Tutorial 4. Differential Geometry I - Curves

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1 23686 Numerical Geometry of Images Tutorial 4 Differential Geometry I - Curves Anastasia Dubrovina c 22 / 2 Anastasia Dubrovina CS Tutorial 4 - Differential Geometry I - Curves

2 Differential Geometry - Introduction Differential geometry - study of local properties (and sometimes global..) of curves and surfaces, using methods of differential calculus Planar curve Curve in R 3 Surface in R Figure: Examples 2 / 2 Anastasia Dubrovina CS Tutorial 4 - Differential Geometry I - Curves

3 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 Tutorial 4 - Differential Geometry I - Curves

4 Examples of parametric curves in R 2 and R 3 Which curve is differentiable and which is not? () C(p) = (p 3, p 2 ) (2) C(p) = (cos(5p), sin(5p), p) (3) C(p) = (p 3 4p, p 2 4) (4) C(p) = (p, p ) 4 / 2 Anastasia Dubrovina CS Tutorial 4 - Differential Geometry I - Curves

5 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 Tutorial 4 - Differential Geometry I - Curves

6 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 Tutorial 4 - Differential Geometry I - Curves

7 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 Tutorial 4 - Differential Geometry I - Curves

8 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 Tutorial 4 - Differential Geometry I - Curves

9 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 Tutorial 4 - Differential Geometry I - Curves

10 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 Tutorial 4 - Differential Geometry I - Curves

11 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 Tutorial 4 - Differential Geometry I - Curves

12 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 Tutorial 4 - Differential Geometry I - Curves

13 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 Tutorial 4 - Differential Geometry I - Curves

14 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 Tutorial 4 - Differential Geometry I - Curves

15 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 Tutorial 4 - Differential Geometry I - Curves

16 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 Tutorial 4 - Differential Geometry I - Curves

17 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 Tutorial 4 - Differential Geometry I - Curves

18 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 Tutorial 4 - Differential Geometry I - Curves

19 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 Tutorial 4 - Differential Geometry I - Curves

20 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 Tutorial 4 - Differential Geometry I - Curves