The Central Idea

A quadratic surface is a surface in three-dimensional space described by an equation containing terms of degree at most two in \(x\), \(y\), and \(z\).

A general quadratic equation in three variables can contain

\[ x^2,\quad y^2,\quad z^2,\quad xy,\quad xz,\quad yz, \]

together with linear and constant terms.

The standard forms in this lesson have axes aligned with the coordinate directions and no mixed quadratic terms.

From Conic Sections to Quadratic Surfaces

In two dimensions, second-degree equations produce conic sections.

In three dimensions, related equations produce:

Why Cross-Sections Matter

A three-dimensional surface can be understood by intersecting it with planes.

These intersections are called traces or cross-sections.

Common choices are

\[ x=c,\qquad y=c,\qquad z=c. \]

The resulting two-dimensional equations are often conic sections.

Coordinate Traces

Three especially useful traces occur in the coordinate planes:

These traces provide an initial geometric skeleton of the surface.

Ellipsoid

The standard ellipsoid centered at the origin is

\[ \boxed{ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1 }. \]

Here \(a,b,c>0\). The surface is closed and bounded.

Intercepts of an Ellipsoid

Setting two coordinates equal to zero gives the six principal intercepts:

\[ (\pm a,0,0),\qquad (0,\pm b,0),\qquad (0,0,\pm c). \]

The quantities \(a\), \(b\), and \(c\) are the semiaxis lengths.

Ellipsoid Traces

Setting

\[ z=0 \]

gives

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1, \]

an ellipse.

The other coordinate-plane traces are also ellipses.

Sphere as a Special Ellipsoid

If

\[ a=b=c=R, \]

then

\[ \boxed{x^2+y^2+z^2=R^2}. \]

The sphere is the fully symmetric special case of the ellipsoid.

Translated Ellipsoid

An ellipsoid centered at

\[ (h,k,\ell) \]

has equation

\[ \boxed{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} + \frac{(z-\ell)^2}{c^2} =1 }. \]

Elliptic Paraboloid

A standard elliptic paraboloid opening in the positive \(z\)-direction is

\[ \boxed{ z= \frac{x^2}{a^2} + \frac{y^2}{b^2} }. \]

It has one vertex at the origin and extends upward without bound.

Horizontal Traces of an Elliptic Paraboloid

Set

\[ z=c. \]

Then

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} =c. \]

If \(c>0\), the trace is an ellipse. If \(c=0\), it is the vertex. If \(c<0\), there are no real points.

Vertical Traces of an Elliptic Paraboloid

Setting

\[ y=0 \]

gives

\[ z=\frac{x^2}{a^2}, \]

a parabola in the \(xz\)-plane.

Setting \(x=0\) gives a parabola in the \(yz\)-plane.

Circular Paraboloid

If

\[ a=b, \]

the horizontal traces are circles.

For example,

\[ \boxed{z=x^2+y^2} \]

is a paraboloid of revolution about the \(z\)-axis.

Downward-Opening Elliptic Paraboloid

The equation

\[ \boxed{ z= - \frac{x^2}{a^2} - \frac{y^2}{b^2} } \]

opens downward.

Hyperbolic Paraboloid

A standard hyperbolic paraboloid is

\[ \boxed{ z= \frac{x^2}{a^2} - \frac{y^2}{b^2} }. \]

It is often called a saddle surface.

Why It Is Called a Saddle

Along

\[ y=0, \]

we obtain

\[ z=\frac{x^2}{a^2}, \]

which opens upward.

Along

\[ x=0, \]

we obtain

\[ z=-\frac{y^2}{b^2}, \]

which opens downward.

Horizontal Traces of a Hyperbolic Paraboloid

Set

\[ z=c. \]

Then

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} =c. \]

For \(c\neq0\), the trace is a hyperbola.

At \(c=0\), the trace becomes two intersecting lines.

Hyperboloid of One Sheet

A standard hyperboloid of one sheet is

\[ \boxed{ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} =1 }. \]

It is connected and extends infinitely in both directions along its principal axis.

The Negative Term Identifies the Axis

In

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

the negative term is the \(z^2\)-term.

The principal axis is therefore the \(z\)-axis.

Horizontal Traces of a One-Sheet Hyperboloid

Set

\[ z=k. \]

Then

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1+\frac{k^2}{c^2}. \]

Every horizontal trace is an ellipse.

Vertical Traces of a One-Sheet Hyperboloid

Set

\[ y=0. \]

Then

\[ \frac{x^2}{a^2} - \frac{z^2}{c^2} =1, \]

which is a hyperbola.

Hyperboloid of Two Sheets

A standard hyperboloid of two sheets is

\[ \boxed{ \frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} =1 }. \]

It has two disconnected components.

The Positive Term Identifies the Axis

A two-sheet hyperboloid has exactly one positive squared term when normalized to \(1\).

In

\[ \frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} =1, \]

the sheets lie along the \(z\)-axis.

Why There Is a Gap Between the Sheets

Rearranging gives

\[ \frac{z^2}{c^2} = 1+ \frac{x^2}{a^2} + \frac{y^2}{b^2}. \]

Therefore,

\[ |z|\ge c. \]

No surface points occur for

\[ -c

Horizontal Traces of a Two-Sheet Hyperboloid

Set

\[ z=k. \]

Then

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{k^2}{c^2}-1. \]

Real elliptical traces occur only when

\[ |k|>c. \]

Elliptic Cone

A standard elliptic cone is

\[ \boxed{ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} =0 }. \]

It has two nappes meeting at the origin.

Cone as a Homogeneous Quadratic Surface

Every term in the cone equation has degree two.

If \((x,y,z)\) lies on the cone, then

\[ (\lambda x,\lambda y,\lambda z) \]

also lies on the cone for every real \(\lambda\).

This scaling property explains the straight generating lines through the vertex.

Horizontal Traces of an Elliptic Cone

Set

\[ z=k. \]

Then

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{k^2}{c^2}. \]

For \(k\neq0\), the trace is an ellipse.

At \(k=0\), it collapses to the vertex.

Circular Cone

If \(a=b\), horizontal traces become circles.

For example,

\[ \boxed{x^2+y^2=z^2} \]

is a circular double cone around the \(z\)-axis.

Quadratic Cylinders

If an equation is independent of one coordinate, the surface extends parallel to that coordinate axis.

For example,

\[ \boxed{ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1 } \]

contains no \(z\).

Therefore, the ellipse extends unchanged along the \(z\)-direction.

Elliptic Cylinder

The equation

\[ \boxed{ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1 } \]

describes an elliptic cylinder parallel to the \(z\)-axis.

If \(a=b=R\), it becomes

\[ x^2+y^2=R^2. \]

Parabolic Cylinder

The equation

\[ \boxed{y=x^2} \]

contains no \(z\).

Therefore, the parabola extends unchanged parallel to the \(z\)-axis.

Hyperbolic Cylinder

The equation

\[ \boxed{ \frac{x^2}{a^2} - \frac{y^2}{b^2} =1 } \]

contains no \(z\).

The corresponding hyperbola extends parallel to the \(z\)-axis.

How to Recognize a Cylinder Quickly

If one coordinate does not appear at all, first suspect a cylinder.

The missing variable identifies the direction in which the surface extends.

For example,

\[ x^2+z^2=9 \]

is a circular cylinder parallel to the \(y\)-axis.

Comparing the Major Standard Forms

A useful recognition strategy is to inspect:

  • how many variables are squared;
  • which signs appear;
  • whether the equation equals \(1\), \(0\), or an isolated variable;
  • whether one coordinate is missing.

All Positive Squared Terms

The pattern

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1 \]

is the standard form of an ellipsoid.

Two Positive and One Negative

The sign pattern

\[ +,+,- \]

with the equation equal to \(1\), gives a one-sheet hyperboloid.

One Positive and Two Negative

The sign pattern

\[ +,-,- \]

with the equation equal to \(1\), gives a two-sheet hyperboloid.

The positive term identifies the principal axis.

Mixed Signs Equal to Zero

If the quadratic form has mixed signs and equals zero, the standard result is a cone.

For example,

\[ x^2+y^2-z^2=0. \]

One Variable Isolated Linearly

Equations such as

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

and

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

produce paraboloids.

Translation of Quadratic Surfaces

Replacing coordinates by shifted expressions moves the surface.

For example,

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

is centered at

\[ \boxed{(h,k,\ell)}. \]

Example: Translated Paraboloid

Consider

\[ z-3=(x-1)^2+2(y+4)^2. \]

The vertex is

\[ \boxed{(1,-4,3)}. \]

The surface is an elliptic paraboloid opening in the positive \(z\)-direction.

Completing Squares in Three Variables

Expanded quadratic equations can often be converted to standard form by grouping coordinate terms and completing squares.

Consider

\[ x^2+y^2+z^2-4x+6y-2z-2=0. \]

Group:

\[ (x^2-4x)+(y^2+6y)+(z^2-2z)=2. \]

Example: Completing Squares for a Sphere

Complete the squares:

\[ (x-2)^2-4 + (y+3)^2-9 + (z-1)^2-1 =2. \]

Therefore,

\[ \boxed{ (x-2)^2+(y+3)^2+(z-1)^2=16 }. \]

The surface is a sphere centered at

\[ (2,-3,1) \]

with radius \(4\).

Symmetry of Standard Quadratic Surfaces

Squared coordinates create sign symmetry.

If an equation contains only

\[ x^2,\qquad y^2,\qquad z^2, \]

then replacing any coordinate by its negative leaves the equation unchanged.

Rotational Symmetry as a Special Case

If two perpendicular coordinates appear with equal coefficients, the surface may have rotational symmetry about the remaining axis.

For example,

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

is rotationally symmetric about the \(z\)-axis.

Ellipsoid Versus Elliptic Paraboloid

The ellipsoid

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1 \]

is closed and bounded.

The elliptic paraboloid

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

has one vertex and extends without bound in one direction.

One-Sheet Versus Two-Sheet Hyperboloid

One sheet:

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

Two sheets:

\[ \frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} =1. \]

The first is connected. The second has two disconnected components.

Cone Versus One-Sheet Hyperboloid

Compare

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} =0 \]

with

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

The first is a cone. The second is a one-sheet hyperboloid.

Quadratic Surfaces and Level Surfaces

A quadratic surface can often be viewed as a level surface

\[ F(x,y,z)=c. \]

For example,

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

is a level surface of

\[ F(x,y,z)=x^2+y^2+z^2. \]

Quadratic Surfaces and Matrices

A quadratic expression can be represented using a symmetric matrix.

For example,

\[ ax^2+by^2+cz^2 \]

can be written

\[ \boxed{\mathbf x^T A\mathbf x} \]

with

\[ \mathbf x= \begin{pmatrix} x\\y\\z \end{pmatrix}, \qquad A= \begin{pmatrix} a&0&0\\ 0&b&0\\ 0&0&c \end{pmatrix}. \]

Why Eigenvalues Matter

For a rotated quadratic surface, the quadratic-form matrix may not initially be diagonal.

A suitable orthogonal change of coordinates can diagonalize a real symmetric matrix.

Geometrically, this aligns the coordinates with the principal axes of the surface.

Mixed Terms Indicate Possible Rotation

Terms such as

\[ xy,\qquad xz,\qquad yz \]

often indicate that the principal axes are not aligned with the coordinate axes.

Removing these terms generally requires a rotation of coordinates.

Degenerate Quadratic Surfaces

Not every quadratic equation describes one of the nondegenerate standard surfaces.

Degenerate cases can include:

  • a point;
  • a line;
  • a plane;
  • two planes;
  • no real points.

Example: A Single Point

Consider

\[ x^2+y^2+z^2=0. \]

The only real solution is

\[ \boxed{(0,0,0)}. \]

Example: No Real Points

Consider

\[ x^2+y^2+z^2=-1. \]

The left side is always nonnegative for real coordinates.

Therefore, the equation has

\[ \boxed{\text{no real points}}. \]

Example: Two Planes

Consider

\[ x^2-y^2=0. \]

Factor:

\[ (x-y)(x+y)=0. \]

Therefore,

\[ \boxed{x=y\quad\text{or}\quad x=-y}. \]

Since \(z\) is unrestricted, these are two planes intersecting along the \(z\)-axis.

Common Error: Identifying a Surface from One Trace Only

A single cross-section may not uniquely determine a surface.

Use several traces together with the full equation.

Common Error: Forgetting the Missing Variable in a Cylinder

In three dimensions,

\[ x^2+y^2=9 \]

is not merely a circle.

Since \(z\) is unrestricted, it is a circular cylinder.

Common Error: Confusing One-Sheet and Two-Sheet Hyperboloids

One sheet:

\[ +,+,-=1. \]

Two sheets:

\[ +,-,-=1. \]

The count of positive and negative squared terms is the fastest standard distinction.

Common Error: Confusing a Cone with a Hyperboloid

Compare

\[ x^2+y^2-z^2=0 \]

and

\[ x^2+y^2-z^2=1. \]

The first is a cone. The second is a one-sheet hyperboloid.

Common Error: Assuming Every Paraboloid Is Bowl-Shaped

An elliptic paraboloid is bowl-like.

A hyperbolic paraboloid is saddle-shaped.

The difference comes from whether the squared terms have the same sign or opposite signs.

Common Error: Ignoring Translation

In

\[ \frac{(x-2)^2}{4} + \frac{(y+1)^2}{9} + \frac{(z-5)^2}{16} =1, \]

the center is

\[ \boxed{(2,-1,5)}. \]

Common Error: Ignoring Feasibility of Cross-Sections

A trace equation can have no real points for some plane positions.

For the two-sheet hyperboloid

\[ \frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} =1, \]

horizontal traces do not exist for

\[ |z|

A Reliable Procedure for Identifying a Quadratic Surface

  1. Move the equation into a convenient algebraic arrangement.
  2. Complete squares if linear terms are present.
  3. Check whether one variable is missing.
  4. Count the squared variables.
  5. Inspect the signs of the squared terms.
  6. Normalize the equation when appropriate.
  7. Identify the center, vertex, or principal axis.
  8. Compute coordinate intercepts when useful.
  9. Examine several planar traces.
  10. Check for degeneracy or absence of real points.

Example: Full Identification Workflow

Identify

\[ 4x^2+9y^2-z^2=36. \]

Divide by \(36\):

\[ \frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} =1. \]

There are two positive squared terms and one negative squared term.

Therefore, the surface is a

\[ \boxed{\text{hyperboloid of one sheet}}. \]

The negative term is the \(z^2\)-term, so the principal axis is the \(z\)-axis.

Example: Identify from Traces

Suppose a surface has:

  • elliptical traces for \(z=c>0\);
  • a single point at \(z=0\);
  • parabolic traces in vertical coordinate planes;
  • no real points for \(z<0\).

These features identify an

\[ \boxed{\text{elliptic paraboloid opening upward}}. \]

Practice 1: Ellipsoid

Identify:

\[ \frac{x^2}{9} + \frac{y^2}{16} + \frac{z^2}{25} =1. \]

Practice 2: Elliptic Paraboloid

Identify:

\[ z=4x^2+y^2. \]

Practice 3: Hyperbolic Paraboloid

Identify:

\[ z=x^2-3y^2. \]

Practice 4: One-Sheet Hyperboloid

Identify the surface and principal axis:

\[ \frac{x^2}{4} + \frac{z^2}{9} - \frac{y^2}{16} =1. \]

Practice 5: Two-Sheet Hyperboloid

Identify the surface and principal axis:

\[ \frac{x^2}{25} - \frac{y^2}{9} - \frac{z^2}{4} =1. \]

Practice 6: Cone

Identify:

\[ x^2+y^2=4z^2. \]

Practice 7: Cylinder

Identify the surface and its axis direction:

\[ \frac{x^2}{9} + \frac{z^2}{4} =1. \]

Practice 8: Cross-Section

For

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

describe the trace at

\[ z=9. \]

Practice 9: Translation

Find the center of

\[ \frac{(x+1)^2}{4} + \frac{(y-2)^2}{9} + \frac{(z+5)^2}{16} =1. \]

Practice 10: Degenerate Surface

Classify the real point set:

\[ x^2+y^2+z^2=0. \]

Practice Answers

Practice 1

All three squared terms are positive and the equation equals \(1\).

\[ \boxed{\text{ellipsoid}}. \]

Practice 2

Both squared terms have the same sign and \(z\) is isolated linearly.

\[ \boxed{\text{elliptic paraboloid opening upward}}. \]

Practice 3

The squared terms have opposite signs.

\[ \boxed{\text{hyperbolic paraboloid}}. \]

Practice 4

Two squared terms are positive and one is negative.

\[ \boxed{\text{hyperboloid of one sheet}}. \]

The negative term is \(y^2\), so the principal axis is the

\[ \boxed{y\text{-axis}}. \]

Practice 5

One squared term is positive and two are negative.

\[ \boxed{\text{hyperboloid of two sheets}}. \]

The positive term is \(x^2\), so the principal axis is the

\[ \boxed{x\text{-axis}}. \]

Practice 6

Rewrite:

\[ x^2+y^2-4z^2=0. \]

Mixed signs equal to zero give a

\[ \boxed{\text{circular double cone}}. \]

Practice 7

The equation contains no \(y\), so it extends parallel to the \(y\)-axis.

\[ \boxed{\text{elliptic cylinder parallel to the }y\text{-axis}}. \]

Practice 8

Set \(z=9\):

\[ x^2+y^2=9. \]

The trace is a

\[ \boxed{\text{circle of radius }3}. \]

Practice 9

The center is

\[ \boxed{(-1,2,-5)}. \]

Practice 10

The sum of three squares is zero only when all three coordinates are zero.

\[ \boxed{\text{the single point }(0,0,0)}. \]

What to Remember

Quadratic surfaces are the three-dimensional counterparts of conic sections.

  • Ellipsoid: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1. \]
  • Elliptic paraboloid: \[ z= \frac{x^2}{a^2} + \frac{y^2}{b^2}. \]
  • Hyperbolic paraboloid: \[ z= \frac{x^2}{a^2} - \frac{y^2}{b^2}. \]
  • One-sheet hyperboloid: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} =1. \]
  • Two-sheet hyperboloid: \[ \frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} =1. \]
  • Elliptic cone: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} =0. \]

The most useful recognition tools are: signs, missing variables, translations, intercepts, and planar traces.

Where We Go Next

Quadratic surfaces show how three-dimensional form can be understood through families of planar slices. The next lesson generalizes that idea beyond quadratic equations.

In Level Curves and Level Surfaces, we will study sets of points satisfying

\[ F(x,y)=c \]

and

\[ F(x,y,z)=c, \]

using contour geometry to visualize functions and spatial structure.