Distance, Spheres, Cylinders, and Quadric Surfaces

students, in this lesson you will learn how to describe and recognize common geometric shapes in three dimensions using coordinates and equations ✨ These ideas are the foundation for many topics in multivariable calculus, because they help us understand shapes, surfaces, and motion in space.

What you will learn

By the end of this lesson, you should be able to:

find the distance between two points in three-dimensional space,
recognize and write equations of spheres and cylinders,
identify several standard quadric surfaces,
connect algebraic equations to geometric shapes in space,
use examples to explain how these surfaces appear in real life.

A big idea in multivariable calculus is that equations in $x$, $y$, and $z$ describe surfaces in space. Instead of just graphing lines on a plane, we now study 3D shapes like balls, tunnels, bowls, and saddles. That is why these formulas matter so much 🌍

Distance in three dimensions

In two dimensions, the distance between $(x_1,y_1)$ and $(x_2,y_2)$ comes from the Pythagorean Theorem. In three dimensions, we use the same idea, but now there are three directions: left-right, forward-back, and up-down.

If two points are $P_1=(x_1,y_1,z_1)$ and $P_2=(x_2,y_2,z_2)$, then the distance between them is

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$

This formula is really just the Pythagorean Theorem applied twice. Imagine walking from one point to another in a room. You may move in the $x$ direction, then the $y$ direction, then the $z$ direction. The total straight-line distance is the diagonal through space.

Example

Find the distance between $A=(1,2,3)$ and $B=(4,6,3)$.

Use the distance formula:

$$d=\sqrt{(4-1)^2+(6-2)^2+(3-3)^2}$$

$$d=\sqrt{3^2+4^2+0^2}$$

$$d=\sqrt{25}=5$$

Notice something important: the $z$-coordinates are the same, so the points lie in a horizontal plane. The distance becomes the same as the 2D $3$-$4$-$5$ triangle. That is a good example of how three-dimensional geometry still builds on familiar ideas.

Spheres: all points the same distance from a center

A sphere is the 3D version of a circle. A circle is all points the same distance from a center in a plane. A sphere is all points the same distance from a center in space.

If the center is $(h,k,l)$ and the radius is $r$, then the equation of the sphere is

$$ (x-h)^2+(y-k)^2+(z-l)^2=r^2 $$

This equation comes directly from the distance formula. A point $(x,y,z)$ lies on the sphere exactly when its distance from the center equals $r$.

Example

The sphere

$$ (x-2)^2+(y+1)^2+(z-4)^2=9 $$

has center $(2,-1,4)$ and radius $3$, because $9=3^2$.

A sphere can model many real objects, such as a ball, a bubble, or a planet. In physics and engineering, spheres appear when something spreads evenly in all directions from a center.

How to recognize a sphere

If the equation has $x^2$, $y^2$, and $z^2$ with the same coefficient, and no mixed terms like $xy$, $xz$, or $yz$, then it may be a sphere after completing the square. For example:

$$x^2+y^2+z^2-6x+4y+2z=11$$

Group terms and complete the square:

$$ (x^2-6x)+(y^2+4y)+(z^2+2z)=11 $$

$$ (x-3)^2-9+(y+2)^2-4+(z+1)^2-1=11 $$

$$ (x-3)^2+(y+2)^2+(z+1)^2=25 $$

So the sphere has center $(3,-2,-1)$ and radius $5$.

Cylinders: a surface made by extending a curve

A cylinder is a surface formed when a curve is extended in a straight direction. The curve is called the directrix, and the moving line is parallel to a fixed direction.

A simple example is a circular cylinder. Its equation might be

$$x^2+y^2=4$$

This is not a sphere because there is no $z$ in the equation. That means the shape does not change as $z$ changes. The circle $x^2+y^2=4$ in the $xy$-plane is stretched endlessly up and down along the $z$-axis.

Example

The equation

$$ (x-1)^2+(y+2)^2=16 $$

describes a cylinder centered on the line parallel to the $z$-axis passing through $(1,-2)$ in the $xy$-plane. Its radius is $4$.

Key idea

If one variable is missing from the equation, the surface is often a cylinder parallel to the axis of the missing variable. For example:

$x^2+y^2=9$ is a cylinder along the $z$-axis,
$x^2+z^2=9$ is a cylinder along the $y$-axis,
$y^2+z^2=9$ is a cylinder along the $x$-axis.

This is useful because it helps you picture 3D shapes by starting with a familiar 2D curve and extending it through space.

Quadric surfaces: the major families of second-degree surfaces

Quadric surfaces are surfaces described by second-degree equations in $x$, $y$, and $z$. They are the 3D analogs of conic sections in 2D, and they include ellipsoids, hyperboloids, paraboloids, and more.

A general second-degree equation may look like

$$Ax^2+By^2+Cz^2+Dx+Ey+Fz+G=0$$

When the equation is put into a standard form, you can identify the surface more easily. The most common quadric surfaces are listed below.

1. Ellipsoid

An ellipsoid is like a stretched sphere. Its standard form is

$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}+\frac{(z-l)^2}{c^2}=1$$

If $a=b=c$, then the ellipsoid is actually a sphere.

Example:

$$\frac{x^2}{9}+\frac{y^2}{4}+\frac{z^2}{16}=1$$

This is centered at the origin. It stretches farthest along the $z$-axis because $16$ is the largest denominator.

2. Hyperboloid of one sheet

A hyperboloid of one sheet has the form

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$$

or any version with one negative term and two positive terms. It looks like a connected hourglass-like surface.

Example:

$$x^2+y^2-z^2=1$$

This surface gets wider as $|z|$ increases.

3. Hyperboloid of two sheets

A hyperboloid of two sheets has the form

$$-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$

or any version with one positive term and two negative terms. It has two separate pieces.

Example:

$$z^2-x^2-y^2=1$$

This means $|z|\ge 1$, so the surface does not exist near the center in the same way a one-sheet hyperboloid does.

4. Elliptic paraboloid

An elliptic paraboloid has one squared term on one side and two squared terms on the other side, often written as

$$z=\frac{x^2}{a^2}+\frac{y^2}{b^2}$$

It looks like a bowl. If the signs are reversed, it opens downward.

Example:

$$z=x^2+4y^2$$

This surface opens upward, and cross-sections for fixed $z$ are ellipses.

5. Hyperbolic paraboloid

A hyperbolic paraboloid is a saddle-shaped surface. A common form is

$$z=\frac{x^2}{a^2}-\frac{y^2}{b^2}$$

Example:

$$z=x^2-y^2$$

This surface curves upward in one direction and downward in the other. A saddle on a horse is a good real-world visual 🐎

How to identify these surfaces

A smart way to study quadric surfaces is to look at the signs and the shape of the equation.

If all three squared variables are present and equal to a constant, the surface may be an ellipsoid or sphere.
If one variable is missing, the surface is often a cylinder.
If only one variable is not squared, the surface is likely a paraboloid.
If the squared terms have mixed signs and equal $1$ on one side, the surface may be a hyperboloid.

Cross-sections are also very helpful. A cross-section is the intersection of a surface with a plane such as $x=c$, $y=c$, or $z=c$. By slicing a 3D surface, you can better understand its shape, just like cutting through a fruit to see the inside.

For example, for the surface

$$z=x^2+y^2$$

if $z=c$, then

$$x^2+y^2=c$$

which is a circle when $c>0$. This helps show why the surface is a bowl.

Conclusion

students, distance, spheres, cylinders, and quadric surfaces are all ways of turning algebra into geometry in three dimensions. The distance formula extends the Pythagorean Theorem into space. A sphere is defined by equal distance from a center. A cylinder appears when a curve is extended in one direction. Quadric surfaces come from second-degree equations and include many important shapes such as ellipsoids, paraboloids, and hyperboloids.

These ideas are more than just formulas. They help you visualize space, interpret equations, and prepare for later multivariable calculus topics like level surfaces, partial derivatives, and integrals over regions in space.

Study Notes

The distance between $P_1=(x_1,y_1,z_1)$ and $P_2=(x_2,y_2,z_2)$ is

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$

A sphere with center $(h,k,l)$ and radius $r$ has equation

$$ (x-h)^2+(y-k)^2+(z-l)^2=r^2 $$

A cylinder is formed by extending a curve in a straight direction; a missing variable often tells you the cylinder’s axis.
Common cylinder examples:
$x^2+y^2=r^2$ along the $z$-axis,
$x^2+z^2=r^2$ along the $y$-axis,
$y^2+z^2=r^2$ along the $x$-axis.
Quadric surfaces are second-degree surfaces in $x$, $y$, and $z$.
Ellipsoid standard form:

$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}+\frac{(z-l)^2}{c^2}=1$$

Hyperboloid of one sheet has one negative squared term and two positive squared terms.
Hyperboloid of two sheets has two negative squared terms and one positive squared term.
Elliptic paraboloid example:

$$z=x^2+y^2$$

Hyperbolic paraboloid example:

$$z=x^2-y^2$$

Cross-sections help reveal the shape of a surface by slicing it with planes like $x=c$, $y=c$, or $z=c$.
These surfaces connect algebraic equations to real 3D geometry in multivariable calculus.