In the blog  Plots of twisted ribbons, the author gave an interesting description of plotting twisted ribbons. In this blog , we give a similar description of twisted ribbons and give the geometrical interpretations of this definition.

Let r(phi)=[a*cos(phi), a*sin(phi), 0] (phi=0..2*Pi) be a circle in the xy-plane, and P be a point on the circle. Let QR be a line segment (with length of 2) passing through the point P and let P be the middle point of QR. Also, QR is coplanar with the z-axis.

Now let P rotate about the z-axis at the angular velocity of phi, where phi is the angle between OP and the x-axis. At the same time, the line segment QR is rotating about its middle point, P, at the angular velocity of theta (where theta is the angle between PQ and the z-axis and theta is dependent on phi, eg, theta=k*phi). In the whole process, QR will remain coplanar with the z-axis.

Apparently, the locus of the line segment QR is a twisted ribbon. When theta=phi/2, we have the Mobius strip.

2. The equation of the twisted ribbon

Now we try to find the equation of the surface. Clearly,

vector(OP)= [a*cos(phi), a*sin(phi), 0].

And with some geometrical manipulations, we have

vector(PQ)=[sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)].

So the vector equation of the twisted ribbon is

V(phi, t)=vector(OP)+t*vector(PQ)

And the parametric equation is
x=a*cos(phi)+t*sin(theta)*cos(phi),
y=a*sin(phi)+t*sin(theta)*sin(phi)

z=t*cos(theta)

(where theta=k*phi (k a constant), phi=0..2*Pi and t=-b..b (b determines the width of the ribbon.))

Or
x=(a+t*sin(k*phi))*cos(phi),
y=(a+t*sin(k*phi))*sin(phi)

z=t*cos(k*phi)

(where k a constant, phi=0..2*Pi and t= -b..b)

When we take k=1/2, 1, 3/2, 2,…, we have different twisted ribbons.

When k=1/2, we have the equation of the Mobius strip:
x=(a+t*sin(phi/2))*cos(phi),
y=(a+t*sin(phi/2))*sin(phi)

z=t*cos(ph)/2

(phi=0..2*Pi and t= -b..b) .

k=1

k=2