Chapter 5

Planes and Standard Surfaces

We have spent the first four chapters building the geometry of points and vectors. We can now put those tools to work describing subsets of space. Our first examples are lines and planes, which are given by linear equations involving only the coordinate variables, constants, scalar multiplication, and addition.

We will then return to circles and spheres, and finish with a collection of standard surfaces described by quadratic equations. The goal is not to memorize a catalog of pictures, but to learn how an equation reveals the shape it describes.

5.1Normal Vectors and Planes

We begin with a familiar equation in the plane.

Definition 5.1 (Lines in $\RR^2$). A line in the plane is given by an equation of the form

𝑎 𝑥 + 𝑏 𝑦 = 𝑑,

where $𝑎$ and $𝑏$ are not both zero. When $𝑏 \neq 0$ , this can be written in slope-intercept form as

𝑦 = - 𝑎 𝑏 𝑥 + 𝑑 𝑏 .

The coefficients do more than determine the slope. Taken together as the vector $⃗ 𝑛 = ⟨ 𝑎, 𝑏 ⟩$ , they point perpendicular to the line.

Theorem 5.2 (Normal Direction to a Line). The line $𝑎 𝑥 + 𝑏 𝑦 = 𝑑$ is orthogonal to the coefficient vector

⃗ 𝑛 = ⟨ 𝑎, 𝑏 ⟩ .

How can we confirm this? Choose two points $𝑝 = (𝑝 𝑥, 𝑝 𝑦)$ and $𝑞 = (𝑞 𝑥, 𝑞 𝑦)$ on the line. Their displacement vector is $𝑞 - 𝑝 = ⟨ 𝑞 𝑥 - 𝑝 𝑥, 𝑞 𝑦 - 𝑝 𝑦 ⟩$ , and

⃗ 𝑛 \cdot (𝑞 - 𝑝) = 𝑎 (𝑞 𝑥 - 𝑝 𝑥) + 𝑏 (𝑞 𝑦 - 𝑝 𝑦) = (𝑎 𝑞 𝑥 + 𝑏 𝑞 𝑦) - (𝑎 𝑝 𝑥 + 𝑏 𝑝 𝑦) = 𝑑 - 𝑑 = 0 .

Thus every displacement along the line is orthogonal to $⃗ 𝑛$ .

Finding a direction along a line by first finding two points is tedious, so luckily we do not need to do this every time! Since the line is orthogonal to $⟨ 𝑎, 𝑏 ⟩$ , it points in the direction $⟨ - 𝑏, 𝑎 ⟩$ (which is itself orthogonal to $⟨ 𝑎, 𝑏 ⟩$ ).

The same idea works one dimension higher. A plane is determined by a point it passes through and a direction perpendicular to it.

Definition 5.3 (Point-Normal Form of a Plane). Let $𝑝 = (𝑥 0, 𝑦 0, 𝑧 0)$ be a point and let $⃗ 𝑛 = ⟨ 𝑎, 𝑏, 𝑐 ⟩$ be a nonzero normal vector. A variable point $𝑋 = (𝑥, 𝑦, 𝑧)$ lies in the plane through $𝑝$ perpendicular to $⃗ 𝑛$ exactly when the displacement $𝑋 - 𝑝$ is orthogonal to $⃗ 𝑛$ . Thus the plane is determined by

⃗ 𝑛 \cdot (𝑋 - 𝑝) = 0,

or, in coordinates,

𝑎 (𝑥 - 𝑥 0) + 𝑏 (𝑦 - 𝑦 0) + 𝑐 (𝑧 - 𝑧 0) = 0 .

Distributing and collecting all the constants on the right-hand side, we see that planes are given by simple linear equations in three variables.

Definition 5.4 (Scalar Equation of a Plane). A plane in $ℝ 3$ is specified by a scalar equation of the form

𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 = 𝑑,

where $𝑎, 𝑏$ , and $𝑐$ are not all zero. The plane is orthogonal to the normal vector

⃗ 𝑛 = ⟨ 𝑎, 𝑏, 𝑐 ⟩ .

The figure below places the two-dimensional and three-dimensional versions side by side. Change the coefficients by dragging the normal vectors, and change the constant $𝑑$ to see what each part of the equation controls.

Figure 5.1 The coefficient vector stays perpendicular to the line or plane. Changing

𝑑

slides the line or plane along this normal direction without changing its orientation.

For example, the plane through $𝑝 = (1, 2, - 1)$ with normal vector $⃗ 𝑛 = ⟨ 2, - 1, 3 ⟩$ has point-normal equation

2 (𝑥 - 1) - (𝑦 - 2) + 3 (𝑧 + 1) = 0 .

Expanding gives the scalar equation

2 𝑥 - 𝑦 + 3 𝑧 = - 3 .

5.1.1A Plane Through Three Points

Three noncollinear points $𝑝, 𝑞$ , and $𝑟$ determine a plane. The previous three chapters tell us how to find its equation.

First, subtract points to find two displacement vectors lying in the plane:

⃗ 𝑢 = 𝑞 - 𝑝, ⃗ 𝑣 = 𝑟 - 𝑝 .

Their cross product

⃗ 𝑛 = ⃗ 𝑢 \times ⃗ 𝑣

is orthogonal to both directions and is therefore normal to the plane. Finally, use any one of the three points in point-normal form.

Theorem 5.5 (Plane Through Three Points). If $𝑝, 𝑞$ , and $𝑟$ are noncollinear points, define

⃗ 𝑢 = 𝑞 - 𝑝, ⃗ 𝑣 = 𝑟 - 𝑝, ⃗ 𝑛 = ⃗ 𝑢 \times ⃗ 𝑣 .

Then the plane through the three points is

⃗ 𝑛 \cdot (𝑋 - 𝑝) = 0 .

Figure 5.2 The displacements

𝑞 - 𝑝

and

𝑟 - 𝑝

lie in the plane, so their cross product is normal to it. If the three points become collinear, the cross product becomes zero and no unique plane is determined.

For a concrete example, take

𝑝 = (1, 2, 0), 𝑞 = (0, 5, 0), 𝑟 = (2, 1, 1) .

Then

⃗ 𝑢 = 𝑞 - 𝑝 = ⟨ - 1, 3, 0 ⟩, ⃗ 𝑣 = 𝑟 - 𝑝 = ⟨ 1, - 1, 1 ⟩,

⃗ 𝑛 = ⃗ 𝑢 \times ⃗ 𝑣 = ⟨ 3, 1, - 2 ⟩ .

Using $𝑝$ in point-normal form gives

3 (𝑥 - 1) + (𝑦 - 2) - 2 𝑧 = 0,

3 𝑥 + 𝑦 - 2 𝑧 = 5 .

5.2Fixed-Distance Sets and Cylinders

In Chapter 1, we defined circles and spheres as sets of points a fixed distance from a center. We can now record their equations for an arbitrary center and radius.

Definition 5.6 (Circle). The circle $𝐶$ of radius $𝑅$ centered at a point $𝑝$ in the plane is the set of all points which lie at distance $𝑅$ from $𝑝$ :

𝐶 = {𝑞 \in ℝ 2 ∣ d i s t (𝑝, 𝑞) = 𝑅} .

If $𝑝 = (𝑝 𝑥, 𝑝 𝑦)$ , its equation is

(𝑥 - 𝑝 𝑥) 2 + (𝑦 - 𝑝 𝑦) 2 = 𝑅 2 .

Definition 5.7 (Sphere). The sphere $𝑆$ of radius $𝑅$ centered at $𝑝 \in ℝ 3$ is the set of all points in three-dimensional space which lie at distance $𝑅$ from $𝑝$ :

𝑆 = {𝑞 \in ℝ 3 ∣ d i s t (𝑝, 𝑞) = 𝑅} .

If $𝑝 = (𝑝 𝑥, 𝑝 𝑦, 𝑝 𝑧)$ , its equation is

(𝑥 - 𝑝 𝑥) 2 + (𝑦 - 𝑝 𝑦) 2 + (𝑧 - 𝑝 𝑧) 2 = 𝑅 2 .

Now consider the equation

𝑥 2 + 𝑦 2 = 1 .

In $ℝ 2$ , this is a circle. But what shape does the same equation determine in $ℝ 3$ ? Since $𝑧$ does not appear, it is free to take any value. For every fixed $𝑧$ , the cross-section is another copy of the same circle, so in space the equation describes a circular cylinder.

Figure 5.3 An equation in

𝑥

and

𝑦

describes a curve in the plane. Read in space, the same equation says nothing about

𝑧

, so the curve is stacked along the

𝑧

-direction to make a cylinder.

This is the missing-variable principle: an equation which does not involve one coordinate extends unchanged in that coordinate direction.

For example,

𝑦 2 + 𝑧 2 = 4

describes a circular cylinder of radius $2$ around the $𝑥$ -axis because $𝑥$ is missing. The equation

𝑦 = 𝑥 2

describes a parabolic cylinder extending in the $𝑧$ -direction.

The same principle produces several families of cylinders:

| Equation in $ℝ 3$ | Surface | Free direction | |---|---|---| | $𝑥 2 𝑎 2 + 𝑦 2 𝑏 2 = 1$ | elliptic cylinder | $𝑧$ | | $𝑦 = 𝑥 2 𝑎 2$ | parabolic cylinder | $𝑧$ | | $𝑥 2 𝑎 2 - 𝑦 2 𝑏 2 = 1$ | hyperbolic cylinder | $𝑧$ |

Permuting $𝑥, 𝑦$ , and $𝑧$ changes the direction in which the cylinder extends.

5.3Standard Quadric Surfaces

The remaining standard surfaces are quadrics: surfaces described by second-degree equations in $𝑥, 𝑦$ , and $𝑧$ . Their three-dimensional pictures can be difficult to remember, but their two-dimensional cross-sections are familiar curves. In the standard forms below, $𝑎, 𝑏$ , and $𝑐$ are positive.

A trace of a surface is its intersection with a plane such as $𝑥 = 𝑘$ , $𝑦 = 𝑘$ , or $𝑧 = 𝑘$ . To identify an unfamiliar surface, fix one variable in its equation and study the curve left behind. Several well-chosen traces usually reveal the entire shape.

5.3.1Ellipsoids

Other shapes that will be useful are ellipsoids, which are squashed or stretched spheres.

Definition 5.8 (Ellipsoid). An ellipsoid is given by the equation

𝑥 2 𝑎 2 + 𝑦 2 𝑏 2 + 𝑧 2 𝑐 2 = 1 .

It meets the coordinate axes at $𝑥 = \pm 𝑎$ , $𝑦 = \pm 𝑏$ , and $𝑧 = \pm 𝑐$ .

Fixing $𝑧 = 𝑘$ leaves

𝑥 2 𝑎 2 + 𝑦 2 𝑏 2 = 1 - 𝑘 2 𝑐 2,

an ellipse when $| 𝑘 | < 𝑐$ , a point when $| 𝑘 | = 𝑐$ , and no trace at all when $| 𝑘 | > 𝑐$ . This tells us the ellipsoid is bounded and closes up at its top and bottom.

5.3.2Paraboloids

Definition 5.9 (Elliptic Paraboloid). An elliptic paraboloid has the form

𝑧 = 𝑥 2 𝑎 2 + 𝑦 2 𝑏 2 .

Its horizontal traces are ellipses, while its vertical traces are upward-facing parabolas. When $𝑎 = 𝑏$ , these ellipses are circles.

Definition 5.10 (Hyperbolic Paraboloid). A saddle-shaped surface, or hyperbolic paraboloid, has the form

𝑧 = 𝑥 2 𝑎 2 - 𝑦 2 𝑏 2 .

It has an upward-facing parabolic trace in the $𝑥 𝑧$ -plane and a downward-facing parabolic trace in the $𝑦 𝑧$ -plane. Its horizontal traces are hyperbolas, except at $𝑧 = 0$ , where the trace is a pair of crossing lines.

5.3.3Cones and Hyperboloids

The other recurring quadrics can be recognized by the signs in their equations and by how their traces change.

| Surface | Standard equation | What to notice | |---|---|---| | Elliptic cone | $𝑥 2 𝑎 2 + 𝑦 2 𝑏 2 = 𝑧 2 𝑐 2$ | Two halves meet at the origin; horizontal traces are ellipses whose size grows with $| 𝑧 |$ . | | Hyperboloid of one sheet | $𝑥 2 𝑎 2 + 𝑦 2 𝑏 2 - 𝑧 2 𝑐 2 = 1$ | One connected surface with an elliptical waist at $𝑧 = 0$ . | | Hyperboloid of two sheets | $𝑧 2 𝑐 2 - 𝑥 2 𝑎 2 - 𝑦 2 𝑏 2 = 1$ | Two disconnected pieces; there are no points when $| 𝑧 | < 𝑐$ . |

For a one-sheeted hyperboloid, the variable with the negative squared term gives the axis of the surface. For a two-sheeted hyperboloid, the variable with the positive squared term gives the axis.

The interactive atlas below collects these standard forms. Do not only rotate the surfaces: move the cutting plane and try to identify each trace directly from the equation.

Figure 5.4 The standard quadrics can look similar from a particular viewing angle, but their traces distinguish them. Change the constants to see how they stretch the surface, and slide a coordinate plane through it to watch the cross-sections change.

We have now developed the basic geometry of points, vectors, planes, and standard surfaces. So far, these shapes have mostly been described as sets of points satisfying equations. In the next part of the book, we change our point of view: instead of asking which points belong to a curve, we describe a point moving along it. This is the idea of parameterization.