Evolutes and involutes of fronts Masatomo Takahashi Muroran Institute of Technology

Evolutes and involutes of fronts Masatomo Takahashi Muroran Institute of Technology

evolute involute, evolvent involute : involvo

evolvent : evolvo 6 4 6 4 4 6 4 6

H 3

γ : I R C γ(t) 0, t(t) := γ(t) γ(t), n(t) := J(t(t)) = J( γ(t)) γ(t), J π/ {t(t), n(t)} γ(t) ( ṫ(t) ṅ(t) ) = ( 0 γ(t) κ(t) γ(t) κ(t) 0 ) ( t(t) n(t) ), κ(t) = det( γ(t), γ(t)) γ(t) 3 = ṫ(t) n(t) γ(t) κ(t)

γ(t) = γ, γ : I R γ γ R A b γ(t) = A(γ(t)) + b ( t I). κ : I R κ γ : I R γ(t) = ( ( cos ) κ(t) dt dt, ( sin ) ) κ(t) dt dt.

γ, γ : I R s = γ(t) s = γ(t) κ κ γ γ γ : I R γ Ev(γ) : I R, t γ(t) + κ(t) n(t), κ(t) 0 t 0 I γ Inv(γ, t 0 ) : I R, t γ(t) ( t t 0 γ(t) dt ) t(t).

.0 6 6 4 4 0.5.0 0.5 0.5.0 6 4 4 6 6 4 4 6 0.5 4 4.0 6 6 t 0 = 0 t 0 = π () Ev(Inv(γ, t 0 ))(t) = γ(t). () Inv(Ev(γ), t 0 )(t) = γ(t) + κ(t 0 ) n(t). Ev Inv

γ γ λ : I R, t γ(t) + λn(t) (λ R) γ λ /κ(t) γ λ Ev(γ λ )(t) = Ev(γ)(t)

(γ, ν) : I R S : (γ, ν) : I R S (γ, ν) θ = 0 γ(t) ν(t) = 0 ( t I). θ T R = R S (γ, ν) ( γ(t), ν(t)) (0, 0) (γ, ν) γ : I R ν : I S : C (γ, ν) γ : I R ν : I S ν(t) = n(t) (γ, ν) γ

(n, m) n < m = n + k γ : R R ( γ(t) = n tn, ) m tm ν : R S ν(t) = t k + ( tk, ) (γ, ν) k = (, 3) (3, 4) (, 4) (, 5) γ(t) = ( t, 3 t3) γ(t) = ( 3 t3, 4 t4) γ(t) = ( t, 4 t4) γ(t) = ( t, 5 t5)

(γ, ν) : I R S µ(t) := J(ν(t)) {ν(t), µ(t)} γ(t) ( ) ( ) ( ) ν(t) 0 l(t) ν(t) =, µ(t) l(t) 0 µ(t) l(t) = ν(t) µ(t) γ(t) ν(t) = 0 β : I R γ(t) = β(t)µ(t) (l, β) : I R, t (l(t), β(t)) t

γ κ(t) (γ, n) (l(t), β(t)) l(t) = β(t) κ(t) β(t) = l(t) = κ(t) t 0 γ β(t 0 ) = 0 t 0 γ (γ, ν) l(t 0 ) = 0 (l(t), β(t)) (0, 0) (γ, ν), ( γ, ν) : I R S

(γ, ν) ( γ, ν) R A b γ(t) = A(γ(t)) + b, ν(t) = A(ν(t)) ( t I). (l, β) : I R (l, β) (γ, ν) : I R S ( ( ( γ(t) = ν(t) = ( cos ) β(t) sin l(t) dt ( ) ( l(t) dt, sin dt, β(t) cos )) l(t) dt. ) ) l(t) dt dt,

(γ, ν), ( γ, ν) : I R S (l, β) ( l, β) (γ, ν) ( γ, ν) γ : I R : ν : I S (γ, ν) : I R S γ(t) ν(t) = 0 ( γ(t), ν(t)) (0, 0) t I. ( ν(t) µ(t) ) = ( (l(t), β(t)) (0, 0). 0 l(t) l(t) 0 ) ( ν(t) µ(t) ), γ(t) = β(t)µ(t)

(γ, ν) : I R S l(t) 0 γ Ev(γ)(t) := γ(t) β(t) l(t) ν(t). ( ) t γ Inv(γ, t 0 )(t) := γ(t) β(t) dt µ(t). t 0 () Ev(γ)(t) ν Ev(γ) (t) := J(ν(t)) = µ(t) () Inv(γ, t 0 )(t) ν Inv(γ,t0 ) (t) := J (ν(t)) = µ(t)

() Ev(Inv(γ, t 0 ))(t) = γ(t). () Inv(Ev(γ), t 0 )(t) = γ(t) β(t 0) l(t 0 ) ν(t). t 0 γ Inv(Ev(γ), t 0 )(t) = γ(t) t 0 γ () t 0 Ev(γ) γ 3/ t 0 () t 0 Ev(γ) Ev(Ev(γ)) γ 4/3 t 0

γ t 0 3/ γ(t 0 ) = 0, det ( γ(t 0 ),... γ (t 0 )) 0. γ t 0 4/3 γ(t 0 ) = γ(t 0 ) = 0, det ( γ (3) (t 0 ), γ (4) ) (t 0 ) 0. 4 4 4 4 4 4 4 4 3/

4/3 γ(t) = ( 3 t3, 4 t4) () t 0 γ Inv(γ, t 0 ) 3/ t 0 () t 0 γ 3/ Inv(γ, t 0 ) 4/3 t 0 t 0 γ d dt ( ) β (t 0 ) = 0 d l dt Ev(t 0) = 0.

(γ, ν) : I R S () γ γ () γ 3/ γ (d/dt)ev(t) = 0

(Inv(γ, t 0 ), µ) : I R S () Inv(γ, t 0 ) γ () Inv(γ, t 0 ) 3/ γ Ev(γ) Inv(γ, t 0 ) (γ, ν) : I R S (l, β) l(t) 0

( ) () γ Ev (γ)(t) = Ev(Ev(γ))(t) = Ev(γ)(t) () γ t 0 d l(t) dt ( ) β l (t) µ(t). Inv (γ, t 0 )(t) = Inv((Inv(γ), t 0 ), t 0 )(t) ( ( t ) ) t = Inv(γ, t 0 )(t) β(t)dt l(t) dt ν(t). t 0 t 0 Ev 0 (γ) := γ, Ev (γ) := Ev(γ), Ev n (γ) := Ev(Ev n (γ)) Inv 0 (γ, t 0 ) := γ, Inv (γ, t 0 ) := Inv(γ, t 0 ), Inv n (γ, t 0 ) := Inv(Inv n (γ, t 0 ), t 0 )

β 0 (t) := β(t), β n (t) := β n (t), β n (t) := l(t) ( t t 0 β n+ (t)dt ) l(t) (n ) () γ n Ev n (γ)(t) = Ev n (γ)(t) β n (t)j n (ν(t)), () γ n Inv n (γ, t 0 )(t) = Inv n (γ, t 0 ) + ( t t 0 β n+ (t)dt ) J n (ν(t)). J k J k J π/

(Inv, ν) invo (Inv, µ) invo (γ, ν) evo (Ev, µ) evo (l, β) (Ev, ν)

(Inv, ν) invo (Inv, µ) invo (γ, ν) evo (Ev, µ) evo (Ev, ν) (l, β ) (l, β ) (l, β) (l, β ) (l, β ) l(t) 0 l(t) = β(t) 0 ( t ) t t β(t)dt dt β(t)dt β(t) β (t) β (t) t 0 t 0 t 0