23686 Numerical Geometry of Images Tutorial 4 Differential Geometry I - Curves Anastasia Dubrovina c 22 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Differential Geometry - Introduction Differential geometry - study of local properties (and sometimes global..) of curves and surfaces, using methods of differential calculus..9.8 2.7.6.5.4 5 5.3.2..5.5 2.5.5 Planar curve Curve in R 3 Surface in R 3.5.5 3 2 5 5 2 25 Figure: Examples 2 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Parametric curves A parametric curve in R n is a map C : I R n on an interval I R into R n. Explicitly, C is written as C(p) = (C (p), C 2 (p),..., C n (p)), where p I is the parameter of the curve and C,..., C n : I R. The curve C(p) is called differentiable if C i (p), i n are differentiable functions w.r.t. the parameter p. The set of all values {C(p) ; p I } is called the trace of the curve. 3 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Examples of parametric curves in R 2 and R 3 Which curve is differentiable and which is not?.9.8 2.7.6.5.4.3.2..5.5 () C(p) = (p 3, p 2 ) (2) C(p) = (cos(5p), sin(5p), p) 2.5.5.5.5 3 2.9.8.7.6.5 2 3.4.3.2. 4 5 5.5.5 (3) C(p) = (p 3 4p, p 2 4) (4) C(p) = (p, p ) 4 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Tangent and arclength Tangent (velocity) vector: C p (p) = ( C (p), C 2 (p),..., C n (p)) Points with C p (p) = are called singular. At regular (non-singular) points of C(p), the unit tangent vector is defined as T (p) = Cp C p. A differentiable parametric curve C(p) satisfying C p (p) = for all p I is said to be regular. Question: which one of the 4 curves from the previous slide is regular and which is not? Arclength: for a curve parameterized on (, P), the length from to p is given by s(p) = p C p ( p) d p ; p (, P) 5 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Arclength parameterization Let r(p) : (, ) (, R) be a monotonously increasing (decreasing) function with the inverse p(r). The curve C(r) C(p(r)) has a trace equal to that of C(p), yet, the corresponding tangent vectors C r (r) and C p (p) are different. A special case of such a re-parameterization is the arclength s(p). C(s) = C(p(s)) is called arclength parameterization of the curve. C(s) also satisfies: C s (s) =. It corresponds to traveling along the curve with unit velocity. s(p) = p C p ( p) d p, ds dp = C p, C s = dc ds = C dp p ds = C p C p Different parameterizations correspond to traveling along the curve with different speeds. 6 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Re-parameterization example What are the traces of the following two curves? D : [, 2π] R 2 defined as D(p) = (r cos(p), r sin(p)) C : [, 2πr] R 2 defined as ( ( s ) ( s )) C(s) = r cos, r sin r r Their tangent vectors D p and C s : D p (p) = ( r sin(p), r cos(p)) D p (p) = r ( ( s ) ( s )) C s (s) = sin, cos C s (s) = r r C is the arclength re-parameterization of D: C(s) = D(p(s)). 7 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Normal and Curvature - Planar curves Let C(p) : I R 2 be a regular curve, with T (p) = C p / C p. ( ) Its normal is defined by N(p) = T (p). Let C(s) = C(p(s)) be the arclength parameterization of the curve. There exist a smooth function α : [, L] R such that T (s) = (cos α(s), sin α(s)) C ss (s) = T s (s) = ( sin α(s), cos α(s)) dα(s) ds The curvature of C(s) is defined by κ(s) = dα(s) ds. Show that: C ss (s) = κ(s)n(s), and κ(s) = C ss (s). 8 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Curvature via radius of osculating circle In R 2, κ(s) equals the radius of the osculating circle at C(s): y N T α( p) ρ = κ C( p) x 9 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Curvature of planar curves - examples Straight line Arclength parameterization of a straight line is given by C(s) = as + b, a =, therefore its curvature is κ(s) (prove...) Circle Arclength parameterization of a circle of radius r in R 2 is C(s) = ( r cos ( ) ( s r, r sin s )) r C s (s) = ( sin ( ) ( s r, cos s )) r C ss (s) = ( r cos ( ) s r, r sin ( )) s r Therefore: κ(s) = ( r ) 2 ( cos 2 (s) + sin 2 (s) ) = r What is the sign of κ(s)? / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Curvature of an arbitrarily parameterized curve Let C(p) : I R 2 be a regular curve. κ(p) =? s(p) = p C p ( p) d p ds dp = C p(p) = xp 2 + yp 2 T (p) = (cos(α(p)), sin(α(p)) α(p) = arctan y p(p) x p (p) dα(p) dp = κ(p(s)) = dα(p(s)) ds d(y p /x p ) + (y p /x p ) 2 = y ppx p x pp y p dp xp 2 + yp 2 = dα(p) dp dp ds = dα(p) dp ds dp = y ppx p x pp y p (x 2 p + y 2 p ) 3/2 The latter formula can also be obtained by applying the chain rule of differentiation on C ss (s) (see Numerical Geometry of Images, chapter 2). / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Examples Example Calculate the curvature of a circle C : [, 2π] C(p) = (r cos(p), r sin(p)) : x p = r sin(p), y p = r cos(p) x pp = r cos(p), y pp = r sin(p) = κ(p) = r 2 r 3 = r Example 2 Calculate the curvature of a circle C : [, 2π] C(p) = (r cos( p), r sin( p)). What can you say about the traces of the two curves? 2 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Parameterized curves in R 3 Let C(s) : [, L] R 3 be a regular curve given in arclength parameterization. The curvature of C at s is defined by κ(s) = C ss (s). Hence, κ(s) is always positive. Its normal unit vector is defined by N(s) = C ss(s) κ(s). Note that N(s) is defined only when κ(s). Exercise: show that C s and C ss are orthogonal to each other. C s 2 = d ds C s 2 = 2 C s, C ss = We will say that s [, L] is singular point of order if κ(s) =, and restrict ourselves to curves without such singular points. 3 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Osculating plane and binormal vector Osculating plane: a plane spanned at every point by C s and C ss (we saw that C s C ss ). Normal plane: a plane perpendicular to the tangent vector T. The normal vector N belongs to the normal plane. Binormal vector is a unit vector B defined by B = T N. The vectors T, N and B define a local orthonormal system of coordinates on the curve. T, N and B are related by the equations: T s κ T N s = κ τ N B s τ B known as Frenet formulas., 4 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Frenet formulas - proof T s = κn By definition of κ. B s = τn B s is orthogonal to B: B, B = B s, B = B s is also orthogonal to T : Recall that B, T s = and so B, T = B s, T + B, T s = B s, T = Therefore, B s is parallel to N. τ(s) defined by B s (s) = τ(s)n(s) is called the torsion of the curve. N s = κt + τb Prove! (Differentiate N = B T ) Sometimes τ(s) is defined by Bs (s) = τ(s)n(s). 5 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Torsion Exercise: show that a curve with κ is planar iff τ. Let us assume a planar curve in the arclength parameterization, given w.l.o.g. by C(s) = (x(s), y(s), ). Then, T = (x s, y s, )/ x 2 s + y 2 s N = ( y s, x s, )/ x 2 s + y 2 s and the binormal is constant: B = (,, ). Therefore, B s = and τ =. Now let C(s) be a curve with τ and κ. We readily have that B s =, and therefore the binormal is a constant vector B = B = const. Therefore, s C(s), B = C s (s), B =. It follows that C(s), B = const, therefore, C(s) is contained in a plane normal to B, i.e. C(s) is a planar curve. 6 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Interpretation of curvature and torsion Both curvature and torsion are invariant intrinsic quantities. The curvature measures the rate of change in the tangent direction at s. By definition, κ(s) >. The torsion measures the rate of change of the osculating plane of the curve. τ(s) can be positive or negative. Physically, a 3D curve can be thought of as being obtained from a straight line by bending and twisting: the curvature expresses the bending, the torsion expresses the twisting. 7 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Fundamental theorem of the local theory of curves Given two differentiable functions κ(s) and τ(s), for s I, there exists exactly one 3D curve (determined up to an Euclidean transformation), for which s is the arclength, κ is the curvature and τ is the torsion. In other words, κ and τ describe completely the local behavior of the curve. 8 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Curvature and torsion For a curve given in arclength parameterization: (Cs Css) C (3) κ(s) = C ss (s), τ(s) = C ss 2 Proof (using Frenet equations): C ss = κn C (3) = κ s N + κn s = κ 2 T + κ s N κτb (C s C ss ) C (3) = κ 2 τ For arbitrary parameterization: κ(p) = C p C pp C p, τ(p) = (C p C pp ) C (3) 3 C p C pp 2 9 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves
Exercise Find curvature and torsion of a helix defined as C(p) = (a cos(p), a sin(p), bp), a >, b R Solution: C p (p) = ( a sin(p), a cos(p), b) C pp (p) = ( a cos(p), a sin(p), ) C (3) (p) = (a sin(p), a cos(p), ) C p (p) = a 2 + b 2, C p (p) C pp (p) = ( ab sin(p), ab cos(p), a 2) C p (p) C pp (p) = a a 2 + b 2 κ(p) = (C p C pp ) C (3) = a 2 b a a 2 + b 2, τ(p) = b a 2 + b 2 A helix has fixed curvature and torsion. 2 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves