easy to verify the following results:

The skeleton of a polygon consists only of linear segments and parabolic arcs. If the polygon is convex, the skeleton consists only of linear seg- Fig. 18: A closed curve and its skeleton. ments. 7

Fig. 19: Left: medial axis transform is used for chromosome identification. Right: the skeleton of a figure is very sensitive to small perturba- tions of the boundary of the figure.

Fig. 22: From left to right, from top to bottom: evolution of a closed curve with speed equal to curvature is combined with rescaling.

To solve (10) numerically we ﬁrst replace the time derivative term by

its forward difference approximation

¦¢¡§ ¦¢¡§ kT5¥¤: 0 ¦¢¡§

¤¤ s:9

Fig. 20: A set of points and its Voronoi diagram. ¡

£ ¤

It gives

¦¢¡§ kT5¥¤: ¢ ¦¢¡§ !5¦¤ 9

let us consider a dense set of points on the curve. The inner Voronoi z vertices of the set provides us with a good approximation of the medial axis. We approximate smooth curves by polygonal lines for which we intro-

Fig. 21 demonstrates how the approximation of the skeleton via a sub- duce discrete analogs of the curvature, tangent and normal vectors. For

=¨§ set of the Voronoi diagram of the interpolating polygon depends on the a polygonal curve given by its vertices we consider the discrete

sampling rate of the polygonal interpolation. evolution shifting each vertex according to

# # # E #

! ! ! !

9 5¥¤ ¢

z = =

§ § §

§ (11)

# #

! !

= z §

Here § indicates the location of vertex after iterations, is

# #

! !

= §

the unit normal vector at § , is a discrete approximation of the

! =

curvature at § , the iterative step parameter is sufﬁciently small. See Fig. 23. (j) P Fig. 21: Left: approximation of the skeleton via a subset of the Voronoi i diagram of a sparse set of boundary points. Right: approximation of the (j) (j) P skeleton with a dense set of boundary points. 3 i+1

k (j)

(j) n P Curve Smoothing i−1 i (j+1) Smoothing reduces small, narrow, and edgy protrusions. Therefore pre- Pi liminary curve smoothing is very useful for robust extraction of the medial axis and other purposes. The smoothing method presented below is Fig. 23: Discrete curvature ﬂow applied to a polyline. based on curve evolution by curvature. At present it is considered as one of the best smoothing methods. Consider a smooth curve each point of which moves in the normal Practical Curvature Estimation direction with speed equal to the curvature at that point. It turns out that

such curve evolution has remarkable smoothing properties: Circle approximation for curvature estimation. Let us use the

circle passing through the points , , as an approximation to the – it reduces quickly small-scale curve oscillations; osculating circle to the curve at , see the left image of Fig. 24. Then – a nonconvex curve becomes convex and then it tends to a circle and simultaneously shrinks into a point; 2 k = O a+b – no new curvature extrema are generated. b n a S See Fig. 22 where smoothing by curve evolution with speed equal to B r(s) t curvature is combined with rescaling. c O r(s+ ) A r(s− ) a b Let us describe the curve evolution by curvature mathematically. abc 1

R = k =

¢¡§ k ¡ Consider a family of smooth curves ¦ , where parameterizes the 4S R A B

curve and parameterizes the family, either connecting two given endpoints or closed. We suppose ¡ to be independent of . The family evolves according to the evolution equation

Fig. 24: Circle and angle approximations of curvature.

¦¢¡§ k

¢ ¦¢¡§ (¢ ¦¢¡ ¡ z (10)

the inverse value of the radius will serve as an approximation of the

curvature at . Let denote the area of the triangle . The discrete

¦¢¡§

¦¢¡§ ¦¢¡§

where z is the unit normal vector for , is the curva- curvature at is given by

¦¢¡§

ture of . The family parameter can be considered as the time ¢ (12)

duration of the evolution.

¦ ¤K0 ¦ © ¦

y y Angle approximation for curvature estimation. Another idea Thus the vectors > , , belong to the same plane and, to deﬁne the curvature of a polygonal line is based on the deﬁnition of therefore, are linearly dependent. It gives the following equation

the curvature as the rate of change of the angle between the tangent and

¨~08¨¦ ¡02¡!¦ ¡;0 ¡[¦

> > > i

the positive direction of the -axis when we proceed along the curve. i

¨¦ ¡%¦ ¡[¦

¢¥

i i

c c c

> > >

{ i

Let denote the turn angle at (see Fig. 24). Let us deﬁne the discrete i

¨¦ ¡%¦ ¡[¦

i i

d > d > d >

curvature at as

i i

{ 9

¢ (13)

;5 describing the osculating plane.

Curvature vector approximation via derivatives. One can es- Principal normal. Unlike the planar case, there is an inﬁnity nor-

timate the tangent, normal, and curvature vectors using ﬁnite difference mals to the curve at each point. The line in the osculating plane at =

¦ e approximations of curve derivatives. The ﬁrst derivative of at perpendicular to the tangent line is called the principal normal. In its

can be approximated by direction let us place a unit vector z such that n(P) changes continuously

when = moves along the curve.

¦ 0 ¦ ¦ (0 ¦ ¦ 0 ¦

¢ 5 0

¦ 3¢ ¨¦ ¡ ¦ ¥¡¦

e e e e S

Curvature vector. If the curve S is param-

¦ (¢ ¦ r

y e e e

The second derivative of at can be approximated as follows eterized by arc length e , then is a unit tangent vector:

S S S

¦ ¦ ¦

¡ ¡ ¡

¢ ¢ 0 5 S ¢ S ¢ S ¢ts:9

i i i i

y z

(14)

£ ¦&5B ¦& 5 @ i i i i

i i i i

S S

¢ ¦ r

y e e The vector } belongs to the osculating plane and perpen-

Space Curves dicular to y . We can therefore introduce a proportionality factor

S S

¢ ¦ r ¢ 9

y e e z

Velocity and acceleration. A space curve can be speciﬁed by the }

¡m¢¡ ¦ ¡ ¢¢¡¦

parametric equations ¨8¢u¨¦ , , or, in vector nota-

¢ ¦

S S

tions, . The vector } , which express the rate of change of the tangent when we

S S

If is time, then is the velocity vector of the end point of the proceed along the curve, is called the curvature vector. The factor is

4 4 vector , and is the acceleration vector. called the curvature.

Arc length. Consider a space curve described parametrically Osculating circle. For a space curve, the osculating circle or circle

of curvature can be deﬁned as the limiting position of the circle pass-

¦ ¢ ¨¦ ¡ ¦ £¡[¦ 9

ing through three nearby points of the curve when two of these points =

approaches the third. For a point > on the curve, the center of the os-

If is the arc length from the point on the curve corresponding to the culating circle lies on the principal normal at distance from

= = >

parameter value to the general point of the curve then > , where is the curvature at . The center of the osculating circle is

called the center of curvature and is called the curvature

O!S O!S O!S S

S S

¨ ¡

¡ radius.

4 4 4

¦ ¢ Q S 5 S 5 S S A9 ¢ Q

i i

a a

e It can be shown that the best approximation of the curve by a circle is

i i

TP !P %P i

i given by the osculating circle. ag a/g Torsion. The curvature measures the rate of change of the tangent Osculating plane. For a curve, the tangent can be deﬁned as the when moving along the curve. Let us now introduce, the torsion, a limiting position of the line passing through two nearby points of the quantity measuring the rate of change of the osculating plane. curve when one of these points approaches the second. For a space A curve normal orthogonal to the osculating plane is called the bi-

curve, the osculating plane can be deﬁned as the limiting position of ¢

S yS z normal. It is given by .

the plane passing through three nearby points of the curve when two of r

e z It is easy to show that is parallel to :

these points approaches the third.

S S S S S

¢ ¦ ¢ ¦ ¢ ¦

= = =

? ?

Let , 4 , be three points on a curve

y z y z z

S S S S S

¢ ¢ 5 ¢

2 z y y 6y

given parametrically

e e e e e

¦ "¢ ¨¦ ¡ ¦ £¡[¦

The torsion is deﬁned by the formula

¤ ¤¢

="? = =

Consider the plane passing through the points , 4 , .

v S ¢t0 9

¤ ¢¦©¨T ¡% ¥¡#

Here the vector deﬁnes a generic point = on the plane, the

vector ¤ is orthogonal to the plane, and is constant. Then the function ¢q

If for every point on a given space curve, then the curve is

¤§¦ (¢ ¦ ¤ 0

planar.

(¢6 (¢6

¢ b

? has zeros at , 4 , and :

curve

(¢ ¤§¦ (¢¥ ¤§¦ (¢ 9

¤§¦ Normal

4 Tangential plane Hence, according to Rolle’s theorem, plane

osculating

¡ ¡ ¡ ¡

¦ (¢ ¢¥¤ ¦¥¦B ¤ ¦¥§ (¢ ¤ circle

¨ ¨ ¨

¦ ¥¦ ¦ ¥§ ¦ ¥¦@

= =

? ?

4 4 4

for some , , and . When , ,

= => ? >

approaches some as , we obtain, for the limiting ¤ Osculating k

values of and , the conditions: plane

¤§¦ ¢ ¦ ¤ 0 ¢

> >

¦ ¤ ¢ ¦ ¢

¤ Fig. 25: Geometric attributes of space curve.

> >

¡ ¡

¤ ¦ ¢ © ¦ ¤ ¢

> > v 9

Frenet formulas. The orthogonal frame Local shape of space curve. Let us approximate shape of a space

¦ ¦

e e e

curve parameterized by arc length . By expanding into the

By z

Taylor series, with as the origin, we get:

is known as the Frenet frame at the point on the curve. The Frenet

¡ ¡ ¡ ¡ ¡ ¡

e e e

5 59A9@9 ¢ 0 5 9B9@9 5 ¦ "¢ 5 S

frame rotates as the point moves alongS the curve.

e y e e

e The orthonormal expansion of z relative to the Frenet frame

yields

S O"S O3S O3S

| | |

z z z z

e e e

S S S S

5 ¢ 5 5 9 5 569@9@9 5 569@9@9

y y z z z

v P v P v P

e e e e

Note that

S S S S

S where a prime mark indicates differentiation with respect to arc length

e y z

y z z

z . To express the shape of the curve near the origin with , and as

S S S S S

¢t0 ¢u0 ¢ ¢t0 ¢ 9

y z z z

v v v z

the coordinate axes, let us take the ﬁrst term of each coefﬁcient of y ,

e e e e e

¨ ¡ ¡

y z and in the previous equation and replace , and by , and

Thus S respectively. It gives

S ¢u0 5 9

| |

e e

¡I¢ 9 ¨ ¢ ¡ ¢

for the space curves:

S S

r Eliminating we obtain

S S

y e z

r ¢ 0 5

S S

z e y

s s

| |

r ¢ 0 9

4 4

¡ ¢ ¨ ¡ ¢ ¡ ¡ ¢ ¨ 9

e [z

Curvature and torsion for arbitrary parameterized curve. From these equations we can analyze curve local shape. The projection

¦ m¢ ¨¦ k¡!¦ ¥¡[¦

¨[¡

Consider a space curvS e parameterized by a of the curve on the plane (the osculating plane) is a symmetrical

¦ r (¢

parameter . Since e , where is arc length, we have second-degree curve tangent to the axis at the origin. The projection

¡ ¡

on the plane (the normal plane) is symmetrical with respect to the

¦ ¦ © ¦

¡ ¡¨

¢ ¢ 9

¢ axis and has a cusp at the origin. The projection on the plane

y z y

¦ ¦ © ¦

(the tangential plane) is anti-symmetrical with respect to the origin and tangent to the ¨ axis. See Fig. 26 below.

Notice that

O!S S S S S

4 4 4 4

e e e

S S S S 4 S

z y

(15)

4 4 4

e e

S ¢ © y

Thus and therefore the curvature vector is given by

0 ¦ ©

y y

¢ ¢

} z 4

Fig. 26: Projection of a space curve onto the osculating plane (left),

| normal plane (middle), and tangential plane (right). ¦

The curvature can be also computed if we multiply (15) by z

S S

| |

¢ ¦ r 4&¢

ymz e

Notice also that y and thus

¦ A grey-scale image consists of an 2D array of pixels where every pixel

(16)

¦ is assigned a positive number characterizing the pixel intensity. For ex-

ample, the following image ¦

The torsion is computed in a similar way. Differentiating (15)

by yields

S S S S S S S

consists of pixels and is represented by the array shown below.

z y e e e e

S S S S S S S

z y

4 4

e e

£¢ £¢

¤ ¤ ¤ ¥ ¥ ¥

£¢ £¢

¤ ¥

g gfgjg g g gfgfgjg g g gfg g gfgfg g g gjgfg g

£¢ £¢ ¥

In view of the Frenet formulas the scalar product between the above ¤

g g g g g gfgfg g g g g gIg

£¢ £¢ £¢

¤ ¤ ¤ ¥ ¥

g g g gfg g g g gfg g g g g g g

£¢ £¢

¤ ¥

equation and yields ¤

g g g gfg g g g gfg g g g g g g g

£¢ £¢

¤ ¤ ¥ © ¢ 9

g g g g gfgfg g g g g g g g

£¢ £¢

¤ ¤ ¤ ¤ ¥ ¥ ¥ v

g g g gfg g gfgfg gfg g g g g g g

g g g g gfgfg g g g gIg

Taking into account (16) we have g

g gfgjg g g gfgfgjg g g gfg g gfgfg g g gjgfg g

For the above image, the intensity values vary from (black) to

| |

¢ ¢

(white). Usually the range of intensity values extends from (black)

§¦¨¦ to (white). and ﬁnally arrive at Often it is convenient to forget that a grey-scale image has a discrete

nature and think that the image is given by a positive continuous function

¦©¨§ k¡ #

¦ ¦ © ¦

deﬁned on a rectangular domain where the intensity

¢ 9

¦©¨§ ¡ ¦©¨§ ¡

¦ © ¦

4 (brightness) at is given .

A binary image is a special case where at each pixel the intensity

¦ ¦

Notice that is a second-derivative quantity while is a third- function takes on one of two values, black and white. Plane ﬁgures are derivative quantity. often represented by binary images.

¦©¨§ ¡ ¦©¨§ ¡

Active contour models (snakes) in image processing. The skeleton of > is given by all points such that

D Curvature-driven curve evolutions are used in image processing for de- D for all such that .

tection of object boundaries.

¦©¨§ k¡[

Let us introduce a positive function that achieves its minimal

¦©¨§ ¡

values where is large. For example, such a function can be

deﬁned by

¦©¨§ k¡ ¢ 9

s"5 ¦©¨§ k¡

¦¢¡

Consider a contour ¡ oriented by its inner normal and enclosing an

object whose boundary we want to detect. Let us move all the points of

¦©¨T ¡ A¦ 5 @

the contour along the contour normals with speed equal to

where is the curvature of the contour and is a positive constant, see Fig. 27.

deformations along normals I(x,y) Additional Sources of Information with speed g(k+c) n 1. Internet x,y g(x,y) 2. M. Hosaka. Modeling of Curves and Surfaces in CAD/CAM. Springer-Verlag, 1992. Chapter 5. x,y 3. D. J. Struik. Lectures on Classical Differential Geometry. Dover, 1988. Chapter 1. Fig. 27: Geometric idea of using active contours for detection of object boundaries. 4. J. Gallier. Geometric Methods and Applications for Computer Sci-

ence and Engineering. Springer-Verlag, 2000.

¦¢¡T

We can represent the contour evolution as a family of contours

¡ ¦¢¡§

where parameterizes the family and parameterizes the curve . 5. M. Berger and B. Gostiaux. Differential Geometry: Manifolds,

¦¢¡§

The family of contours satisfy the following initial value problem Curves, and Surfaces. Graduate Texts in Mathematics, Vol 115.

Springer-Verlag, 1988.

¡ ¡

r ¢ ¦ A¦ ¦ 5¥B ¦

¦¢¡§ ' ¢ ¦¢¡%

! (17) 6. J. W. Bruce and P. J. Giblin. Curves and Singularities: A Geomet- rical Introduction to Singularity Theory. Cambridge University

where is constant. Press, 1992.

¦¢¡§ k ¦¢¡%

!¤£

Let the evolving contour converges to a limit contour .

# #

!¤£ !¥£ Thus along . So is close to object boundary. A similar but more complex contour evolution provides a user with an efﬁcient tool for detecting objects in grey-scale images. Fig. 28 demonstrates tumor area detection in a shaded MRI image of human brain.

Fig. 28: Tumor area detection with active contours.

Practical skeleton extraction. Let a ﬁgure ¥ is represented by a

¥ © inside and ¦©¨§ ¡ ¢us (white) otherwise. The distance transform is an operator that maps a binary image into a grey-level image. The pixel values of the grey level image represent a distance from the border of the binary image. The three most common distances for this transform are: Euclidean, city-block (4-connected),

and chess-board (8-connected) distances.

Let > be a binary image. It is used to compute grey-scale im-

? D ages 4 according to the following iterative

algorithm:

*,Y

D > D

1. ? , where minimum is taken

for all such that

D D 2. Terminate if ? . 11

Problems ¦

1. Consider a closed curve from the ﬁgure below. The curve is |

oriented with respect to its inward normal. Let ¦ be the curva- | ture of the curve. Sketch the graph of ¦ . Indicate the curvature maxima, minima, and zeros in ﬁgure below. tangent

normal

2. Find the curvature and an equation for the circle of curvature of the

+ * Y

¡£¢ ¨ ¦ r BsB curve at the point . 3. Fig. 3 shows curves reconstructed from the following curvature

functions

0 s

| | | | + *,Y

¦ (¢ ¦ (¢ ¦ ¢ 0 ¦ (¢ ¦ 9

e e e e e e e e e

56s

4 e

Determine which curve corresponds to which curvature. Brieﬂy

explain your reasoning.

\AX+ + * Y

¦ (¢¦ kA 9

4. Find the curvature of the curve

5. Consider a family of segments of length such that for any segment one of its end-point lies on the ¨ -axis and another end-point lies on the ¡ -axis. Find the curvature of the curve consisting of the centers of the segments.

6. A circular disk of radius s in the plane rolls without slipping along the ¨ -axis. The curve described by a point on the circumference is called a cycloid.

(a) Obtain a parameterization of the cycloid. (b) Find the curvature at the highest point of the cycloid. (c) Find the length of a part of the cycloid corresponding to a complete rotation of a disk.

7. Consider a Bezier´ curve of degree and its endpoint. Show that the curvature of the curve at the endpoint is determined by the triangle formed by the endpoint and the next two control points. Find a

geometric formula for the curvature.

¨;0 ¡¢ 4

8. Find the the envelope of the family of straight lines ,

where parameterizes the family.

@¦©¡ 5¥ &06¨ ¢ 9. Find the envelope of the family of curves 4 , where parameterizes the family. 10. Prove that the evolute of an involute of a given curve is the curve itself.

11. Show that the sum of the distances from each point of an ellipse to the foci is constant. 12. Prove that the rays emitted from a focus of an ellipse after being reﬂected by the ellipse will pass through the other focus of the ellipse.

13. Suppose the incident rays are parallel to a given vector ¡ . For a given curve find the caustic generated by the reflected rays. 14. Let the reflector be a circle and a source of light be a point inside the circle. Plot the caustic generated by the reflected rays. 15. Propose a method to estimate the torsion of a space curve approximated by a polyline. 16. A space curve can be reconstructed (up to rigid transformations) from its curvature and torsion functions via the Frenet equations. Describe a curve with constant curvature and torsion.