Radius of curvature of space curves

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Problem 2286 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve

x=t−sin⁡t,y=1−cos⁡t,z=4sin⁡t2. x = t-\sin t, \qquad y = 1-\cos t, \qquad z = 4\sin \frac{t}{2} .
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Problem 2287 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x=ach⁡tcos⁡tx=a\ch t\cos t, y=ach⁡tsin⁡ty=a\ch t\sin t, z=atz=at.

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Problem 2288 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x=etx=e^{t}, y=e−ty=e^{-t}, z=t2z=t\sqrt{2}.

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Problem 2289 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x=etcos⁡tx=e^{t}\cos t, y=etsin⁡ty=e^{t}\sin t, z=etz=e^{t}.

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Problem 2290 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x=acos⁡tch⁡tx=a\dfrac {\cos t}{\ch t}, y=asin⁡tch⁡ty=a\dfrac {\sin t}{\ch t}, z=a(t−th⁡t)z=a(t-\th t).

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Problem 2291 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x2=2azx^{2}=2az, y2=2bzy^{2}=2bz.

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Problem 2292 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x2−y2+z2=1x^{2}-y^{2}+z^{2}=1, y2−2x+z=0y^{2}-2x+z=0 at the point (1,1,1)(1,1,1).

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Problem 2293 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x=3t−t3x=3t-t^{3}, y=3t2y=3t^{2}, z=3t+t3z=3t+t^{3}.

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Problem 2294 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x=a(1−cos⁡2t)x=a(1-\cos 2t), y=asin⁡2ty=a\sin 2t, z=2acos⁡tz=2a\cos t.

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Problem 2295 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x=acos⁡3tx=a\cos^{3}t, y=asin⁡3ty=a\sin^{3}t, z=acos⁡2tz=a\cos 2t.

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Problem 2296 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Find the radius of curvature ρ\rho of the curve x=sin⁡z−zcos⁡zx=\sin z-z\cos z, y=cos⁡z+zsin⁡zy=\cos z+z\sin z.

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References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.