Chapter 2

计算参数曲线的弧长、曲率、Frenet标架¶

Method 1: Using x[t] and y[t]¶

In [18]:

ClearAll["Global`*"]

x[t] = a*Cos[t];
y[t] = b*Sin[t];

x1[t] = D[x[t], t];
y1[t] = D[y[t], t];
x2[t] = D[x1[t], t];
y2[t] = D[y1[t], t];
k[t] = (x1[t]*y2[t] - y1[t]*x2[t])/(x1[t]^2 + y1[t]^2)^(3/2);

Simplify[k[t]]

ClearAll["Global`*"] x[t] = a*Cos[t]; y[t] = b*Sin[t]; x1[t] = D[x[t], t]; y1[t] = D[y[t], t]; x2[t] = D[x1[t], t]; y2[t] = D[y1[t], t]; k[t] = (x1[t]*y2[t] - y1[t]*x2[t])/(x1[t]^2 + y1[t]^2)^(3/2); Simplify[k[t]]

Out[18]:

Method 2: Using r[t]¶

把上述两个方法封装成函数

说明：

• $a(t)=\dfrac{dr}{ds}=\dfrac{dr}{dt}\cdot\dfrac{dt}{ds}$是单位切向量
• $k(t)=\kappa(t)=\lvert\dfrac{da}{ds}\rvert$是曲率

In [44]:

ClearAll["Global`*"]

$Assumptions = {a > 0, b > 0, Element[t, Reals]};
r[t] = {a*Cos[t], b*Sin[t]};

drdt[t] = D[r[t], t]; (* dr/dt *)
dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *)
drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *)
dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *)
k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *)

Simplify[k[t]]

ClearAll["Global`*"] $Assumptions = {a > 0, b > 0, Element[t, Reals]}; r[t] = {a*Cos[t], b*Sin[t]}; drdt[t] = D[r[t], t]; (* dr/dt *) dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *) drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *) dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *) k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *) Simplify[k[t]]

Out[44]:

Method 3: just use ArcCurvature¶

In [48]:

ClearAll["Global`*"]

r[t] = {a*Cos[t], b*Sin[t]};

k[t] = ArcCurvature[r[t], t];

Simplify[k[t]]

ClearAll["Global`*"] r[t] = {a*Cos[t], b*Sin[t]}; k[t] = ArcCurvature[r[t], t]; Simplify[k[t]]

Out[48]:

Questions:¶

2.1¶

In [136]:

ClearAll["Global`*"]

(* 1 *)
$Assumptions = {a > 0, Element[t, Reals]};

r[t] = {t, a*t^2};

(* trusted result *)
Simplify[ArcCurvature[r[t], t]]

(* steps *)
drdt[t] = D[r[t], t]; (* dr/dt *)
dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *)
drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *)
dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *)
k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *)

(* print results *)
Simplify[drds[t]] (* a = dr/ds *)
Simplify[dads[t]] (* da/ds *)
Simplify[k[t]] (* k[t] = norm[da/ds] *)

Block[{Line = Arrow},
    ParametricPlot[{t, t^2}, {t, -1, 1}]
]

ClearAll["Global`*"] (* 1 *) $Assumptions = {a > 0, Element[t, Reals]}; r[t] = {t, a*t^2}; (* trusted result *) Simplify[ArcCurvature[r[t], t]] (* steps *) drdt[t] = D[r[t], t]; (* dr/dt *) dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *) drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *) dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *) k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *) (* print results *) Simplify[drds[t]] (* a = dr/ds *) Simplify[dads[t]] (* da/ds *) Simplify[k[t]] (* k[t] = norm[da/ds] *) Block[{Line = Arrow}, ParametricPlot[{t, t^2}, {t, -1, 1}] ]

Out[136]:

In [103]:

ClearAll["Global`*"]

(* 2 *)
$Assumptions = {a > 0, b > 0, Element[t, Reals]};

r[t] = {a*Cos[t], b*Sin[t]};

(* trusted result *)
Simplify[ArcCurvature[r[t], t]]

(* steps *)
drdt[t] = D[r[t], t]; (* dr/dt *)
dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *)
drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *)
dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *)
k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *)

Simplify[drds[t]] (* a = dr/ds *)
Simplify[dads[t]] (* da/ds *)
Simplify[k[t]] (* k[t] = norm[da/ds] *)

Block[{Line = Arrow},
    ParametricPlot[{Cos[t], Sqrt[3]*Sin[t]}, {t, -Pi, Pi}]
]

ClearAll["Global`*"] (* 2 *) $Assumptions = {a > 0, b > 0, Element[t, Reals]}; r[t] = {a*Cos[t], b*Sin[t]}; (* trusted result *) Simplify[ArcCurvature[r[t], t]] (* steps *) drdt[t] = D[r[t], t]; (* dr/dt *) dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *) drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *) dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *) k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *) Simplify[drds[t]] (* a = dr/ds *) Simplify[dads[t]] (* da/ds *) Simplify[k[t]] (* k[t] = norm[da/ds] *) Block[{Line = Arrow}, ParametricPlot[{Cos[t], Sqrt[3]*Sin[t]}, {t, -Pi, Pi}] ]

Out[103]:

In [154]:

ClearAll["Global`*"]

(* 3 *)
$Assumptions = {a > 0, b > 0, Element[t, Reals]};

(* notice here k[t] shall have a negative sign for rotating counterclock-wise *)
r[t] = {a*Cosh[t],b*Sinh[t]};

(* result *)
Simplify[ArcCurvature[r[t], t]]

(* steps *)
drdt[t] = D[r[t], t]; (* dr/dt *)
dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *)
drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *)
dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *)
k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *)
Simplify[drds[t]] (* a = dr/ds *)
Simplify[dads[t]] (* da/ds *)
Simplify[k[t]] (* k[t] = norm[da/ds] *)

Block[{Line = Arrow},
    ParametricPlot[{Cosh[t],Sqrt[3]*Sinh[t]}, {t, -1, 2}]
]

ClearAll["Global`*"] (* 3 *) $Assumptions = {a > 0, b > 0, Element[t, Reals]}; (* notice here k[t] shall have a negative sign for rotating counterclock-wise *) r[t] = {a*Cosh[t],b*Sinh[t]}; (* result *) Simplify[ArcCurvature[r[t], t]] (* steps *) drdt[t] = D[r[t], t]; (* dr/dt *) dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *) drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *) dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *) k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *) Simplify[drds[t]] (* a = dr/ds *) Simplify[dads[t]] (* da/ds *) Simplify[k[t]] (* k[t] = norm[da/ds] *) Block[{Line = Arrow}, ParametricPlot[{Cosh[t],Sqrt[3]*Sinh[t]}, {t, -1, 2}] ]

Out[154]:

In [169]:

(* 4 *)
r[t] = {t, a*Cosh[t/a]};

$Assumptions = {a > 0, Element[t, Reals]};

(* result *)
Simplify[ArcCurvature[r[t], t]]

(* steps *)
drdt[t] = D[r[t], t]; (* dr/dt *)
dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *)
drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *)
dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *)
k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *)
Simplify[drds[t]] (* a = dr/ds *)
Simplify[dads[t]] (* da/ds *)
Simplify[k[t]] (* k[t] = norm[da/ds] *)

Block[{Line = Arrow},
    ParametricPlot[{t, Sqrt[3]*Cosh[t/Sqrt[3]]}, {t, -Pi, Pi}]
]

(* 4 *) r[t] = {t, a*Cosh[t/a]}; $Assumptions = {a > 0, Element[t, Reals]}; (* result *) Simplify[ArcCurvature[r[t], t]] (* steps *) drdt[t] = D[r[t], t]; (* dr/dt *) dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *) drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *) dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *) k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *) Simplify[drds[t]] (* a = dr/ds *) Simplify[dads[t]] (* da/ds *) Simplify[k[t]] (* k[t] = norm[da/ds] *) Block[{Line = Arrow}, ParametricPlot[{t, Sqrt[3]*Cosh[t/Sqrt[3]]}, {t, -Pi, Pi}] ]

Out[169]:

2.2¶

$r(t) = (x(t), y(t))$的曲率算法，已经在上面的代码中给出了。

In [194]:

r[t] = {x[t], y[t]};
Simplify[ArcCurvature[r[t], t]]

r[t] = {x[t], y[t]}; Simplify[ArcCurvature[r[t], t]]

Out[194]:

2.3¶

极坐标下的曲率算法：

In [183]:

r[t] = {f[t]*Cos[t], f[t]*Sin[t]};
Simplify[ArcCurvature[r[t], t]]

r[t] = {f[t]*Cos[t], f[t]*Sin[t]}; Simplify[ArcCurvature[r[t], t]]

Out[183]:

2.4¶

我们同时计算曲线的挠度，在这里我们队整体这个算法做一步改进，为了计算方便。

前面使用弧长参数的方法主要是便于理解，并让公式更加简洁，实际上计算Frenet标架时，我们只需要计算单位切向量和单位法向量即可，而单位切向量的计算只需要计算一阶导数，而单位法向量的计算只需要计算二阶导数，因此我们可以直接使用参数$t$来计算单位切向量和单位法向量，而不需要使用弧长参数$s$（有时甚至无法给出弧长参数的解析表达式）。

计算方法简要总结如下：

• 计算一阶导数， 二阶导数，三阶导数
• 一阶导数取单位向量，二阶导数取单位向量。分别为单位切向量和单位法向量
• Frenet标架还需要计算单位副法向量，单位副法向量为单位切向量和单位法向量的叉乘
• 给出Frenet标架 $T, N, B$
• 使用如下公式直接给出曲率和挠率

$$ \kappa(t) = \frac {\lvert r^\prime (t) \wedge r^{\prime\prime} (t) \ \rvert} {\lvert r^\prime (t) \rvert^3} $$ $$ \tau(t) = \frac {r^\prime (t) \cdot (r^{\prime\prime} (t) \wedge r^{\prime\prime\prime} (t))} {\lvert r^\prime (t) \wedge r^{\prime\prime} (t) \ \rvert^2} $$

In [280]:

ClearAll["Global`*"]
$Assumptions = {a > 0, b > 0, El
(* 1 *)ement[t, Reals]};

(* 1 *)
r[t] = {a*Cosh[t],a*Sinh[t],b*t};

(* result *)
Simplify[FrenetSerretSystem[r[t], t]]

(* steps *)
r1[t] = D[r[t], t]; (* dr/dt *)
r2[t] = D[r1[t], t]; (* d^2r/dt^2 *)
r3[t] = D[r2[t], t]; (* d^3r/dt^3 *)

(* tangent *)
tangent[t] = r1[t]/Sqrt[r1[t].r1[t]];

(* normal *)
normal[t] = r2[t]/Sqrt[r2[t].r2[t]];

(* binormal *)
binormal[t] = Cross[tangent[t], normal[t]];

(* curvature *)
k[t] = Sqrt[r2[t].r2[t]]/Sqrt[r1[t].r1[t]]^3;

(* torsion *)
torsion[t] = Det[{r1[t], r2[t], r3[t]}]/Sqrt[r1[t].r2[t]]^2;

(* print results *)
Simplify[tangent[t]]
Simplify[normal[t]]
Simplify[binormal[t]]
Simplify[k[t]]
Simplify[torsion[t]]

(* visualize *)
Block[{Line = Arrow},
    ParametricPlot3D[{Cosh[t],Sinh[t],t}, {t, -3, 3}]
]

ClearAll["Global`*"] $Assumptions = {a > 0, b > 0, El (* 1 *)ement[t, Reals]}; (* 1 *) r[t] = {a*Cosh[t],a*Sinh[t],b*t}; (* result *) Simplify[FrenetSerretSystem[r[t], t]] (* steps *) r1[t] = D[r[t], t]; (* dr/dt *) r2[t] = D[r1[t], t]; (* d^2r/dt^2 *) r3[t] = D[r2[t], t]; (* d^3r/dt^3 *) (* tangent *) tangent[t] = r1[t]/Sqrt[r1[t].r1[t]]; (* normal *) normal[t] = r2[t]/Sqrt[r2[t].r2[t]]; (* binormal *) binormal[t] = Cross[tangent[t], normal[t]]; (* curvature *) k[t] = Sqrt[r2[t].r2[t]]/Sqrt[r1[t].r1[t]]^3; (* torsion *) torsion[t] = Det[{r1[t], r2[t], r3[t]}]/Sqrt[r1[t].r2[t]]^2; (* print results *) Simplify[tangent[t]] Simplify[normal[t]] Simplify[binormal[t]] Simplify[k[t]] Simplify[torsion[t]] (* visualize *) Block[{Line = Arrow}, ParametricPlot3D[{Cosh[t],Sinh[t],t}, {t, -3, 3}] ]

Out[280]:

In [311]:

ClearAll["Global`*"]
$Assumptions = {a > 0, b > 0};

(* 3 *)
r[t] = {a*(1-Sin[t]),a*(1-Cos[t]),b*t};

(* result *)
Simplify[ArcCurvature[r[t], t]]

(* steps *)
drdt[t] = D[r[t], t]; (* dr/dt *)
dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *)
drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *)
dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *)
k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *)
n[t] = dads[t]/k[t]; (* n[t] = da/ds / norm[da/ds] *)
b[t] = Cross[drds[t], n[t]]; (* b[t] = dr/ds x n*)
dnds[t] = D[n[t], t]/dsdt[t]; (* dn/ds *)
tao[t] = dnds[t].b[t]; (* tao[t] = dn/ds . b[t] *)


Simplify[drds[t]] (* a = dr/ds *)
Simplify[dads[t]] (* da/ds *)
Simplify[k[t]] (* k[t] = norm[da/ds] *)
Simplify[n[t]] (* n[t] = da/ds / norm[da/ds] *)
Simplify[b[t]] (* b[t] = dr/ds x da/ds *)
Simplify[tao[t]] (* tao[t] = dn/ds . b[t] *)

(* visualize *)
Block[{Line = Arrow},
    ParametricPlot3D[{1-Sin[t],1-Cos[t],t}, {t, -3, 3}]
]

ClearAll["Global`*"] $Assumptions = {a > 0, b > 0}; (* 3 *) r[t] = {a*(1-Sin[t]),a*(1-Cos[t]),b*t}; (* result *) Simplify[ArcCurvature[r[t], t]] (* steps *) drdt[t] = D[r[t], t]; (* dr/dt *) dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *) drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *) dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *) k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *) n[t] = dads[t]/k[t]; (* n[t] = da/ds / norm[da/ds] *) b[t] = Cross[drds[t], n[t]]; (* b[t] = dr/ds x n*) dnds[t] = D[n[t], t]/dsdt[t]; (* dn/ds *) tao[t] = dnds[t].b[t]; (* tao[t] = dn/ds . b[t] *) Simplify[drds[t]] (* a = dr/ds *) Simplify[dads[t]] (* da/ds *) Simplify[k[t]] (* k[t] = norm[da/ds] *) Simplify[n[t]] (* n[t] = da/ds / norm[da/ds] *) Simplify[b[t]] (* b[t] = dr/ds x da/ds *) Simplify[tao[t]] (* tao[t] = dn/ds . b[t] *) (* visualize *) Block[{Line = Arrow}, ParametricPlot3D[{1-Sin[t],1-Cos[t],t}, {t, -3, 3}] ]

Out[311]:

2.6¶

In [448]:

ClearAll["Global`*"]
$Assumptions = {t>-1 && t<1};

r[t] = {(1+t)^(3/2)/3,(1-t)^(3/2)/3,t/Sqrt[2]}

(* result *)
Simplify[ArcCurvature[r[t], t]]

(* steps *)
drdt[t] = D[r[t], t]; (* dr/dt *)
dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *)
drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *)
dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *)
k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *)
n[t] = dads[t]/k[t]; (* n[t] = da/ds / norm[da/ds] *)
b[t] = Cross[drds[t], n[t]]; (* b[t] = dr/ds x n*)
dnds[t] = D[n[t], t]/dsdt[t]; (* dn/ds *)
tao[t] = dnds[t].b[t]; (* tao[t] = dn/ds . b[t] *)


Simplify[drds[t]] (* a = dr/ds *)
Simplify[dads[t]] (* da/ds *)
Simplify[k[t]] (* k[t] = norm[da/ds] *)
Simplify[n[t]] (* n[t] = da/ds / norm[da/ds] *)
Simplify[b[t]] (* b[t] = dr/ds x da/ds *)
Simplify[tao[t]] (* tao[t] = dn/ds . b[t] *)

(* visualize *)
Block[{Line = Arrow},
    ParametricPlot3D[{(1+t)^(3/2)/3,(1-t)^(3/2)/3,t/Sqrt[2]}, {t,-1,1}]
]

ClearAll["Global`*"] $Assumptions = {t>-1 && t<1}; r[t] = {(1+t)^(3/2)/3,(1-t)^(3/2)/3,t/Sqrt[2]} (* result *) Simplify[ArcCurvature[r[t], t]] (* steps *) drdt[t] = D[r[t], t]; (* dr/dt *) dsdt[t] = Sqrt[drdt[t].drdt[t]]; (* ds/dt: *) drds[t] = drdt[t]/dsdt[t]; (* a = dr/ds *) dads[t] = D[drds[t], t]/dsdt[t]; (* da/ds *) k[t] = Sqrt[dads[t].dads[t]]; (* k[t] = norm[da/ds] *) n[t] = dads[t]/k[t]; (* n[t] = da/ds / norm[da/ds] *) b[t] = Cross[drds[t], n[t]]; (* b[t] = dr/ds x n*) dnds[t] = D[n[t], t]/dsdt[t]; (* dn/ds *) tao[t] = dnds[t].b[t]; (* tao[t] = dn/ds . b[t] *) Simplify[drds[t]] (* a = dr/ds *) Simplify[dads[t]] (* da/ds *) Simplify[k[t]] (* k[t] = norm[da/ds] *) Simplify[n[t]] (* n[t] = da/ds / norm[da/ds] *) Simplify[b[t]] (* b[t] = dr/ds x da/ds *) Simplify[tao[t]] (* tao[t] = dn/ds . b[t] *) (* visualize *) Block[{Line = Arrow}, ParametricPlot3D[{(1+t)^(3/2)/3,(1-t)^(3/2)/3,t/Sqrt[2]}, {t,-1,1}] ]

Out[448]:

2.7¶

In [317]:

Clear["Global`*"]

$Assumptions = {t < 0, Element[t, Reals]};

k[t] = Simplify[FrenetSerretSystem[{Exp[-1/(t^2)],t,0},t]]

(* limit t-> 0, k[t]? *)
Limit[k[t], t -> 0]

Clear["Global`*"] $Assumptions = {t < 0, Element[t, Reals]}; k[t] = Simplify[FrenetSerretSystem[{Exp[-1/(t^2)],t,0},t]] (* limit t-> 0, k[t]? *) Limit[k[t], t -> 0]

Limit::alimv: Warning: Assumptions that involve the limit variable are ignored.

Out[317]:

2.13¶

In [40]:

Clear["Global`*"]

k[s] = a/(a^2 + s^2)
(* k[s] = 1/Sqrt[a^2 - s^2] *)
C0 = 0;
C1 = 0;
C2 = 0;
r[s] = {Integrate[Cos[Integrate[k[s], s] + C0], s] + C1, Integrate[Sin[Integrate[k[s], s] + C0], s] + C2};
Simplify[r[s]]
Simplify[ArcCurvature[r[s], s]]



Clear["Global`*"]
r[s] = {a*ArcSin[s/a], Sqrt[a^2+s^2]}
Simplify[ArcCurvature[r[s], s]]

Clear["Global`*"] k[s] = a/(a^2 + s^2) (* k[s] = 1/Sqrt[a^2 - s^2] *) C0 = 0; C1 = 0; C2 = 0; r[s] = {Integrate[Cos[Integrate[k[s], s] + C0], s] + C1, Integrate[Sin[Integrate[k[s], s] + C0], s] + C2}; Simplify[r[s]] Simplify[ArcCurvature[r[s], s]] Clear["Global`*"] r[s] = {a*ArcSin[s/a], Sqrt[a^2+s^2]} Simplify[ArcCurvature[r[s], s]]

Out[40]:

2.20¶

In [73]:

r1[t] = {t+Sqrt[3]*t,2*Cos[t],Sqrt[3]*t-Sin[t]};
r2[t] = {2*Cos[t/2], 2*Sin[t/2], -t};

Simplify[FrenetSerretSystem[r1[t], t]]
Simplify[FrenetSerretSystem[r2[t], t]]

(* plot these two curves
Block[{Line = Arrow},
    ParametricPlot3D[{t+Sqrt[3]*t,2*Cos[t],Sqrt[3]*t-Sin[t]}, {t, -Pi, Pi}]
]
Block[{Line = Arrow},
    ParametricPlot3D[{t, 2*Cos[t/2], 2*Sin[t/2]}, {t, -Pi, Pi}]
] *)

r1[t] = {t+Sqrt[3]*t,2*Cos[t],Sqrt[3]*t-Sin[t]}; r2[t] = {2*Cos[t/2], 2*Sin[t/2], -t}; Simplify[FrenetSerretSystem[r1[t], t]] Simplify[FrenetSerretSystem[r2[t], t]] (* plot these two curves Block[{Line = Arrow}, ParametricPlot3D[{t+Sqrt[3]*t,2*Cos[t],Sqrt[3]*t-Sin[t]}, {t, -Pi, Pi}] ] Block[{Line = Arrow}, ParametricPlot3D[{t, 2*Cos[t/2], 2*Sin[t/2]}, {t, -Pi, Pi}] ] *)

Out[73]:

In [ ]: