Right conoid - Alchetron, The Free Social Encyclopedia

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 z-axis 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 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

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 = c u .

Whitney umbrella: x = v u , y = v , z = u 2 .

Wallis’s conical edge: x = v cos ⁡ u , y = v sin ⁡ u , z = c a 2 − b 2 cos 2 ⁡ u .

Plücker’s conoid: x = v cos ⁡ u , y = v sin ⁡ u , z = c sin ⁡ n u .

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

References

Right conoid Wikipedia

(Text) CC BY-SA