geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a right conoid is a

ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...

generated by a family of

straight line In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimens ...

s that all intersect

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

ly to a fixed straight line, called the ''axis'' of the right conoid. Using a

Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...

three-dimensional space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...

, if we take the to be the axis of a right conoid, then the right conoid can be represented by the

parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...

s: :

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 The helicoid, also known as helical surface, is a smooth Surface (differential geometry), surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its Rotation ...

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

* Whitney umbrella:

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

Wallis's conical edge In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt where , and are constants. Wallis's conical edge is also a kind of right conoid. It is named after the English math ...

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

* Plücker's conoid:

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

hyperbolic paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every pla ...

x=v, y=u, z=uv

(with x-axis and y-axis as its axes).

External links

*
Right Conoid
from MathWorld.

from MathWorld Surfaces Geometric shapes {{geometry-stub

Examples

See also

External links