One student parametrizes a circle of radius 3 as r(t) = (3cos(t/2), 3sin(t/2), 0) and another as r(t) = (3cos(2t), 3sin(2t), 0). Which statement is true about the curvature κ computed from each?
AThe fast parametrization gives larger curvature because the tangent vector changes direction more quickly in time
BBoth parametrizations give κ = 1/3, because curvature is a property of the curve's shape, not the speed of traversal
CThe slow parametrization gives smaller curvature because direction changes per unit of time are less frequent
DCurvature is undefined unless the curve is parametrized by arc length
Curvature is intrinsic to the curve's geometry, not its parametrization. The formula κ = |r′ × r″|/|r′|³ corrects for speed through the |r′|³ denominator, ensuring both parametrizations yield κ = 1/3. Options A and C confuse dT/dt — which does depend on traversal speed — with dT/ds, which is purely geometric. The whole point of defining curvature via arc length is to make it parametrization-independent.
Question 2 Multiple Choice
The unit normal vector N in the Frenet-Serret frame points:
AIn the direction of the velocity vector, along the tangent to the curve
BToward the center of curvature — the direction the curve is turning
CPerpendicular to the plane of the curve, out of the osculating plane
DIn the direction of maximum torsion
N = (dT/ds)/|dT/ds|: it is the direction in which the unit tangent is changing, which points toward the center of curvature — the curve is, so to speak, turning toward N. It is the binormal B = T × N that points out of the T-N plane (the osculating plane). Confusing N and B is a common error.
Question 3 True / False
A straight line has curvature zero everywhere, because the unit tangent vector does not change direction as you move along it.
TTrue
FFalse
Answer: True
κ = |dT/ds|. For a straight line, T is constant — the direction of travel never changes — so dT/ds = 0 and κ = 0 everywhere. This matches the intuition that a line has no bending. A common misconception is that curvature is 'undefined' for a straight line; it is defined and equals zero.
Question 4 True / False
Torsion τ can primarily be zero or positive; a negative value indicates a computational error in the Frenet-Serret calculations.
TTrue
FFalse
Answer: False
Torsion can be negative. Its sign encodes the handedness of the twist: a right-handed helix has positive torsion and a left-handed helix has negative torsion. Negative torsion is geometrically meaningful, not an error. This is analogous to how the sign of a cross product encodes orientation.
Question 5 Short Answer
Why is curvature defined with respect to arc length s rather than the parameter t, and what goes wrong if you compute dT/dt instead of dT/ds?
Think about your answer, then reveal below.
Model answer: Using dT/dt makes the rate of change of the tangent depend on how fast you traverse the curve, not on its shape. A car driving fast around a circle generates a large dT/dt; the same car crawling around the same circle generates a small dT/dt — yet the circle's geometry is unchanged. Arc length s represents actual distance traveled, so dT/ds measures direction-change per unit of distance, which is purely geometric. The practical formula κ = |r′ × r″|/|r′|³ uses the |r′|³ denominator precisely to convert from parameter-time to arc-length, making curvature intrinsic to the curve's shape regardless of how fast it is traversed.
The intrinsic vs. parameter-dependent distinction is the conceptual core of curvature. Students who miss this compute 'curvature' that changes when they reparametrize the same curve — a sign that they are measuring something about their description of the curve rather than the curve itself.