3-axis ellipsoid with a circle as a plane section
A circuit section plane is in the geometry of a plane (in 3-dimensional space) which is a quadric ( ellipsoid , hyperboloid , ...) in a circle intersect. A sphere is cut in a circle by every plane with which it has at least 2 points in common. The position is also simple for rotational quadrics (rotational ellipsoid, hyperboloid, paraboloid, cylinder, ...): They are cut in circles by all planes that are perpendicular to the axis of rotation, if they have at least 2 points in common. The position of 3-axis ellipsoids, genuinely elliptical hyperboloids, paraboloids, cylinders, ... is no longer obvious , although there are even more intersection circles in these asymmetrical cases. The following applies:
- Every quadric (area in 3-dimensional space) that contains ellipses also contains circles (see below).
Quadrics that are not included are: 1) parabolic cylinder , 2) hyperbolic cylinder, and 3) hyperbolic paraboloid . A comprehensive discussion of all cases is e.g. B. in the book by Grotemeyer (see literature ) included.
Circular sections of quadrics were previously used to create models (see #Weblinks ).
Circular sectional planes also play a role in crystallography .
Description of the method
To find the planes that intersect a quadric in a circle, two essential observations are used:
-
(K :) If the intersection of a quadric with a sphere (auxiliary sphere ) lies in a pair of planes (two intersecting planes), the intersection consists of two circles.
-
(P :) If a plane intersects a quadric in a circle, this is also the case for all planes parallel to it , which have at least two points in common with the quadric.
To find circles on a quadric, it is sufficient to find an auxiliary sphere that intersects the quadric in a pair of planes. Then the planes parallel to the planes provide two sets of intersection circles.
3-axis ellipsoid with circular cuts (blue and green) and the auxiliary sphere (red) that cuts the ellipsoid in the blue circles
Figure 1: Ellipsoid cut with spheres:
3-axis ellipsoid
For the ellipsoid with the equation
![{\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} + {\ frac {z ^ {2}} {c ^ {2}}} = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/68f83fa62cd61acf1ca6e1a1ab024fc7bf317df1)
and the semi-axes one uses an auxiliary sphere with the equation
![{\ displaystyle a> b> c> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/291666363e8b41ff5b645aa60b7bbfe5a136356e)
![{\ displaystyle x ^ {2} + y ^ {2} + z ^ {2} = r ^ {2} \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8213cdf43240ef6dfedc85eb7b80d252b16301)
The sphere radius must now be determined in such a way that the intersection of the ellipsoid and the sphere lies in a pair of planes through the origin . So that the absolute term falls out, one subtracts the spherical equation from the factor of the ellipsoid equation. It turns out
![r ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a363a15442d031416d1eb62254a9c726e3f6c66c)
![{\ displaystyle \ left ({\ frac {r ^ {2}} {a ^ {2}}} - 1 \ right) \; x ^ {2} + \ left ({\ frac {r ^ {2}} {b ^ {2}}} - 1 \ right) \; y ^ {2} + \ left ({\ frac {r ^ {2}} {c ^ {2}}} - 1 \ right) \; z ^ {2} = 0 \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5425f968aace7b6fb108aa81e5cb31f148abd931)
This equation describes a pair of planes only if one of the three coefficients is zero. Both for and for there are equations that are only satisfied by points on the -axis or -axis. Only the case leads to a pair of planes with the equation
![{\ displaystyle r = a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e193cdce556664593cd0a8347617f5284d0364c9)
![{\ displaystyle r = c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/251cfade2ce6c9bc930a895410b1699bb7d5aa83)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![{\ displaystyle r = b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1214af636c70a8359956e08b2c20c64b59c6071a)
![{\ displaystyle \ left ({\ frac {b ^ {2}} {a ^ {2}}} - 1 \ right) \; x ^ {2} + \ left ({\ frac {b ^ {2}} {c ^ {2}}} - 1 \ right) \; z ^ {2} = 0 \ \ quad \ Leftrightarrow \ quad z = \ pm {\ frac {c} {a}} {\ sqrt {\ frac { a ^ {2} -b ^ {2}} {b ^ {2} -c ^ {2}}}} \; x \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/425addb3002a4839c47235f5cd50844776ca720c)
because only in this case do the remaining coefficients have different signs (because of ).
![a> b> c](https://wikimedia.org/api/rest_v1/media/math/render/svg/3756bee44d7cb7e6221499eedb579fb848d2e5ac)
Figure 1 shows how the cuts with unsuitable spheres look like: The radius is too large (magenta) or too small (cyan).
If the values of the semi-axes and approach, the two sets of circles also approach. For (ellipsoid of revolution) all circular planes are orthogonal to the axis of rotation.
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
![a = b> c](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4e2c770be564fcc272802932f2139280fb5720)
Proof of property (P):
If the ellipsoid is rotated around the -axis so that one of the two blue circles lies in the -plane, the ellipsoid suffices for one equation
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
![xy](https://wikimedia.org/api/rest_v1/media/math/render/svg/3129cb3620bd9f38d0304a0fca719644d7d2d265)
![{\ displaystyle Ax ^ {2} + By ^ {2} + Cz ^ {2} + D {\ color {red} xz} = E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/467f59bda765e63a8cb817ae2b62f11a24e23115)
and for results . For this to be a circular equation, it must be true. If one now cuts the ellipsoid with a plane parallel to the plane with the equation , the result is
![z = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3)
![{\ displaystyle Ax ^ {2} + By ^ {2} = E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f90e0294c72d58a6f0588b71fc776b1e216089)
![{\ displaystyle A = B \ neq 0, \ E> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c16c385dc3d4779fcc753406b3425f9d408244)
![xy](https://wikimedia.org/api/rest_v1/media/math/render/svg/3129cb3620bd9f38d0304a0fca719644d7d2d265)
![z = z_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8e63a3f2769739a78c3c24a091f778fdc72dfe)
-
.
This equation describes a circle or a point or the empty set. (The center and radius are obtained after completing the square .)
Elliptical single-shell hyperboloid
For the hyperboloid with the equation
![{\ displaystyle {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} - {\ frac {z ^ {2} } {c ^ {2}}} = 1 \, \ quad a> b \, c> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/259e5da3b16eb2728c19eedcac0fb56896203671)
one obtains (as with the ellipsoid) the equation
for the intersection with a sphere![x ^ {2} + y ^ {2} + z ^ {2} = r ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e71876d62bb22964e3c7042447a6541f963993)
![{\ displaystyle \ left ({\ frac {r ^ {2}} {a ^ {2}}} - 1 \ right) \; x ^ {2} + \ left ({\ frac {r ^ {2}} {b ^ {2}}} - 1 \ right) \; y ^ {2} - \ left ({\ frac {r ^ {2}} {c ^ {2}}} + 1 \ right) \; z ^ {2} = 0 \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aad7899d71b3419277c52572fd5ebde1f65f3845)
This results in a pair of levels only for :
![{\ displaystyle r = a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e193cdce556664593cd0a8347617f5284d0364c9)
![{\ displaystyle \ left ({\ frac {a ^ {2}} {b ^ {2}}} - 1 \ right) \; y ^ {2} - \ left ({\ frac {a ^ {2}} {c ^ {2}}} + 1 \ right) \; z ^ {2} = 0 \ \ quad \ Leftrightarrow \ quad z = \ pm {\ frac {c} {b}} {\ sqrt {\ frac { a ^ {2} -b ^ {2}} {a ^ {2} + c ^ {2}}}} \; y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4036dfbfb0a938707995d57e9269b9f872896310)
Elliptical cylinder
For the elliptical cylinder with the equation
![{\ displaystyle {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} = 1 \, \ quad a> b \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d24f645b6024ef26066f6e82c50f861ecb09a007)
one obtains the equation
![{\ displaystyle \ left ({\ frac {r ^ {2}} {a ^ {2}}} - 1 \ right) \; x ^ {2} + \ left ({\ frac {r ^ {2}} {b ^ {2}}} - 1 \ right) \; y ^ {2} -z ^ {2} = 0 \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38d45c31917ed2e1c2c0b1f2b1cd0108d0bba016)
There is a pair of levels only for :
![{\ displaystyle r = a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e193cdce556664593cd0a8347617f5284d0364c9)
![{\ displaystyle \ left ({\ frac {a ^ {2}} {b ^ {2}}} - 1 \ right) \; y ^ {2} -z ^ {2} = 0 \ \ quad \ Leftrightarrow \ quad z = \ pm {\ frac {\ sqrt {a ^ {2} -b ^ {2}}} {b}} \; y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0afe7c741b74d68cb9d0acf2fe5614d36832e3c4)
Remarks:
- Since the above single-shell hyperboloid becomes the cylinder of this section, the circular planes of the cylinder result from those of the hyperboloid in this way.
![c \ to \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/532c440a13204c3dabd0b254a735638eb11b9fa1)
- An elliptical cylinder can therefore always be understood as an oblique circular cylinder .
Elliptical paraboloid
Figure 2: Elliptical paraboloid
For the elliptical paraboloid with the equation
![{\ displaystyle ax ^ {2} + by ^ {2} -z = 0 \, \ quad a \, {\ color {red} {<}} \, b \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6aa63a2e03e4c22c1172a703028495863df49f1)
choose a sphere through the vertex with the center on the axis:
![{\ displaystyle x ^ {2} + y ^ {2} + (zr) ^ {2} = r ^ {2} \ quad \ Leftrightarrow \ quad x ^ {2} + y ^ {2} + z ^ {2 } -2rz = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd145c861689f1be8beb9ac86a65874059d6c413)
After eliminating the linear term, the equation results
![{\ displaystyle (2ra-1) \; x ^ {2} + (2rb-1) \; y ^ {2} -z ^ {2} = 0 \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7de9547048270855f4e3bda47b2309e119f2c5)
There is a pair of levels only for :
![{\ displaystyle r = {\ tfrac {1} {2a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78a2f3169efd9b81e9bc0c5fb4b4260f4ad2f3ef)
![{\ displaystyle \ left ({\ frac {b} {a}} - 1 \ right) y ^ {2} -z ^ {2} = 0 \ quad \ Leftrightarrow \ quad z = \ pm {\ sqrt {\ frac {ba} {a}}} \; y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e629193269450895e6cbdf32ea56a0d48a086531)
Note:
The radius of the sphere is equal to the radius of the curvature of the further parabola (see Figure 2).
Elliptical double-shell hyperboloid
Figure 3: Elliptical double-shell hyperboloid
The double-shell hyperboloid with the equation
![{\ displaystyle - {\ frac {x ^ {2}} {a ^ {2}}} - {\ frac {y ^ {2}} {b ^ {2}}} + {\ frac {z ^ {2 }} {c ^ {2}}} = 1 \, \ quad a> b \, \ c> 0 \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c83fe92212dc8dd036285a84d766fe323872f7c5)
it is advisable to move it so that a vertex is the origin (see Figure 3):
![{\ displaystyle - {\ frac {x ^ {2}} {a ^ {2}}} - {\ frac {y ^ {2}} {b ^ {2}}} + {\ frac {(z + c ) ^ {2}} {c ^ {2}}} = 1 \ quad \ Leftrightarrow \ quad \ - {\ frac {x ^ {2}} {a ^ {2}}} - {\ frac {y ^ { 2}} {b ^ {2}}} + {\ frac {z ^ {2}} {c ^ {2}}} + {\ frac {2z} {c}} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0df1cc229161926688bb1346387371a9217d398d)
The sphere is also chosen in such a way that it contains the origin and its center point lies on the axis:
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![{\ displaystyle x ^ {2} + y ^ {2} + (zr) ^ {2} = r ^ {2} \ quad \ Leftrightarrow \ quad x ^ {2} + y ^ {2} + z ^ {2 } -2zr = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18a8c92d8915e9a387252422f3f95b58cf6bf3b8)
After eliminating the linear term, the equation results
![{\ displaystyle \ left (- {\ frac {r} {a ^ {2}}} + {\ frac {1} {c}} \ right) \; x ^ {2} + \ left (- {\ frac {r} {b ^ {2}}} + {\ frac {1} {c}} \ right) \; y ^ {2} + \ left ({\ frac {r} {c ^ {2}}} + {\ frac {1} {c}} \ right) \; z ^ {2} = 0 \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b594ec787f40d07bfa8f0664c503f39f08aaebd)
There is a pair of levels only for :
![{\ displaystyle r = {\ tfrac {a ^ {2}} {c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5d3e173f2cee2f3d382c2b7ee1b31cd770c10df)
![{\ displaystyle \ left (- {\ frac {a ^ {2}} {b ^ {2} c}} + {\ frac {1} {c}} \ right) \; y ^ {2} + \ left ({\ frac {a ^ {2}} {c ^ {3}}} + {\ frac {1} {c}} \ right) \; z ^ {2} = 0 \ \ quad \ Leftrightarrow \ quad z = \ pm {\ frac {c} {b}} {\ sqrt {\ frac {a ^ {2} -b ^ {2}} {a ^ {2} + c ^ {2}}}} \; y }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3de959eebda1c7d8a894273f2c174af408d4454d)
Note:
The radius of the sphere is equal to the radius of the circle of curvature of the further hyperbola (see Figure 3).
Elliptical cone
Figure 4: Elliptical cone
The elliptical cone with the equation
![{\ displaystyle {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} - z ^ {2} = 0 \, \ quad a> b \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d649df73345646b51a042f6c6106ca9a04c4eea)
it is advisable to move it so that its tip is not at the origin (see Figure 4):
![{\ displaystyle {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} - (z-1) ^ {2} = 0 \ quad \ Leftrightarrow \ quad {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} - z ^ {2 } + 2z = 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03ca3a647e6e772515ac18071769401e4db13c74)
Now you can use a sphere around the origin:
![x ^ {2} + y ^ {2} + z ^ {2} = r ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e71876d62bb22964e3c7042447a6541f963993)
The elimination of results in:
![x ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93)
![{\ displaystyle \ left ({\ frac {a ^ {2}} {b ^ {2}}} - 1 \ right) \; y ^ {2} - (1 + a ^ {2}) \; z ^ {2} + 2a ^ {2} z = a ^ {2} -r ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08f4eece54ea974e1123583f8ff2623b234d6584)
Since planes are to be expected that do not go through the origin, one carries out a quadratic addition and obtains:
![{\ displaystyle {\ frac {a ^ {2} -b ^ {2}} {b ^ {2}}} \; y ^ {2} - (1 + a ^ {2}) \ left (z- { \ frac {a ^ {2}} {1 + a ^ {2}}} \ right) ^ {2} = a ^ {2} - {\ frac {a ^ {4}} {1 + a ^ {2 }}} - r ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33cd1e7e3b3188ba6c61ae24e03e187608810a70)
For this equation to describe a pair of planes, the right-hand side must be zero; H.
If you then dissolve, you get:
![{\ displaystyle r = {\ tfrac {a} {\ sqrt {1 + a ^ {2}}}} \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a07cb1a239beac98d272d17fc498b87571671a18)
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![{\ displaystyle z = {\ frac {a ^ {2}} {1 + a ^ {2}}} \ pm {\ frac {1} {b}} {\ sqrt {\ frac {a ^ {2} - b ^ {2}} {1 + a ^ {2}}}} \; y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e1a93c90f78dda946793d73f34484201795f56)
Note:
An elliptical cone can always be understood as an oblique circular cone . One should note, however, that the straight line through the apex and the center of the circle is not the axis of the cone (axis of symmetry).
literature
-
KP Grotemeyer : Analytical Geometry. Göschen-Verlag, 1962, p. 143.
- H. Scheid, W. Schwarz: Elements of linear algebra and analysis. Spektrum, Heidelberg, 2009, ISBN 978-3-8274-1971-2 , p. 132.
Individual evidence
-
^ WH Westphal: Physical Dictionary: Two parts in one volume. Springer-Verlag, 1952, ISBN 978-3-662-12707-0 , p. 350.
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↑ H. Tertsch: The strength phenomena of the crystals. Springer-Verlag, Vienna, 1949, ISBN 978-3-211-80120-8 , p. 87.
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^ G. Masing: Textbook of general metal science. Springer-Verlag, Berlin, 1950, ISBN 978-3-642-52-993-1 , p. 355.
Web links