Space Curves: The Osculating Plane, the Principal Normal, and the Binormal

Using the third Frenet formula, show that a curve whose torsion is identically zero lies in a plane.

Find the osculating plane of the curve

at the point $(1,1,1)$.

Find the osculating plane of the curve

Find the osculating plane of the curve

Find the osculating plane of the curve $x=e^{t}$, $y=e^{-t}$, $z=t\sqrt{2}$.

Find the osculating plane of the curve $x=a\cos t$, $y=a\sin t$, $z=bt$.

Find the osculating plane of the curve $x=at^{2}+bt+c$, $y=a_{1}t^{2}+b_{1}t+c_{1}$, $z=a_{2}t^{2}+b_{2}t+c_{2}$.

Prove that the osculating plane of the curve

lying on the conoidal surface $z=\varphi \left(\dfrac {y}{x}\right)$, coincides with the tangent plane to this surface.

Find the equations of the principal normal and the binormal to the curve $x=a\cos t$, $y=a\sin t$, $z=bt$.

Find the equations of the principal normal and the binormal to the curve $x=y^{2}$, $z=x^{2}$ at the point $(1,1,1)$.

Find the equations of the principal normal and the binormal to the curve $x=\dfrac {t^{4}}{3}$, $y=\dfrac {t^{2}}{3}$, $z=\dfrac {t^{2}}{2}$.

Find the equations of the principal normal and the binormal to the curve $x=\dfrac {t^{2}}{2}$, $y=\dfrac {2t^{3}}{3}$, $z=\dfrac {t^{4}}{2}$ at the point $\left(\dfrac {1}{2},\dfrac {2}{3},\dfrac {1}{2}\right)$.

Find the equations of the spherical indicatrices of the principal normal and the binormal of the helix (see problems 2265, 2275).

The curve

lies on the surface of revolution $x^{2}+y^{2}=a^{2}\ch^{2}\dfrac {z}{a}$, called a catenoid. Prove that at every point of the curve the binormal coincides with the normal to the surface.

Prove that one of the bisectors of the angles between the tangent and the binormal to the curve

has a constant direction.

A curve has as its projection onto the plane $xOy$ the sinusoid $y=\sin x$. What relation must the applicates $z$ of its points satisfy so that the principal normals are parallel to the plane $yOz$?

Along the principal normals of the helix

segments of given length $l$ are laid off. Find the locus of their endpoints.

Find the surface on which all the principal normals of the helix of the preceding problem lie.

The same question (find the surface on which the lines lie) as in the preceding problem, for the binormals.

Find the radius of curvature $\rho$ of the curve

Find the radius of curvature $\rho$ of the curve $x=a\ch t\cos t$, $y=a\ch t\sin t$, $z=at$.

Find the radius of curvature $\rho$ of the curve $x=e^{t}$, $y=e^{-t}$, $z=t\sqrt{2}$.

Find the radius of curvature $\rho$ of the curve $x=e^{t}\cos t$, $y=e^{t}\sin t$, $z=e^{t}$.

Find the radius of curvature $\rho$ of the curve $x=a\dfrac {\cos t}{\ch t}$, $y=a\dfrac {\sin t}{\ch t}$, $z=a(t-\th t)$.

Find the radius of curvature $\rho$ of the curve $x^{2}=2az$, $y^{2}=2bz$.

Find the radius of curvature $\rho$ of the curve $x^{2}-y^{2}+z^{2}=1$, $y^{2}-2x+z=0$ at the point $(1,1,1)$.

Find the radius of curvature $\rho$ of the curve $x=3t-t^{3}$, $y=3t^{2}$, $z=3t+t^{3}$.

Find the radius of curvature $\rho$ of the curve $x=a(1-\cos 2t)$, $y=a\sin 2t$, $z=2a\cos t$.

Find the radius of curvature $\rho$ of the curve $x=a\cos^{3}t$, $y=a\sin^{3}t$, $z=a\cos 2t$.

Find the radius of curvature $\rho$ of the curve $x=\sin z-z\cos z$, $y=\cos z+z\sin z$.

Find the torsion of the curve

Find the torsion of the curve $x=3t-t^{3}$, $y=3t^{2}$, $z=3t+t^{3}$.

Find the torsion of the curve $x=\sin u-u\cos u$, $y=\cos u+u\sin u$, $z=u$.

Find the torsion of the curve $x=a(1-\cos t)$, $y=a\sin 2t$, $z=2a\cos t$.

Find the curvature and the torsion of the curve

The same question (find the curvature and the torsion) as in the preceding problem, for the curve

The same question (find the curvature and the torsion) as in the preceding problem, for the curve

Prove the equality of the radius of curvature and the radius of torsion for the curve $x=\ch z$, $y=\sh z$.

Prove that for the moving trihedron of the curve

each of its edges makes a constant angle with $Oz$.

Under what condition does the center of curvature of the helix

lie on the same cylinder as the helix itself?

Prove that for the curve

the segment of the (non-principal!) normal from a point of the curve to the axis $Oz$ equals the radius of torsion.

A sphere can be drawn through four points of a curve. If these points tend to a single point, then, under appropriate conditions, this sphere tends to a certain limiting sphere, called the osculating sphere. Find its center and radius.

Find, to an accuracy of higher-order small terms, the approximate equations of the projections of a curve onto the three planes of the moving trihedron near the point of tangency.

From the center of a sphere, radii are drawn parallel to the principal normals of a closed curve having no singular points. Prove that the locus of their endpoints divides the surface of the sphere into two parts of equal area. (Jacobi.)