Fundamental Forms and Curvatures Quiz

The Second Fundamental Form of a surface is used to describe:

• Intrinsic curvature

• Extrinsic curvature

• Euclidean distance

• Metric tensor

The principal curvatures of a surface are:

• The maximum and minimum normal curvatures

• Always equal for any surface

• Related to the Euler’s theorem

• Independent of the Second Fundamental Form

If the two principal curvatures at a point are equal, the surface is said to be:

• A plane

• A sphere

• A developable surface

• A ruled surface

The lines of curvature on a surface are:

• Curves along which the normal curvature is extremal

• Always straight lines

• The intersection of the surface with a plane

• The same as geodesics

A surface is developable if:

• Its Gaussian curvature is zero

• Its mean curvature is zero

• It can be locally approximated by a sphere

• It has nonzero principal curvatures

A developable surface can be generated by:

• The motion of a straight line

• The intersection of two spheres

• The motion of a circle

• None of the above

Which of the following is an example of a minimal surface?

• Helicoid

• Sphere

• Ellipsoid

• Cone

A ruled surface is defined as:

• A surface generated by moving a straight line

• A surface with constant curvature

• A surface of revolution

• A surface with no straight lines

An example of a ruled surface is:

• A hyperboloid

• A sphere

• A torus

• A paraboloid

The shape operator (Weingarten map) is related to:

• The Second Fundamental Form

• The Gauss Map

• The first Fundamental Form

• None of the above