Question

For the following, you can either provide a vector equation for the curve, or you can describe the curve in words with sufficient details.

a) Describe a curve that has curvature zero.
b) Describe a curve that has torsion zero.
c) Describe a curve that has constant nonzero curvature.
d) Describe a curve that has constant nonzero torsion.
e) Describe a curve that has zero curvature and zero torsion.

Answer 1

(a) ax +by +c = 0 shows a straight line,

The shortest distance between two points is a straight line.

The curve show a straight line has Zero curvature on it

paramtetric form of curvature k(t) =I f ' (t) I / [1+{f ' (t)}² ]^3/2

if f '(t) = 0 then k (t ) = 0

so curvature is zero

(b) The Planer curve has zero Torsion, this curve belongs to fixed plane

A planar curve is one that lies in a plane , A plane curve may be open or closed

for example line and parabola is open curve

and circle and ellipse is closed curve

A planer curve shown as f'(x,y) =0

f'(x,y,z) =0

(c) A circular helix has curvature is constant and non zero

an object having a three-dimensional shape like that of a wire wound uniformly in a single layer around a cylinder or cone, as in a corkscrew or spiral staircase is called helix

(d) Any space curve whose curvature and Torsion are both constant and non zero like as helix

The torsion is +ve for right handed helix

and torsion is -ve for left handed helix

(e) A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane

A planar curve connected via a linear segment to another planar curve lying in a different plane would still have zero torsion everywhere.

The Frenet–Serret formulas are: for property of curve as torsion of curve

• T is the unit vector tangent to the curve, pointing in the direction of motion.
• N is the normal unit vector, the derivative of T with respect to the arclength parameter of the curve, divided by its length.
• B is the binormal unit vector, the cross product of T and N.