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For a wide range of choices of the area parameter, the optimal solution to this problem will be a curved triangle whose three sides are circular arcs with equal radii. In particular, when the three points are equidistant from each other and the area is that of the Reuleaux triangle, the Reuleaux triangle is the optimal enclosure.
Circular triangles are triangles with circular-arc edges, including the Reuleaux triangle as well as other shapes. The deltoid curve is another type of curvilinear triangle, but one in which the curves replacing each side of an equilateral triangle are concave rather than convex. It is not composed of circular arcs, but may be formed by rolling one circle within another of three times the radius.
In plane geometry the Blaschke—Lebesgue theorem, named after Wilhelm Blaschke and Henri Lebesgue, states that the Reuleaux triangle has the least area of all curves of given constant width. By the isoperimetric inequality, the curve of constant width with the largest area is a circle. In Ohmann proved the analogue of the Blaschke—Lebesgue theorem for Minkowski planes which uses a concept analogous to that of the Reuleaux triangle and constructed using the triangle equilateral relative to the given gauge body. In geometry, a curve of constant width is a convex planar shape whose width defined as the perpendicular distance between two distinct parallel lines each having at least one point in common with the shape's boundary but none with the shape's interior is the same regardless of the orientation of the curve.
More generally, any compact convex planar body D has one pair of parallel supporting lines in any given direction. A supporting line is a line that has at least one point in common with the boundary of D but no points in common with the interior of D. The width of the body is defined as before. If the width of D is the same in all directions, the body is said to have constant width and its boundary is a curve of constant width ; the planar body itself is called an orbiform.
The width of a circle is constant: its diameter.
Reuleaux tetrahedron - WikiVisually
Thus the question arises: if a given shape's width is constant in all directions, is it necessarily a circle? The surprising answer is that there are many non-circular shapes of constant width. A nontrivial example is the Reuleaux triangle. The resulting figure is of constant width. The Reuleaux triangle lacks tangent continuity at three points, but constant-width curves can also be constructed without such discontinuities as shown in the second illustration on the right.
Curves of constant width can be generated by joining circular arcs centered on the vertices of a regular or irregular convex polygon with an odd number of sides triangle, pentagon, heptagon, etc. In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle.
It can also be defined as the longest chord of the circle.
On the Development of the Constraint Motion Theory of Franz REULEAUX— An Overview
Both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of the diameter rather than a diameter which refers to the line itself , because all diameters of a circle or sphere have the same length, this being twice the radius r.
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For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers. For a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance.
For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing through the center of the ellipse. For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one.
The longest diameter is called the major axis. He was often called the father of kinematics. He was a leader in his profession, contributing to many important domains of science and knowledge. Today, he may be best remembered for the Reuleaux triangle, a curve of constant width that he helped develop as a useful mechanical form. The intermittent mechanism or intermittent movement is the device by which film is regularly advanced and then held in place for a brief duration of time in a movie camera or movie projector. This is in contrast to a continuous mechanism, whereby the film is constantly in motion and the image is held steady by optical or electronic methods.
The reason the intermittent mechanism "works" for the viewer is because of a phenomenon called persistence of vision. In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent.
Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape and which are in turn named for a bird. Kites are also known as deltoids, but the word "deltoid" may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be either convex or concave, but the word "kite" is often restricted to the convex variety. A concave kite is sometimes called a "dart" or "arrowhead", and is a type of pseudotriangle.
This is an alphabetical index of articles related to curves. See also curve, list of curves, and list of differential geometry topics. This is a list of two-dimensional geometric shapes in Euclidean and other geometries. The perimeter will be three nonconcentric arcs. This is a reuleaux triangle. It is not a circle, but, like a circle, it has constant width, no matter how it is oriented. It is not difficult to see this property, but you should prove it.
I made some sketches of the reuleaux and discovered some other interesting properties. It can roll uphill, in a manner of speaking. If you have the ability to view Java, click on the Animate button at right to see a rolling reuleaux triangle. Notice that as it rolls, its height is constant, but the height of its centroid changes.
If it had mass, the centroid would be the center of mass. Imagine that it is standing on one vertex, so that the centroid is at its highest, and imagine that the surface is very slightly inclined. If it moves forward one sixth turn, the centroid will fall. So although the surface rises, the reuleaux is actually falling.
Reuleaux's Revolution Redux
The figure has constant width and constant height. It should be possible to inscribe one into a square and to turn it freely. Here is a sketch to simulate this. Animate the image.
When I first saw it turning within the square, something looked very familiar about it. It looked a lot like those diagrams for the Wankel rotary engine. A Web search confirmed that this is the shape used for the rotor in the engine. The square drill bit was not my idea either.
Notice that as the reuleaux turns inside the square, its trace nearly fills the entire square. This shape can be used to drill square actually squarish holes, and in fact someone did manufacture a drill bit base on this concept. It is not just a simple matter of fitting a bit into a drill though.
It required a more complex mechanism. As the reuleaux turns in the square, watch the motion of the centroid. For every rotation of the reuleaux, the centroid makes three revolutions in the opposite direction, and its path is not circular. Your next assignment is to design a mechanism that would make it work.