Right conoid explained

In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the axis of the right conoid.

Using a Cartesian coordinate system in three-dimensional space, if we take the to be the axis of a right conoid, then the right conoid can be represented by the parametric equations:

x = v\cos u y = v\sin u z = h(u)

where is some function for representing the height of the moving line.

Examples

A typical example of right conoids is given by the parametric equations

x = v\cos u, y = v\sin u, z =2 \sin u

The image on the right shows how the coplanar lines generate the right conoid.

Other right conoids include:

• Helicoid

x = v\cos u, y = v\sin u, z = cu.

• Whitney umbrella

x = vu, y = v, z = u ^{2.}

• Wallis's conical edge

x = v\cos u, y = v \sin u, z = c\sqrt{a ^{2-b} ^{2\cos} ^{2u}.}

• Plücker's conoid

x = v\cos u, y = v\sin u, z = c\sin nu.

• hyperbolic paraboloid

x = v, y = u, z = uv (with x-axis and y-axis as its axes).

See also

• Conoid
• Helicoid
• Whitney umbrella
• Ruled surface

External links

• • Right Conoid from MathWorld.
• Plücker's conoid from MathWorld