Difference between revisions of "Surface of constant width"
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'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
==Meissner solids== | ==Meissner solids== | ||
− | [[File:Sail two views.png| | + | [[File:Sail two views.png|550px|right]] |
− | [[File:Limon surface.png| | + | [[File:Limon surface.png|550px|right]] |
+ | [[File:Meissner solids 1 and 2.png|550px|right]] | ||
Let <math>ABCD</math> be the regular tetrahedron. | Let <math>ABCD</math> be the regular tetrahedron. | ||
− | The «sail» <math>ABD</math> is the piece of sphere centered at the vertex <math>C</math> with radii <math>AB = AC</math> bounded by the planes <math>ACD, BCD,</math> and <math>ADF,</math> where <math>F</math> is the mipoint of <math>BC.</math> | + | The «sail» <math>ABD</math> is the piece of sphere centered at the vertex <math>C</math> |
+ | |||
+ | with radii <math>AB = AC</math> bounded by the planes <math>ACD, BCD,</math> and <math>ADF,</math> | ||
+ | |||
+ | where <math>F</math> is the mipoint of <math>BC.</math> | ||
The view along line <math>DF</math> is shown at the left part of diagram. | The view along line <math>DF</math> is shown at the left part of diagram. | ||
− | Let point <math>E \in \overset{\Large\frown} {AD}</math> be the center of arc <math>\overset{\Large\frown} {BC}</math> with radii <math>CE = BE = AB</math> shown by red. | + | Let point <math>E \in \overset{\Large\frown} {AD}</math> be the center of |
+ | |||
+ | arc <math>\overset{\Large\frown} {BC}</math> with radii <math>CE = BE = AB</math> shown by red. | ||
+ | |||
+ | The set of such arcs centered at points lying on the arc <math>\overset{\Large\frown} {AD}</math> | ||
+ | |||
+ | create a part of lemon-shaped solid between arc <math>\overset{\Large\frown} {BC}</math> of the plane <math>BCD</math> | ||
− | + | and arc <math>\overset{\Large\frown} {BC}</math> of the plane <math>BCA,</math> shown at right part of diagram. | |
There are two possibilityes to create Meissner solids using four sails and three lemon-shaped surfaces. | There are two possibilityes to create Meissner solids using four sails and three lemon-shaped surfaces. |
Latest revision as of 13:03, 10 August 2024
A curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. A surface of constant width (orbiform) is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes.
Contents
Reuleaux triangle
The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle.
Let be equilateral triangle.
Let be the arc centered at with radius
Arcs and define similarly.
All points on this arcs are equidistant from the opposite vertex.
Distance is
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Reuleaux triangle analogue
Let be the boundary of a Reuleaux triangle, be the centroid of
Let be the arc centered at with radius points and are collinear.
Let be the arc centered at with radius points are collinear.
Similarly define arcs from point to
The width of this curve is
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Rotation of the Reuleaux triangle
The left part of diagram shows the curve which is a Reuleaux triangle determined by the center and the vertex
The Reuleaux triangle is constructed by the rotation of the curve around axis
The right part of the diagram shows the surface which arose as the result of the rotation
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Rotation of the Reuleaux triangle analogue
The left part of diagram shows the curve which is a Reuleaux triangle analogy determined by the center the vertex and point on curve analogue.
Similarly, we use the Reuleaux triangle is constructed by the rotation of the curve around axis and get the second position of the Reuleaux triangle analogue.
The middle part of the diagram shows part of the surface. We can see all four parts of this surface shown by different colors.
The right part of the diagram shows the surface which arose as the result of the rotation curve It is impossible see the down side of the surface (blue) in this view.
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Meissner solids
Let be the regular tetrahedron.
The «sail» is the piece of sphere centered at the vertex
with radii bounded by the planes and
where is the mipoint of
The view along line is shown at the left part of diagram.
Let point be the center of
arc with radii shown by red.
The set of such arcs centered at points lying on the arc
create a part of lemon-shaped solid between arc of the plane
and arc of the plane shown at right part of diagram.
There are two possibilityes to create Meissner solids using four sails and three lemon-shaped surfaces.
If lemon-shaped surfaces made triangle we name this solid as first Meissner solid.
It is shown in left part of diagram.
If lemon-shaped surfaces have the common point we name this solid as second Meissner solid.
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