2.6 Quadric Surfaces

← Chapter 2 §2.6 Quadric Surfaces 二次曲面 · Chapter 2 · Vectors in Space

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Objective	Takeaways; say them first, then click
2.6.1 Identify a cylinder as a type of three-dimensional surface.	a missing variable → the curve slid along that axis; rulings run along the missing variable
2.6.2 Recognize the main features of ellipsoids, paraboloids, and hyperboloids.	right side 1: count minus signs (0 ellipsoid, 1 one sheet, 2 two sheets); right side 0: cone; a first power: paraboloid; axis = the odd variable out
2.6.3 Use traces to draw the intersections of quadric surfaces with the coordinate planes.	set $x$ , $y$ or $z=k$ ; each trace is a conic; sketch from the traces in the coordinate planes

Mistake	Result	Fix
$x^2+y^2=9$ "is a circle" in space	misses every height	a missing variable is a ruling direction: a cylinder
Axis of a hyperboloid = a positive variable	wrong axis for one sheet	the odd sign: the minus for one sheet, the plus for two
Reading $16x^2+9y^2+16z^2=144$ as semi-axes 4, 3, 4	semi-axes 3, 4, 3 instead	divide to get 1 first
$9(x^2-2x+1)$ adds 1	it adds 9	subtract the coefficient times the square
$(y+2)^2$ read as centre $y=2$	the book's Example 2.59b	centre $y=-2$
Cone and hyperboloid confused	$=0$ versus $=1$	right side 0: cone
$z=x^2-y^2$ called elliptic	it is a saddle	opposite signs: hyperbolic paraboloid
$x^2=25z$ gives focal length 25	focus at 25	$4p=25$ , $p=6.25$

#	task	answer
2.52	graph the cylinder $z=y^2$	parabola $z=y^2$ in the $yz$ -plane, rulings along $x$
2.53	traces of $\frac{x^2}{3^2}+\frac{y^2}{2^2}-\frac{z^2}{5^2}=1$	ellipses ∥ $xy$ -plane; hyperbolas $\frac{x^2}{9}-\frac{z^2}{25}=1$ , $\frac{y^2}{4}-\frac{z^2}{25}=1$
2.54	$9x^2+y^2-z^2+2z-10=0$	$x^2+\frac{y^2}{9}-\frac{(z-1)^2}{9}=1$ : one sheet, centre $(0,0,1)$

#	task	answer
303	$x^2+z^2=1$	cylinder, rulings along $y$
305	$z=\cos\left(\frac\pi2+x\right)$	$z=-\sin x$ : cylinder, rulings along $y$
307	$z=9-y^2$	cylinder, rulings along $x$
313, 315, 317	match: two sheets, elliptic paraboloid, one sheet	b, d, a
319	$-x^2+36y^2+36z^2=9$	$-\frac{x^2}{9}+\frac{y^2}{1/4}+\frac{z^2}{1/4}=1$ : one sheet, axis $x$
321	$-3x^2+5y^2-z^2=10$	$-\frac{x^2}{10/3}+\frac{y^2}{2}-\frac{z^2}{10}=1$ : two sheets, axis $y$
323	$5y=x^2-z^2$	$y=\frac{x^2}{5}-\frac{z^2}{5}$ : hyperbolic paraboloid, axis $y$
325	$x^2+5y^2+3z^2-15=0$	$\frac{x^2}{15}+\frac{y^2}{3}+\frac{z^2}{5}=1$ : ellipsoid
327	$x^2+5y^2-8z^2=0$	$\frac{x^2}{40}+\frac{y^2}{8}-\frac{z^2}{5}=0$ : cone, axis $z$
329	$6x=3y^2+2z^2$	$x=\frac{y^2}{2}+\frac{z^2}{3}$ : elliptic paraboloid, axis $x$
331	$x^2+z^2+4y=0$ , $z=0$	parabola $y=-\frac{x^2}{4}$
333	$-4x^2+25y^2+z^2=100$ , $x=0$	ellipse $\frac{y^2}{4}+\frac{z^2}{100}=1$
339	$x^2+2z^2+6x-8z+1=0$	$\frac{(x+3)^2}{16}+\frac{(z-2)^2}{8}=1$ : cylinder, rulings along $y$
341	$x^2+4y^2-4z^2-6x-16y-16z+5=0$	$\frac{(x-3)^2}{4}+(y-2)^2-(z+2)^2=1$ : one sheet, centre $(3,2,-2)$
343	$x^2+\frac{y^2}{4}-\frac{z^2}{3}+6x+9=0$	$(x+3)^2+\frac{y^2}{4}-\frac{z^2}{3}=0$ : cone, vertex $(-3,0,0)$
345	ellipsoid through $(2,0,0)$ , $(0,0,1)$ , $\left(\frac12,\sqrt{11},\frac12\right)$	$\frac{x^2}{4}+\frac{y^2}{16}+z^2=1$
347	cone $x^2-y^2-z^2=0$ and line $\frac{x-1}{2}=\frac{y+1}{3}=z$	$(1,-1,0)$ and $\left(\frac{13}{3},4,\frac53\right)$
349	equidistant from $(0,-1,0)$ and $y=1$	$x^2+z^2+4y=0$ : elliptic paraboloid
351	focus of $400z=x^2+y^2$	$(0,0,100)$
357	cylinder $9x^2+4y^2=18$ and ellipsoid $36x^2+16y^2+9z^2=144$	ellipses $\frac{x^2}{2}+\frac{y^2}{9/2}=1$ in $z=\pm2\sqrt2$

Quadric Surfaces

In space, $x^2+y^2=9$ is a cylinder, not a circle

A curve plus parallel lines through it makes a cylinder

Any curve works: find the missing variable and slide the curve along it

$z=2x^2-y$ has no missing variable, so no ruling runs along an axis

Second-degree equations in $x,y,z$ give quadric surfaces

Slicing never raises the degree

Right side 1: count the minus signs — none, one, two

A hyperboloid of one sheet is built from straight lines

Right side 0 gives a cone; a first-power variable gives a paraboloid

$x^2+y^2-z^2=c$ : one sheet, then a cone, then two sheets as $c$ goes 1, 0, −1

Name a quadric with three questions

Divide by 144 first: an ellipsoid with semi-axes 3, 4, 3

Complete the square: an elliptic paraboloid with vertex $(1,-2,0)$

Flip one sign or one power and the surface gets a new name

Complete each square: the centre moves, the name stays

Completing the square in $z$ gives a hyperboloid of one sheet centred at $(0,0,1)$

Some second-degree equations fall apart into planes

A trace is the curve a plane parallel to a coordinate plane cuts out

Slice any quadric: the trace is a conic, or a point, lines or nothing

Three elliptical traces frame the ellipsoid $\frac{x^2}{2^2}+\frac{y^2}{3^2}+\frac{z^2}{5^2}=1$

Slices away from the centre shrink: at $z=3$ the trace is $\frac{x^2}{4}+\frac{y^2}{9}=\frac{16}{25}$

Ellipses across, parabolas along: $x^2+\frac{y^2}{4}=\frac z5$ is an elliptic paraboloid

The reflector $\frac{x^2}{100}+\frac{y^2}{100}=\frac z4$ focuses energy at $(0,0,6.25)$

One minus sign: ellipses across, hyperbolas along

A quadric is the conics it is made of

Common mistakes

Practice and answers