Question

The surface with parametric equations

\(x = 2cos\theta + rcos\left( {\frac{\theta }{2}} \right)\)

\(y = 2sin\theta + rcos\left( {\frac{\theta }{2}} \right)\)

\(z = rsin\left( {\frac{\theta }{2}} \right)\)

Where \( - \frac{1}{2} \le r \le \frac{1}{2}\)and\(0 \le \theta \le 2\pi \), is called a mobius strip. Graph this surface with several viewpoints. What is the unusual about it?

Short answer

Graph of the parametric equation

Step-by-step solution

Using parametric equation of the surface

Let, we know the parametric equation of the surface replaced\(\cos \theta \)by\(\sin \theta \)

\(\begin{array}{c}r(u,v) = \left( {2\cos \theta + r\cos \left( {\frac{\theta }{2}} \right),2\sin \theta + r\cos \left( {\frac{\theta }{2}} \right),r\sin \left( {\frac{\theta }{2}} \right)} \right)\\ - \frac{1}{2} \le r \le \frac{1}{2}\\0 \le \theta \le 2\pi \end{array}\)

The coordinate values are

\(\begin{array}{c}x = 2\cos \theta + r\cos \left( {\frac{\theta }{2}} \right)\\y = 2\sin \theta + r\cos \left( {\frac{\theta }{2}} \right)\\z = r\sin \left( {\frac{\theta }{2}} \right)\end{array}\)

Diagram of the parametric surface

Parametric equation of the surface\(Plot3d\left( {\left( {2\cos \theta + r\cos \left( {\frac{\theta }{2}} \right),2\sin \theta + r\cos \left( {\frac{\theta }{2}} \right),r\sin \left( {\frac{\theta }{2}} \right)} \right),u = - \frac{1}{2}..\frac{1}{2},\theta = 0..\frac{{pi}}{2},lables = \left( {x,y,z} \right)} \right);\)Diagram of the parametric equation by using computer

The unusual about the graph surface is non-orientable.