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 {\displaystyle x=v\cos u}
y = v sin u {\displaystyle y=v\sin u}
z = h ( u ) {\displaystyle 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 {\displaystyle 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 . {\displaystyle x=v\cos u,y=v\sin u,z=cu.}
Whitney umbrella: x = v u , y = v , z = u 2 . {\displaystyle x=vu,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 . {\displaystyle x=v\cos u,y=v\sin u,z=c{\sqrt {a^{2}-b^{2}\cos ^{2}u}}.}
Plücker's conoid: x = v cos u , y = v sin u , z = c sin n u . {\displaystyle x=v\cos u,y=v\sin u,z=c\sin nu.}
hyperbolic paraboloid: x = v , y = u , z = u v {\displaystyle 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 "Conoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Right Conoid from MathWorld. Plücker's conoid from MathWorld