Right conoid



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: $$x=v\cos u, y=v\sin u, z=cu.$$
 * 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).