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 S respectively. The distance between and is given by ¦ | { e ¦ (¢ 9 S e )*,+- ¦ "¢. ¦/¨ 102¨ $3 456¦©¡# 708¡'$3 4:9 e 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 ¦ D 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 N O OTS O O!S 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 D ¦ ~¢M ¨¦ ¡ ¦ S e e e Let a curve be parameterized by arc length. The ¦ ¢ e ¦ ¦ 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 N O!S O!S S S ¢us ¢ y y y y S ¨ ¡ 4 4 uct gives and, therefore, the vectors and are v v S )*,+- V:W,X:Y[Z \@]^_'` e e S 5 S ¦ (¢baQ \@X:+ + *,Y y TP P S ¢o y { { mutually perpendicular: y . Since , see Fig. 2, e then S S S S + *,Y \@X+ a y S ¢u0 %¢ ¨ ¡ ¨ 4 { { 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 ¨¦ ¡ ¦ e > > 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 S ¦ h h e S S Q S S ¦ (¢ ¨ 5 ¡ ¢ ¦ (¢ ¨ 5 ¡ ¢ji i 4 4 4 4 a 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 ag ¦ ¨¦ ¡ ¦ 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 ¦©¨¦ ¡ ¦ ¦ h ©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 ¨ ¡ h | 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 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 2 O S S S S s s ¡ { { c ¢ ¢ ¢ S S S S h Curve Quality Evaluation via Curvature Profile v v v P ¨ 4 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 ¨ ¡ 4 e ity evaluation. Fig. 4 displays an original curve (the left image) c c d d c c d d c | ¢ ¢ ¦ ¦ h ¨ 5 ¡ v ¨ v e z e 4 4 4 ¨ 5 ¡ 4 ¦ and the curve together with its curvature vectors attached to ¨ 5 ¡ 4 4 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 h ¡ ¡ ¢ ,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 by ¡ ¡ ¤ ¦©¨ | ¢ ¢ (2) ¡ 4 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 © © (we use subindices to denote partial derivatives: means © , 1 _ ¦ 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 ¥§¦ ¥ ¥§¦¤¦ ¥§¦ ¥§¦ ¥ ¥ 4 5 ¤ ¢ 5 ¤ 5 ¤ 5 ¤ 5 ¤ ¢ © © © © © © be approximated by a circle. To do this let us consider a general circle ¡ ¡ ¡ 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.