Math Problems Vault

Problem 2370 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the following family of surfaces: spheres $(x-a)^{2}+y^{2}+z^{2}=1$ .

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Remark.

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Surfaces $f(x,y,z,a)=0$ , whose equations contain a coefficient $a$ varying from one surface to another, form a one-parameter family of surfaces. If two infinitely close surfaces meet in a certain curve, then the coordinates of the points of this curve, called the characteristic curve, satisfy the equations:

f(x,y,z,a) = 0, \qquad \frac{\partial f}{\partial a} = 0 .

Eliminating the parameter $a$ from these equations, we find an equation $F(x,y,z)=0$ . If the given family has an envelope surface, then it lies entirely on the surface obtained.

Similarly, the equation $f(x,y,z,a,b)=0$ represents a two-parameter family of surfaces. The limiting position of the point of intersection of three nearby surfaces

f(x,y,z,a,b) = 0, \qquad f(x,y,z,a+\Delta a,b) = 0, \qquad f(x,y,z,a,b+\Delta b) = 0

as $\Delta a\to 0$ and $\Delta b\to 0$ , i.e.the characteristic point of the family of surfaces, satisfies the equations:

f(x,y,z,a,b) = 0, \qquad \frac{\partial f}{\partial a} = 0, \qquad \frac{\partial f}{\partial b} = 0 .

Eliminating the parameters $a$ and $b$ from these, we obtain the equation of a surface $F(x,y,z)=0$ . If the given family has an envelope surface, then all its points satisfy the equation obtained; but other points may satisfy it as well.

Problem 2371 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the following family of surfaces (see problem 2370): planes passing through the point $(\sqrt{2},0,0)$ and at a distance equal to unity from the origin.

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Problem 2372 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the spheres

(x-lt)^{2}+(y-mt)^{2}+(z-nt)^{2} = a^{2},

where $t$ is a variable parameter.

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Problem 2373 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the spheres whose great circles lie on the paraboloid $z=x^{2}+y^{2}$ .

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Problem 2374 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

The same question (find the envelope of the spheres whose great circles lie on the paraboloid) as in the preceding problem, for the paraboloid $2z=\dfrac {x^{2}}{p}+\dfrac {y^{2}}{q}$ .

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Problem 2375 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the spheres of radius $a$ whose centers lie on the circle

x^{2}+y^{2} = r^{2}, \qquad z = 0 .

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Problem 2376 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the planes tangent to the parabolas

y^{2} = 2x, \quad z = 0 \qquad \text{and} \qquad y^{2} = 2z, \quad x = 0 .

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Problem 2377 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the spheres

(x-lt)^{2}+(y-mt)^{2}+(z-nt)^{2} = p^{2}t^{2},

where $t$ is a variable parameter.

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Problem 2378 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the ellipsoids $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ subject to the condition $a^{2}+b^{2}+c^{2}=1$ .

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Problem 2379 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

The same (find the envelope of the ellipsoids $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ , see problem 2378) subject to the condition $a+b+c=1$ .

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Problem 2380 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the planes that, together with the coordinate planes, form a tetrahedron of constant volume.

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Problem 2381 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Find the envelope of the planes for which the segments $a$ , $b$ , $c$ on the axes are related by the equation $a^{n}+b^{n}+c^{n}=1$ .

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Problem 2382 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.10 "Tangent Planes and Normals. Envelopes"

Show that the circles

y = tx, \qquad x^{2}+y^{2}+z^{2}-2f(t)x-a^{2} = 0

are the characteristics of the family of spheres

x^{2}+y^{2}+z^{2}-2\bigl(f(t)+tf'(t)\bigr)x-2f'(t)y-a^{2} = 0 .

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Envelope of a one-parameter family of surfaces

References