Right conoid
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Short description: Ruled surface made of lines orthogonal to an axis
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:
- [math]\displaystyle{ x=v\cos u }[/math]
- [math]\displaystyle{ y=v\sin u }[/math]
- [math]\displaystyle{ z=h(u) }[/math]
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
- [math]\displaystyle{ x=v\cos u, y=v\sin u, z=2\sin u }[/math]
The image on the right shows how the coplanar lines generate the right conoid.
Other right conoids include:
- Helicoid: [math]\displaystyle{ x=v\cos u, y=v\sin u, z=cu. }[/math]
- Whitney umbrella: [math]\displaystyle{ x=vu, y=v, z=u^2. }[/math]
- Wallis's conical edge: [math]\displaystyle{ x=v\cos u, y=v \sin u, z=c\sqrt{a^2-b^2\cos^2u}. }[/math]
- Plücker's conoid: [math]\displaystyle{ x=v\cos u, y=v\sin u, z=c\sin nu. }[/math]
- hyperbolic paraboloid: [math]\displaystyle{ x=v, y=u, z=uv }[/math] (with x-axis and y-axis as its axes).
See also
External links
- Hazewinkel, Michiel, ed. (2001), "Conoid", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/c025210
- Right Conoid from MathWorld.
- Plücker's conoid from MathWorld
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