Right conoid
Encyclopedia
In geometry
, a right conoid
is a ruled surface
generated by a family of straight lines that all intersect perpendicularly a fixed straight line, called the axis of the right conoid.
In the space Cartesian coordinate system
, if we take the z-axis as the axis of a right conoid, then the right conoid can be represented by the following parametric equations
where h(u) is some function for representing the height of the moving line.
Figure 2 shows how the coplanar lines generate the right conoid.
Other right conoids include:
1. Helicoid
:
2. Whitney umbrella
:
3. Wallis’s conical edge
:
4. Plücker’s conoid
:
5. hyperbolic paraboloid:
(with x-axis and y-axis as its axes).
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a right conoid
Conoid
In geometry, a conoid is a Catalan surface all of whose rulings intersect a fixed line, called the axis of the conoid. If all its rulings are perpendicular to its axis, then the conoid is called a right conoid....
is a ruled surface
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
generated by a family of straight lines that all intersect perpendicularly a fixed straight line, called the axis of the right conoid.
In the space Cartesian coordinate system
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
, if we take the z-axis as the axis of a right conoid, then the right conoid can be represented by the following parametric equations
where h(u) is some function for representing the height of the moving line.
Examples
A typical example of right conoids is given by the parametric equations:
Figure 2 shows how the coplanar lines generate the right conoid.
Other right conoids include:
1. Helicoid
Helicoid
The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...
:
2. Whitney umbrella
Whitney umbrella
right|frame|240px|Section of the surfaceIn mathematics, the Whitney umbrella is a self-intersecting surface placed in three dimensions...
:
3. Wallis’s conical edge
Wallis’s conical edge
Wallis's conical edge is a ruled surface given by the parametric equations:where a, b and c are constants.Wallis's conical edge is also a kind of right conoid.Figure 2 shows that the Wallis's conical edge is generated by a moving line....
:
4. Plücker’s conoid
Plücker’s conoid
In geometry, the Plücker’s conoid is a ruled surface named after the German mathematician Julius Plücker. It is also called a cylindroid or .The Plücker’s conoid is defined by the function of two variables:...
:
5. hyperbolic paraboloid:
(with x-axis and y-axis as its axes).External links
- Right Conoid from MathWorld.
- Plücker's conoid from MathWorld