Nested Helix
A helix spiraling up another helix. sliders were included to adjust the number of rotations and the radius of curvature for both helices.
Scroll(h,0,5,.1,2)
Scroll(w,1,10,.1,2)
Scroll(k,0,50,.1,25)
Scroll(r1,0,1,.001,0.1)
v1=[cos(2\pi*u*w),sin(2\pi*u*w),h*u]
T1=[D(v1(1),u),D(v1(2),u),D(v1(3),u)]/(D(v1(1),u)^2+D(v1(2),u)^2+D(v1(3),u)^2)^(1/2)
N1=[D(T1(1),u),D(T1(2),u),D(T1(3),u)]/(D(T1(1),u)^2+D(T1(2),u)^2+D(T1(3),u)^2)^(1/2)
B1=Cross(T1,N1)
v2=[v1(1)+r1*(N1(1)*cos(k*2\pi*u)+B1(1)*sin(k*2\pi*u)),v1(2)+r1*(N1(2)*cos(k*2\pi*u)+B1(2)*sin(k*2\pi*u)),v1(3)+r1*(N1(3)*cos(k*2\pi*u)+B1(3)*sin(k*2\pi*u))]
MultiPlot3D(ParametricPlot3D(v2(1),v2(2),v2(3),[u,0,1,2000],[v,0,1,10],lines=0,color=red),ParametricPlot3D(v1(1),v1(2),v1(3),[u,0,1,256],[v,0,1,1],lines=0))
