4.4: The Paraboloid
- Page ID
- 6808
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(\text{IV.4}\).
