Who discovered the catenoid?
Leonhard Euler
… horizontal axis is called a catenoid. The catenoid was discovered in 1744 by the Swiss mathematician Leonhard Euler and it is the only minimal surface, other than the plane, that can be obtained as a surface of revolution.
Is a catenoid a ruled surface?
Another, more interesting, ruled minimal surface is the catenoid. Not counting the plane it was the first minimal surface which was discovered and it turns out that with the plane it is the the only rotationally symmetric minimal surface.
Why is a catenoid a minimal surface?
The catenoid is the surface of revolution generated by the rotation of a catenary around its base. Since the mean curvature is zero at all points, it is a minimal surface; for that matter, it is the only minimal surface of revolution.
What is minimum surface area?
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term “minimal surface” is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint.
What is catenary equation?
The catenary is described by the equation: y=a2(ex/a+e−x/a)=acoshxa. where a is a constant. The lowest point of the chain is at (0,a). This curve is called a catenary.
What is a Tractrix curve?
A tractrix (from the Latin verb trahere “pull, drag”; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller …
Who examined soap films?
The study of soap films is believed to have started around the time of Leonardo da Vinci. Since then, research has proceeded in two distinct directions. On the one hand were the mathematicians, who were concerned with finding the shapes of these surfaces by minimizing their area, given some boundary.
How do you parameterize a Helicoid?
A parametrized helicoid. The function Φ(u,v)=(ucosv,usinv,v) parametrizes a helicoid when 0≤u≤1 and 0≤v≤2π. You can drag the cyan and magenta points on the sliders to change the values of u and v. Or, you can drag the blue point on the helix directly, which will then change u and v so that the blue point is at Φ(u,v).
How do you find the minimum area?
To find the minimum possible area, subtract the greatest possible error from each measurement, then multiply. To find the maximum possible area, add the greatest possible error to each measurement, then multiply.
How do you find the minimum volume?
To find the minimum possible volume, subtract the greatest possible error from each measurement, then multiply.
What do you mean by catenary?
catenary, in mathematics, a curve that describes the shape of a flexible hanging chain or cable—the name derives from the Latin catenaria (“chain”). Any freely hanging cable or string assumes this shape, also called a chainette, if the body is of uniform mass per unit of length and is acted upon solely by gravity.
What is the meaning of a catenoid?
A catenoid is a type of surface, arising by rotating a catenary curve about an axis.
How do you make a catenoid model?
A physical model of a catenoid can be formed by dipping two circular rings into a soap solution and slowly drawing the circles apart. The catenoid may be also defined approximately by the Stretched grid method as a facet 3D model.
When was the first catenoid invented?
It was formally described in 1744 by the mathematician Leonhard Euler . Soap film attached to twin circular rings will take the shape of a catenoid. Because they are members of the same associate family of surfaces, a catenoid can be bent into a portion of a helicoid, and vice versa.
Why can a catenoid be bent into a helicoid?
Because they are members of the same associate family of surfaces, a catenoid can be bent into a portion of a helicoid, and vice versa. The catenoid was the first non-trivial minimal surface in 3-dimensional Euclidean space to be discovered apart from the plane. The catenoid is obtained by rotating a catenary about its directrix.
