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Difference between revisions of "Surface of constant width"

(Created page with "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 const...")
 
(→‎Meissner solids)
 
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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.  
 
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.  
 
==Reuleaux triangle==
 
==Reuleaux triangle==
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[[File:PascalS Lemoine.png|390px|right]]
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[[File:Reuleaux triangle.png|350px|right]]
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The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle.  
+
The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle.
 +
 
Let <math>\triangle ABC</math> be equilateral triangle.  
 
Let <math>\triangle ABC</math> be equilateral triangle.  
  
 
Let <math>\overset{\Large\frown} {AC}</math> be the arc centered at <math>B</math> with radius <math>BB' = BA, \angle ABC = 60^\circ.</math>
 
Let <math>\overset{\Large\frown} {AC}</math> be the arc centered at <math>B</math> with radius <math>BB' = BA, \angle ABC = 60^\circ.</math>
  
−
Arcs <math>AB</math> and <math>BC</math> define similarly.
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Arcs <math>\overset{\Large\frown} {AB}</math> and <math>\overset{\Large\frown} {BC}</math> define similarly.
  
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All points on this arcs are equidistant from the opposite vertex. Distance is <math>|AB|.</math>
+
All points on this arcs are equidistant from the opposite vertex.
 +
 
 +
Distance is <math>|AB|.</math>
 +
 
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
 +
==Reuleaux triangle analogue==
 +
[[File:Reuleaux triangle analogue.png|350px|right]]
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Let <math>ABCD</math> be the boundary of a Reuleaux triangle, <math>O</math> be the centroid of <math>\triangle ABC.</math>
 +
 
 +
Let <math>\overset{\Large\frown} {FF_0}</math> be the arc centered at <math>A</math> with radius <math>AA', \angle FAF_0 = 60^\circ,</math> points <math>F, A, C</math> and <math>F_0, A, B</math> are collinear.
 +
 
 +
Let <math>\overset{\Large\frown} {F_0F'}</math> be the arc centered at <math>B</math> with radius <cmath>|AA'| + |AB|,</cmath>
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<math>\angle F'AF_0 = \angle ABC = 60^\circ,</math> points <math>F', C, B</math> are collinear.
 +
 
 +
Similarly define arcs from point <math>F'</math> to <math>F.</math>
 +
 
 +
The width of this curve is <cmath>|AB| + 2|AA'|.</cmath>
 +
 
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
 +
 
 +
==Rotation of the Reuleaux triangle==
 +
[[File:Solid Reuleaux triangle rotation.png|430px|right]]
 +
The left part of diagram shows the curve <math>ABC</math> which is a Reuleaux triangle determined by the center <math>O</math> and the vertex <math>A.</math>
 +
 
 +
The Reuleaux triangle <math>AB'C'</math> is constructed by the rotation of the curve <math>ABC</math> around axis <math>AO.</math>
 +
 
 +
The right part of the diagram shows the surface which arose as the result of the rotation <math>ABC.</math>
 +
 
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
 +
==Rotation of the Reuleaux triangle analogue==
 +
[[File:Surface Reuleaux triangle analogy construction.png|490px|right]]
 +
The left part of diagram shows the curve <math>G</math> which is a Reuleaux triangle analogy determined by the center <math>O,</math> the vertex <math>A</math> and point <math>G</math> on curve analogue.
 +
 
 +
Similarly, we use the Reuleaux triangle <math>AB'C'</math> is constructed by the rotation of the curve <math>G</math> around axis <math>AO</math> 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 <math>G.</math> It is impossible see the down side of the surface (blue) in this view.
 +
 
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
 +
==Meissner solids==
 +
[[File:Sail two views.png|550px|right]]
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[[File:Limon surface.png|550px|right]]
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[[File:Meissner solids 1 and 2.png|550px|right]]
 +
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 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.
 +
 
 +
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.
 +
 
 +
If lemon-shaped surfaces made triangle  <math>ABC</math> we name this solid as first Meissner solid.
 +
 
 +
It is shown in left part of diagram.
 +
 
 +
If lemon-shaped surfaces have the common point <math>(B)</math> we name this solid as second Meissner solid.
 +
 
 +
 
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''

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.

Reuleaux triangle

Reuleaux triangle.png

The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle.

Let $\triangle ABC$ be equilateral triangle.

Let $\overset{\Large\frown} {AC}$ be the arc centered at $B$ with radius $BB' = BA, \angle ABC = 60^\circ.$

Arcs $\overset{\Large\frown} {AB}$ and $\overset{\Large\frown} {BC}$ define similarly.

All points on this arcs are equidistant from the opposite vertex.

Distance is $|AB|.$

vladimir.shelomovskii@gmail.com, vvsss

Reuleaux triangle analogue

Reuleaux triangle analogue.png

Let $ABCD$ be the boundary of a Reuleaux triangle, $O$ be the centroid of $\triangle ABC.$

Let $\overset{\Large\frown} {FF_0}$ be the arc centered at $A$ with radius $AA', \angle FAF_0 = 60^\circ,$ points $F, A, C$ and $F_0, A, B$ are collinear.

Let $\overset{\Large\frown} {F_0F'}$ be the arc centered at $B$ with radius \[|AA'| + |AB|,\] $\angle F'AF_0 = \angle ABC = 60^\circ,$ points $F', C, B$ are collinear.

Similarly define arcs from point $F'$ to $F.$

The width of this curve is \[|AB| + 2|AA'|.\]

vladimir.shelomovskii@gmail.com, vvsss

Rotation of the Reuleaux triangle

Solid Reuleaux triangle rotation.png

The left part of diagram shows the curve $ABC$ which is a Reuleaux triangle determined by the center $O$ and the vertex $A.$

The Reuleaux triangle $AB'C'$ is constructed by the rotation of the curve $ABC$ around axis $AO.$

The right part of the diagram shows the surface which arose as the result of the rotation $ABC.$

vladimir.shelomovskii@gmail.com, vvsss

Rotation of the Reuleaux triangle analogue

Surface Reuleaux triangle analogy construction.png

The left part of diagram shows the curve $G$ which is a Reuleaux triangle analogy determined by the center $O,$ the vertex $A$ and point $G$ on curve analogue.

Similarly, we use the Reuleaux triangle $AB'C'$ is constructed by the rotation of the curve $G$ around axis $AO$ 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 $G.$ It is impossible see the down side of the surface (blue) in this view.

vladimir.shelomovskii@gmail.com, vvsss

Meissner solids

Sail two views.png
Limon surface.png
Meissner solids 1 and 2.png

Let $ABCD$ be the regular tetrahedron.

The «sail» $ABD$ is the piece of sphere centered at the vertex $C$

with radii $AB = AC$ bounded by the planes $ACD, BCD,$ and $ADF,$

where $F$ is the mipoint of $BC.$

The view along line $DF$ is shown at the left part of diagram.

Let point $E \in \overset{\Large\frown} {AD}$ be the center of

arc $\overset{\Large\frown} {BC}$ with radii $CE = BE = AB$ shown by red.

The set of such arcs centered at points lying on the arc $\overset{\Large\frown} {AD}$

create a part of lemon-shaped solid between arc $\overset{\Large\frown} {BC}$ of the plane $BCD$

and arc $\overset{\Large\frown} {BC}$ of the plane $BCA,$ 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 $ABC$ we name this solid as first Meissner solid.

It is shown in left part of diagram.

If lemon-shaped surfaces have the common point $(B)$ we name this solid as second Meissner solid.


vladimir.shelomovskii@gmail.com, vvsss

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