Paraboloid

A paraboloid is in the form of a quadric surface that has exactly one axis of symmetry and no center of symmetry. We have derived the term "paraboloid"  from parabola, which is similar to a conic section. Such that it has a similar property of symmetry. A surface whose equation is of the second degree is called a conicoid. The standard form of a non-central conicoid equation is \[ax^{2} + by^{2} + 2wz +d = 0\]. If the value of w = 0, then the equation represents a cylinder or a pair of straight lines. When the value of w 0, then the surface represented by the equation is called a paraboloid. In simple words, we can say that a paraboloid is an open surface generated by rotating a parabola about its axis. The standard equation of a paraboloid is represented by the equation \[ax^{2} + by^{2} = 2wz\] , w   ≠  0. 

In this article, we will study paraboloid definition and different types of paraboloid equations. We will select multiple sets of parallel planes to intercept the paraboloid based on the use of the cross-section method, and then discuss the graph and drawing method of the paraboloid.  

Types  of Paraboloid

• Elliptic paraboloid

• hyperbolic paraboloid

We know that the standard equation of a paraboloid is \[ax^{2} + by^{2} = 2wz\], w 0. Based on the sign of a and b we can decide the types of the parabola. When a and b are of the same signs, we get an elliptic paraboloid. When a and b are of the opposite sign then the surface we get is a hyperbolic paraboloid.

Elliptical Paraboloid

If a = b, then the intersections of the surface with planes parallel to and above the xy plane produce circles. The figures generated are the paraboloid of revolution. If the value of a  ≠ b , intersections with planes parallel to the xy plane are ellipses, the surface is an elliptical paraboloid.

If the surface of the paraboloid is defined by the equation, \[\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = z\] cuts parallel to the xz and yz planes producing parabolas of intersection and cutting planes parallel to xy produce hyperbolas. Such a surface is a hyperbolic paraboloid.

Hyperbolic Paraboloid Equation

The intersection of a hyperbolic paraboloid with a plane is a hyperbola, a parabola, or a pair of lines. The two rectilinear generators pass through each point of a hyperbolic paraboloid, which consequently is a ruled surface. In contrast to a hyperbolic paraboloid, an elliptic paraboloid does not intersect every plane in space. When it does intersect a plane, then the intersection is either an ellipse or a parabola. 

Hyperbolic paraboloid formula is given below

\[ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = \frac{z}{c}\]

Given below is a sketch of a typical hyperbolic paraboloid.

(Image will be Updated soon)

These graphs are vaguely saddle-shaped and as with the elliptic paraboloid. The sign of 

c will determine the direction in which the surface will “open up”. The graph above is shown for c positive.

In both the types of paraboloids discussed above, we should note that the surface can be easily moved up or down by adding/subtracting a constant from the left side.

For instance,

\[z = -x^{2} - y^{2} + 6\]

is an elliptic paraboloid that opens downward (sign should be kept carefully, the “-” is on the x and y instead of z), and starts at z=6 instead of z=0.

Here we have provided a couple of quick sketches of this surface.

(Image will be Updated soon)

Note that we have provided two forms of the sketch here. The sketch which is present on the left has the standard set of axes but it is difficult to see the numbers on the axis. The sketch which is present on the right has been “boxed” and this makes it easier to see the numbers to give a sense of perspective to the sketch. In most sketches that actually involve numbers on the axis system, we will give both sketches to get a feel for what the sketch looks like.

Elliptic Paraboloid Equation

Given below is the equation of an elliptic paraboloid.

\[ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = \frac{z}{c}\]

As with cylinders, this has a cross-section of an ellipse and if a=b, it will have a cross-section of a circle. When we deal with these, we will generally be dealing with the kind that has a circle for a cross-section.

Here we have provided a sketch of a typical elliptic paraboloid.

(Image will be Updated soon)

In this case, the variable that is not squared determines the axis upon which the paraboloid opens up. Also, the sign of c will determine the direction that the paraboloid will open. If the sign of c is positive, then it opens up and if it is negative, then it opens down.

Circular Paraboloid Equation

If this parabola is rotated through 360about the z-axis, the figure swept out is a paraboloid of revolution or circular paraboloid. Many telescope mirrors are of this shape. The Equation to the circular paraboloid is \[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{z}{h}\].

Conclusion

Over a small area, a paraboloid differs only slightly from a sphere. A paraboloid mirror can therefore be made by deepening the center of a spherical mirror. A circular or elliptical paraboloid surface may be used as a parabolic reflector. We have studied the paraboloid formula. The applications of this property are used in automobile headlights, solar furnaces, radar, and radio relay stations. Paraboloid surfaces are mostly used in reflecting telescopes and radio telescopes.