In [ ]:
var('t u')
C = vector((2*cos(t), 2*sin(t), 0.5*t))
C1 = C.diff(t)
C2 = C.diff(t,2)
b = C1.cross_product(C2)
T = C1 / norm(C1)
B = b / norm(b)
N = B.cross_product(T)
r = 0.8
ploha = C + r * (-cos(u) * N + sin(u) * B)
In [ ]:
parametric_plot3d(C,(t,0,2*pi),viewer='threejs',online=True,thickness=2)
Out[ ]:
In [ ]:
parametric_plot3d(ploha,(t,0,6*pi), (u,0,2*pi), viewer='threejs',online=True,plot_points=[80,40],mesh=True,color='pink')
Out[ ]:
In [ ]:
ploha
Out[ ]:
(0.800000000000000*(8*(cos(t)^2 + sin(t)^2)*cos(t)/(sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2)*sqrt(4*abs(cos(t))^2 + 4*abs(sin(t))^2 + 0.250000000000000)) + 0.500000000000000*cos(t)/(sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2)*sqrt(4*abs(cos(t))^2 + 4*abs(sin(t))^2 + 0.250000000000000)))*cos(u) + 0.800000000000000*sin(t)*sin(u)/sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2) + 2*cos(t), 0.800000000000000*(8*(cos(t)^2 + sin(t)^2)*sin(t)/(sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2)*sqrt(4*abs(cos(t))^2 + 4*abs(sin(t))^2 + 0.250000000000000)) + 0.500000000000000*sin(t)/(sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2)*sqrt(4*abs(cos(t))^2 + 4*abs(sin(t))^2 + 0.250000000000000)))*cos(u) - 0.800000000000000*cos(t)*sin(u)/sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2) + 2*sin(t), 3.20000000000000*(cos(t)^2 + sin(t)^2)*sin(u)/sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2) + 0.500000000000000*t)
In [ ]:
ploha.simplify_full()
Out[ ]:
(2/85*sqrt(17)*(2*sqrt(17)*cos(t)*cos(u) + 2*sin(t)*sin(u) + 5*sqrt(17)*cos(t)), 2/85*sqrt(17)*(2*sqrt(17)*cos(u)*sin(t) - 2*cos(t)*sin(u) + 5*sqrt(17)*sin(t)), 1/170*sqrt(17)*(5*sqrt(17)*t + 32*sin(u)))
In [ ]:
%display latex
In [ ]:
ploha
Out[ ]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(0.800000000000000 \, {\left(\frac{8 \, {\left(\cos\left(t\right)^{2} + \sin\left(t\right)^{2}\right)} \cos\left(t\right)}{\sqrt{{\left| 4 \, \cos\left(t\right)^{2} + 4 \, \sin\left(t\right)^{2} \right|}^{2} + 1.00000000000000 \, {\left| \cos\left(t\right) \right|}^{2} + 1.00000000000000 \, {\left| \sin\left(t\right) \right|}^{2}} \sqrt{4 \, {\left| \cos\left(t\right) \right|}^{2} + 4 \, {\left| \sin\left(t\right) \right|}^{2} + 0.250000000000000}} + \frac{0.500000000000000 \, \cos\left(t\right)}{\sqrt{{\left| 4 \, \cos\left(t\right)^{2} + 4 \, \sin\left(t\right)^{2} \right|}^{2} + 1.00000000000000 \, {\left| \cos\left(t\right) \right|}^{2} + 1.00000000000000 \, {\left| \sin\left(t\right) \right|}^{2}} \sqrt{4 \, {\left| \cos\left(t\right) \right|}^{2} + 4 \, {\left| \sin\left(t\right) \right|}^{2} + 0.250000000000000}}\right)} \cos\left(u\right) + \frac{0.800000000000000 \, \sin\left(t\right) \sin\left(u\right)}{\sqrt{{\left| 4 \, \cos\left(t\right)^{2} + 4 \, \sin\left(t\right)^{2} \right|}^{2} + 1.00000000000000 \, {\left| \cos\left(t\right) \right|}^{2} + 1.00000000000000 \, {\left| \sin\left(t\right) \right|}^{2}}} + 2 \, \cos\left(t\right),\,0.800000000000000 \, {\left(\frac{8 \, {\left(\cos\left(t\right)^{2} + \sin\left(t\right)^{2}\right)} \sin\left(t\right)}{\sqrt{{\left| 4 \, \cos\left(t\right)^{2} + 4 \, \sin\left(t\right)^{2} \right|}^{2} + 1.00000000000000 \, {\left| \cos\left(t\right) \right|}^{2} + 1.00000000000000 \, {\left| \sin\left(t\right) \right|}^{2}} \sqrt{4 \, {\left| \cos\left(t\right) \right|}^{2} + 4 \, {\left| \sin\left(t\right) \right|}^{2} + 0.250000000000000}} + \frac{0.500000000000000 \, \sin\left(t\right)}{\sqrt{{\left| 4 \, \cos\left(t\right)^{2} + 4 \, \sin\left(t\right)^{2} \right|}^{2} + 1.00000000000000 \, {\left| \cos\left(t\right) \right|}^{2} + 1.00000000000000 \, {\left| \sin\left(t\right) \right|}^{2}} \sqrt{4 \, {\left| \cos\left(t\right) \right|}^{2} + 4 \, {\left| \sin\left(t\right) \right|}^{2} + 0.250000000000000}}\right)} \cos\left(u\right) - \frac{0.800000000000000 \, \cos\left(t\right) \sin\left(u\right)}{\sqrt{{\left| 4 \, \cos\left(t\right)^{2} + 4 \, \sin\left(t\right)^{2} \right|}^{2} + 1.00000000000000 \, {\left| \cos\left(t\right) \right|}^{2} + 1.00000000000000 \, {\left| \sin\left(t\right) \right|}^{2}}} + 2 \, \sin\left(t\right),\,\frac{3.20000000000000 \, {\left(\cos\left(t\right)^{2} + \sin\left(t\right)^{2}\right)} \sin\left(u\right)}{\sqrt{{\left| 4 \, \cos\left(t\right)^{2} + 4 \, \sin\left(t\right)^{2} \right|}^{2} + 1.00000000000000 \, {\left| \cos\left(t\right) \right|}^{2} + 1.00000000000000 \, {\left| \sin\left(t\right) \right|}^{2}}} + 0.500000000000000 \, t\right)\]
In [ ]:
ploha.simplify_full()
Out[ ]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{2}{85} \, \sqrt{17} {\left(2 \, \sqrt{17} \cos\left(t\right) \cos\left(u\right) + 2 \, \sin\left(t\right) \sin\left(u\right) + 5 \, \sqrt{17} \cos\left(t\right)\right)},\,\frac{2}{85} \, \sqrt{17} {\left(2 \, \sqrt{17} \cos\left(u\right) \sin\left(t\right) - 2 \, \cos\left(t\right) \sin\left(u\right) + 5 \, \sqrt{17} \sin\left(t\right)\right)},\,\frac{1}{170} \, \sqrt{17} {\left(5 \, \sqrt{17} t + 32 \, \sin\left(u\right)\right)}\right)\]
In [ ]:
%display plain
In [ ]:
ploha
Out[ ]:
(0.800000000000000*(8*(cos(t)^2 + sin(t)^2)*cos(t)/(sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2)*sqrt(4*abs(cos(t))^2 + 4*abs(sin(t))^2 + 0.250000000000000)) + 0.500000000000000*cos(t)/(sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2)*sqrt(4*abs(cos(t))^2 + 4*abs(sin(t))^2 + 0.250000000000000)))*cos(u) + 0.800000000000000*sin(t)*sin(u)/sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2) + 2*cos(t), 0.800000000000000*(8*(cos(t)^2 + sin(t)^2)*sin(t)/(sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2)*sqrt(4*abs(cos(t))^2 + 4*abs(sin(t))^2 + 0.250000000000000)) + 0.500000000000000*sin(t)/(sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2)*sqrt(4*abs(cos(t))^2 + 4*abs(sin(t))^2 + 0.250000000000000)))*cos(u) - 0.800000000000000*cos(t)*sin(u)/sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2) + 2*sin(t), 3.20000000000000*(cos(t)^2 + sin(t)^2)*sin(u)/sqrt(abs(4*cos(t)^2 + 4*sin(t)^2)^2 + 1.00000000000000*abs(cos(t))^2 + 1.00000000000000*abs(sin(t))^2) + 0.500000000000000*t)
In [ ]: