Download original notes

PAGE 1

13.6 Quadric Surfaces (part 1)

In \(\mathbb{R}^2\) equations like \(y = x^2\) are curves.

In \(\mathbb{R}^3\) equations in terms of \(x, y, z\) are surfaces.

For example:

\[x^2 + y^2 + z^2 = 9 \text{ is a sphere}\]

\[(x - 2) + 3(y - 1) + 5(z + 4) = 0 \text{ is a plane}\]

normal vector: \(\langle 1, 3, 5 \rangle\)
point it goes thru: \((2, 1, -4)\)

Sometimes a variable (or more) is missing (e.g. \(x = 5\))

missing variable is "free"
can take on all values in its domain

PAGE 2

For example, \(z = 5\) is missing \(x, y\) so \(x, y\) can be any real number.

collection of all points \((a, b, 5)\) where \(-\infty < a < \infty, -\infty < b < \infty\)

Figure: 3D coordinate system showing a horizontal plane at z=5 with points (3,5,5) and (7,-1,5) labeled.

Likewise, \(x = 5\)

Figure: 3D coordinate system showing a vertical plane parallel to the yz-plane intersecting the x-axis at 5.

PAGE 3

Cylindrical Surfaces in 3D

The equation \( x^2 + y^2 = 1 \) (in \( \mathbb{R}^2 \) is a circle).

In 3D, \( z \) is missing, so \( -\infty < z < \infty \).

At any \( z \), we have \( x^2 + y^2 = 1 \rightarrow \) circle radius 1.

Figure: 3D coordinate axes with circles of radius 1 drawn at various z-levels: z=5, z=1, z=0, z=-3, and z=-7.

stack them →

Figure: A 3D coordinate system showing a complete vertical cylinder of radius 1 extending along the z-axis.

Next Example

\( x^2 + z^2 = 1 \) ?

PAGE 4

\( x^2 + z^2 = 1 \)

Figure: A 3D coordinate system with a cylinder of radius 1 oriented horizontally along the y-axis.

at any \( y \)

Figure: A 2D coordinate graph of the xz-plane showing a unit circle centered at the origin.

These slices we stack are called traces → intersections w/ a particular \( x, y, \) or \( z \).

intersection w/ xy-plane is called the xy-trace
" " xz-plane " " xz-trace
" " yz-plane " " yz-trace

PAGE 5

Example: Graphing a Surface

\[z = 9 - x^2\]

No \(y\), so \(-\infty < y < \infty\).

At each \(y\), we have a parabola.

Figure: A 2D plot of the parabola z = 9 - x^2 in the xz-plane, opening downwards with vertex at (0,9) and x-intercepts at -3 and 3.

Parabola in the \(xz\)-plane

Figure: A 3D coordinate system showing multiple identical parabolas stacked along the y-axis at y = -2, 0, 1, and 3.

Stacking parabolas along the \(y\)-axis

Stack them →

Figure: A 3D surface plot of a parabolic cylinder extending infinitely along the y-axis, formed by the equation z = 9 - x^2.

Parabolic \(y\)-cylinder

PAGE 6

Example: Analyzing Traces

(Pretend we didn't know this is a sphere)

Let's examine its traces for the equation:

\[x^2 + y^2 + z^2 = 16\]

\(xy\)-trace (intersection w/ \(xy\)-plane)

\(z = 0 \implies x^2 + y^2 = 16\) (circle radius 4)

\(yz\)-trace

\(x = 0 \implies y^2 + z^2 = 16\)

\(xz\)-trace

\(y = 0 \implies x^2 + z^2 = 16\)

Figure: A 3D coordinate system showing a sphere with its circular traces highlighted in the xy, yz, and xz planes.

Trace w/ \(z = k\)

\(x^2 + y^2 = 16 - k^2\)

Circle radius \(\sqrt{16 - k^2}\)

Here, \(-4 \le k \le 4\)

So, constraint on \(z\) is \([-4, 4]\)

At \(z = 3\)

\(x^2 + y^2 = 16 - 3^2 = 7\)

Circle radius \(\sqrt{7}\)

PAGE 7

Example: \(x^2 + y^2 = z^2\)

x-intercept: \(x = 0\)
y-int: \(y = 0\)
z-int: \(z = 0\)

xy-trace: \((z = 0)\) \(x^2 + y^2 = 0\) point at origin
xz-trace: \((y = 0)\) \(x^2 = z^2\) \(z = \pm x\) lines
yz-trace: \((x = 0)\) \(y^2 = z^2\) \(z = \pm y\) lines

Figure: 3D coordinate system showing intersecting lines in the xz and yz planes.

Figure: 2D xz-plane showing two intersecting lines z = x and z = -x.

Figure: 2D yz-plane showing two intersecting lines z = y and z = -y.

Figure: 3D graph of a double cone with highlighted traces in different colors.

try trace w/ \(z = k\)

\[x^2 + y^2 = k^2\]

circle radius \(k\)

Figure: 2D xy-plane showing a circle with radius 2 and labels for z = 2 and z = -2.

PAGE 8

double cone

Example: \(z = x^2 + y^2\)

xy-trace: \(x^2 + y^2 = 0\) point
xz-trace: \(z = x^2\) parabola
yz-trace: \(z = y^2\) parabola

slice at \(z = k\)

\[x^2 + y^2 = k\] circle radius \(\sqrt{k}\)

note \(z \ge 0\)

paraboloid

PAGE 9

Example: \( z = x^2 - y^2 \)

Traces

yz-trace:\( z = -y^2 \)parabola
xz-trace:\( z = x^2 \)parabola
xy-trace:\( x^2 = y^2 \)lines

Level Curves

For \( z = k \), the level curves are given by \( x^2 - y^2 = k \), which are hyperbolas.

Figure: Graph of hyperbolas opening along the x-axis for k > 0 in the xy-plane.

0 in the xy-plane." class="mb-2">

if \( k > 0 \)

Figure: Graph of hyperbolas opening along the y-axis for k < 0 in the xy-plane.

if \( k < 0 \)

same for other perspectives

3D Visualization

hyperbolic paraboloid