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4.4: The Paraboloid

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

The equation $$x^2 = 4qz = 2lz$$ is a parabola in the $$xz$$-plane. The distance between vertex and focus is $$q$$, and the length of the semi latus rectum $$l = 2q$$. The equation can also be written

$\frac{x^2}{a^2} = \frac{z}{h} \label{4.4.1} \tag{4.4.1}$

Here $$a$$ and $$h$$ are distances such that $$x = a$$ when $$z = h$$, and the length of the semi latus rectum is $$l = a^2 /(2h)$$.

If this parabola is rotated through $$360^\circ$$ about 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

$\frac{x^2}{a^2} + \frac{y^2}{a^2} = \frac{z}{h}. \label{4.4.2} \tag{4.4.2}$

The cross-section at $$z = h$$ is a circle of radius $$a$$.

The equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{h} , \label{4.4.3} \tag{4.4.3}$

in which we shall choose the $$x$$- and $$y$$-axes such that $$a > b$$, is an elliptic paraboloid and, if $$a ≠ b$$, is not formed by rotation of a parabola. At $$z = h$$, the cross section is an ellipse of semi major and minor axes equal to $$a$$ and $$b$$ respectively. The section in the plane $$y = 0$$ is a parabola of semi latus rectum $$a^2 /(2h)$$. The section in the plane $$x = 0$$ is a parabola of semi latus rectum $$b^2 /(2h)$$. The elliptic paraboloid lies entirely above the $$xy$$-plane.

The equation

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{h} \label{4.4.4} \tag{4.4.4}$

is a hyperbolic paraboloid, and its shape is not quite so easily visualized. Unlike the elliptic paraboloid, it extends above and below the plane. It is a saddle-shaped surface, with the saddle point at the origin. The section in the plane $$y = 0$$ is the "nose down" parabola $$x^2 = a^2 z / h$$ extending above the xy-plane. The section in the plane $$x = 0$$ is the "nose up" parabola $$y^2 = -b^2 z /h$$ extending below the $$xy$$-plane. The section in the plane $$z = h$$ is the hyperbola

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \label{4.4.5} \tag{4.4.5}$

The section with the plane $$z = −h$$ is the conjugate hyperbola

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1 . \label{4.4.6} \tag{4.4.6}$

The section with the plane $$z = 0$$ is the asymptotes

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 . \label{4.4.7} \tag{4.4.7}$

The surface for $$a = 3$$, $$b = 2$$, $$h = 1$$ is drawn in figure $$\text{IV.4}$$. 