Paraboloid
The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
The paraboloid is elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a section by a tangent plane).
Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder, and has an implicit equation whose part of degree two may be factored over the complex numbers into two different linear factors.
An elliptic paraboloid is shaped like an oval cup and has a maximum or minimum point when its axis is vertical.
In a suitable coordinate system with three axes x, y, and z, it can be represented by the equation[1]
where a and b are constants that dictate the level of curvature in the xz and yz planes respectively.
In this position, the elliptic paraboloid opens upward.
A hyperbolic paraboloid (not to be confused with a hyperboloid) is a doubly ruled surface shaped like a saddle.
In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation[2][3]
In a suitable Cartesian coordinate system, an elliptic paraboloid has the equation
It is a surface of revolution obtained by revolving a parabola around its axis.
This is also true in the general case (see Circular section).
From the point of view of projective geometry, an elliptic paraboloid is an ellipsoid that is tangent to the plane at infinity.
The surface of a rotating liquid is also a circular paraboloid.
The hyperbolic paraboloid is a doubly ruled surface: it contains two families of mutually skew lines.
These properties characterize hyperbolic paraboloids and are used in one of the oldest definitions of hyperbolic paraboloids: a hyperbolic paraboloid is a surface that may be generated by a moving line that is parallel to a fixed plane and crosses two fixed skew lines.
This property makes it simple to manufacture a hyperbolic paraboloid from a variety of materials and for a variety of purposes, from concrete roofs to snack foods.
In particular, Pringles fried snacks resemble a truncated hyperbolic paraboloid.
[4] A hyperbolic paraboloid is a saddle surface, as its Gauss curvature is negative at every point.
From the point of view of projective geometry, a hyperbolic paraboloid is one-sheet hyperboloid that is tangent to the plane at infinity.
can be Saddle roofs are often hyperbolic paraboloids as they are easily constructed from straight sections of material.
is rotated by an angle of π/4 in the +z direction (according to the right hand rule), the result is the surface
which can be thought of as the geometric representation (a three-dimensional nomograph, as it were) of a multiplication table.
are harmonic conjugates, and together form the analytic function
The dimensions of a symmetrical paraboloidal dish are related by the equation
A more complex calculation is needed to find the diameter of the dish measured along its surface.
This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish.
where ln x means the natural logarithm of x, i.e. its logarithm to base e. The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidal wok), is given by
This can be compared with the formulae for the volumes of a cylinder (πR2D), a hemisphere (2π/3R2D, where D = R), and a cone (π/3R2D).
πR2 is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight a reflector dish can intercept.
