℄ ℄ Obzornik mat. fiz. 55 (2008) 3 Franc Savnik 1,2 ◦ ℄ K x 2 +y 2 = a 2 z = 0 S K S K 120 ◦ K x y z O k K T S K O a xy Oxyz S z S k S b k T(r(t)) k z T S T r =f(t) =  (a+bcost)cos3t,(a+bcost)sin3t,bsint  , t∈ [0,2π]. (1.1) r(t) r t+ 2π 3  r t+ 4π 3  t 2π r =f(t)+u f t+ 2π 3  −f(t)  , t∈ [0,2π], u∈ [0,1]. (1.2) Obzornik mat. fiz. 55 (2008) 3 Tri skulpture K n K 2π n r =f(t) =  (a+bcost)cosnt,(a+bcost)sinnt,bsint  , t∈ [0,2π]. (1.3) n r =f(t)+u f t+ 2π n  −f(t)  , t∈ [0,2π], u∈ [0,1]. (1.4) n = 2 t∈ [0,π] x y z a kz Δf(t) x y O -ak x z = 0 r = g(t) − π 2 ≤t≤ π 2 r =f(t) =  k x (a+bcost)cos2t,k y (a+bcost)sin2t,k z bsint  , t∈  − π 2 , 3π 2  . (2.1) 81–90 Franc Savnik − π 2 ≤t≤ π 2 e(t) T(f(t)) T Δf(t) = cos(t)|f ′ (t)Δt|e(t) Δt cost [− π 2 , π 2 ] r =g(t) =  f(t)+Δf(t) t∈  − π 2 , π 2  f(t) t∈  π 2 , 3π 2  , (2.2) P k t = 2kπ n − π 2 k = 0,1,2,...,n t∈ [− π 2 , 3π 2 ] P 0 P n P k r k n k r k O r 0 r max r min 81 18 189 135 216 r k r0 P k P k P k+1 P k r k 2π n Obzornik mat. fiz. 55 (2008) 3 Tri skulpture r k r k =rAsin 4kπ n +B  +C, k = 0,1,2,...,n. (2.3) r n A B C 4kπ n r k r =f(t) = (sin3tcost,sin3tsint,0), t∈  π 2 , 3π 2  , (3.1) y AB AC AB t∈ [ π 2 , 5π 6 ] x y z O 1 A B C x y z O 1 A B B 1 T T 1 S AB y AB1 B − √ 3 2 , 1 2 ,0 B1 − √ 3 4 , 1 2 ,− 3 4 T(f(t)) AB y t− π 2 T 1 s(t) = (f(t)·j)j S(s(t)) T 81–90 Franc Savnik y T T 1 S ρ(t) =f(t)−s(t) r(t) =s(t)+cos t− π 2  ρ(t)+sin t− π 2  ρ(t)×j. t AB 1 f 1 (t) =s(t)+sin(t)ρ(t)−cos(t)ρ(t)×j, t∈  π 2 , 5π 6  . (3.2) y AB AC AB 1 y AC 1 f 2 (t) =  −f 1 (t)·i,f 1 (t)·j,−f 1 (t)·k  , t∈  π 2 , 5π 6  . (3.3) AB 1 OB 1 C 1 B 1 f 3 (t) = 2(f 1 (t)·e)e−f 1 (t), t∈  π 2 , 5π 6  , (3.4) e −−→ OB 1 x y z O A B 1 C 1 1 P AB1 y AC1 OB1 B1C1 C1 AB1 AP PB1 O A B 2 C 2 x 1 y z R Q a = 1,24345655 AB2 AC2 C2B2 10 −6 AB2 AB 1 AC 1 C 1 B 1 A B 1 C 1 A Obzornik mat. fiz. 55 (2008) 3 Tri skulpture B 1 C 1 t at A a a sin 2 t at t ϕ(t) = atsin 2 t a AB 2 g 1 (t) =s(t)+sin[ϕ(t)]ρ(t)−cos[ϕ(t)]ρ(t)×j, t∈  π 2 , 5π 6  . (3.5) y AC 2 OB 2 C 2 B 2 g 2 (t) =  −g 1 (t)·i,g 1 (t)·j,−g 1 (t)·k  , t∈  π 2 , 5π 6  , (3.6) g 3 (t) = 2(g 1 (t)·e)e−g 1 (t), t∈  π 2 , 5π 6  , (3.7) e −−→ OB 2 x y z O 1 A B 2 C 2 AB2 −−→ OC2 AC2 B2C2 −→ OA B 2 C 2 x y O 0.5 r = h3(t) r = h4 4π 3 −t r = χ(t) z = 0 r = χ(t) r =f(t) f ′ (t)·(f ′′ (t)×f ′′′ (t)) f ′ 1 (t)·(f ′′ 1 (t)×f ′′′ 1 (t)) = 0 81–90 Franc Savnik  π 2 , 5π 6  P g ′ 1 (t)·(g ′′ 1 (t)×g ′′′ 1 (t)) = 0  π 2 , 5π 6  Q R a b c −→ OA −−→ OB 2 −−→ OC 2 Δt n 1 (t) n 2 (t) T(g 1 (t)) AB 2 Δg 1 (t) =|g ′ 1 (t)|Δt(αc+βn 1 (t))cos 2 3t. α β π 2 ≤ t ≤ 5π 6 cos 2 3t AC 2 Δg 2 (t) =|g ′ 2 (t)|Δt(αb+βn 2 (t))cos 2 3t B 2 C 2 −→ OA Δg 3 (t) = |g ′ 3 (t)|Δtγacos 2 3t γ P 0 P 120 P 240 P 360 P 0 P 120 P 240 (0,0,10 5 ) P0 P120 P240 z = 0 (10 5 ,0,0) P0 P120 P240 x = 0 3π 360 3π B 2 C 2 h 3 (t) =g 3 (t)+Δg 3 (t), t∈  π 2 , 5π 6  , y Obzornik mat. fiz. 55 (2008) 3 Tri skulpture y h 4 (t) =  −h 3 (t)·i,h 3 (t)·j,−h 3 (t)·k  , t∈  π 2 , 5π 6  . χ(t) = 1 2  h 3 (t)+h 4 ( 4π 3 −t)  , t∈  π 2 , 5π 6  , y h 1 (t) =g 1 (t)+Δg 1 (t), h 2 (t) =g 2 (t)+Δg 2 (t) χ(t), t∈  π 2 , 5π 6  , (3.8) y A B 2 C 2 √ 3 xy y z = 0 z = kd k = 1,...,19 d x =kd k = 1,...,22 d (0,0,10 5 ) (10 5 ,0,0) d y y t t = π 2 +k π 3n 81–90 Franc Savnik k = 0 n t∈  π 2 , 5π 6  n = 120 P 0 P 1 P 2 ...P 360 P 0 P 360 ℄ ℄ ℄ ℄ ℄ ℄ ℄ ℄ Obzornik mat. fiz. 55 (2008) 3