Tangent, Normal and Binormal Vectors — Question 7

Question 7

For $\vec r(t)=\left\langle t,t^3,0\right\rangle$, determine where the Frenet principal normal is undefined. Does the curve still have a tangent there? Analyze the limiting normals on the two sides.

Question 7 – Solution

Strategy Compute $\mathbf T'$ and distinguish loss of curvature from loss of tangent direction.

Tangent $\vec r'(t)=\left\langle 1,3t^2,0\right\rangle$ never vanishes, so the curve is regular and has tangent $\mathbf T(0)=\left\langle 1,0,0\right\rangle$.

Normal failure Differentiating the normalized tangent shows $\mathbf T'(0)=\vec 0$, so $\mathbf N=\mathbf T'/\|\mathbf T'\|$ is undefined at 0. For $t>0$, the normal tends to $\left\langle 0,1,0\right\rangle$; for $t<0$, it tends to $\left\langle 0,-1,0\right\rangle$.

Interpretation The origin is an inflection point: curvature vanishes and the bending side changes.