5.7: Angles between Parameter Curves

To find the angle between the u- and v-parameter curves at a point u0,v0 , we use the formula

    cos(q)= Xu(u,v) ·Xv( u,v)/[sqrt]g11( u,v)[sqrt]g22( u,v)
    =g12( u,v)/[sqrt]g11( u,v)/[sqrt]g22( u,v)
It is important to note right here that if g12(u,v) ≡0 then the parameter curves form an orthogonal net: the intersections of the u- and v-curves are all right angles.

The Angle Between Parameter Lines

To choose a point [u,v] , you may click on the middle mouse button on the desired point in TheDomain window. The value of the point will appear in the Parameter Curves Demo control panel. The point is marked in the domain by a small red circle, and its image under the mapping X(u,v) is also marked by a red circle.

Then, the parameter lines through the chosen point are drawn in green on the surface in order to view the angle in question. The exact value of the angle between the parameter curves is printed in the control panel also. To get a global view of the angle variations on the surface, we have plotted the graph of q(u,v) where the point is also marked by a red circle.

The demonstration shows the domain, the surface in question and the graph of q(u,v) over the domain of the surface.

Finis.