Quadric Surfaces: Recognition, Graphing, and Classification, Study notes of Calculus

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Section 12.2: Quadric Surfaces

Goals : 1. To recognize and write equations of quadric surfaces

2. To graph quadric surfaces by hand

Definitions : 1. A quadric surface is the three-dimensional graph of an equation that can (through

appropriate transformations, if necessary), be written in either of the following forms:

Ax^2^ + By^2 + Cz^2 + J = 0 or Ax^2^ + By^2 + Iz = 0.

2. The intersection of a surface with a plane is called a trace of the surface in the plane.

Notes : 1. There are 6 kinds of quadric surfaces. Scroll down to get an idea of what they look like.

Keep in mind that each graph shown illustrates just one of many possible orientations of

the surface.

2. The traces of quadric surfaces are conic sections (i.e. a parabola, ellipse, or hyperbola).

3. The key to graphing quadric surfaces is making use of traces in planes parallel to the

xy , xz , and yz planes.

4. The following pages are from the lecture notes of Professor Eitan Angel, University of

Colorado. Keep scrolling down (or press the Page Down key) to advance the slide show.

Calculus III – Fall 2008

Lecture – Quadric Surfaces

Eitan Angel

University of Colorado

Monday, September 8, 2008

Introduction

Last time we discussed linear equations. The graph of a linear equation ax + by + cz = d is a plane.

Introduction

Last time we discussed linear equations. The graph of a linear equation ax + by + cz = d is a plane. Now we will discuss second-degree equations (called quadric surfaces). These are the three dimensional analogues of conic sections.

Introduction

Last time we discussed linear equations. The graph of a linear equation ax + by + cz = d is a plane. Now we will discuss second-degree equations (called quadric surfaces). These are the three dimensional analogues of conic sections. To sketch the graph of a quadric surface (or any surface), it is useful to determine curves of intersection of the surface with planes parallel to the coordinate planes. These types of curves are called traces.

Definition

In Calculus II, we discuss second degree equations in x and y of the form

Ax^2 + By^2 + Cxy + Dx + Ey + F = 0, which represents a conic section. If we are allowed to rotate and translate a conic section, it can be written in the standard form

Ax^2 + By^2 + F = 0.

Definition

In Calculus II, we discuss second degree equations in x and y of the form

Ax^2 + By^2 + Cxy + Dx + Ey + F = 0, which represents a conic section. If we are allowed to rotate and translate a conic section, it can be written in the standard form

Ax^2 + By^2 + F = 0.

The most general second degree equation in x, y, and z is

Ax^2 + By^2 + Cz^2 + Dxy + Eyz + F xz + Gx + Hy + Iz + J = 0.

The graphs of such an equations are called quadric surfaces.

Definition

In Calculus II, we discuss second degree equations in x and y of the form

Ax^2 + By^2 + Cxy + Dx + Ey + F = 0, which represents a conic section. If we are allowed to rotate and translate a conic section, it can be written in the standard form

Ax^2 + By^2 + F = 0.

The most general second degree equation in x, y, and z is

Ax^2 + By^2 + Cz^2 + Dxy + Eyz + F xz + Gx + Hy + Iz + J = 0.

The graphs of such an equations are called quadric surfaces. If we are allowed to rotate and translate a quadric surface, it can be written in one of the two standard forms

Ax^2 + By^2 + Cz^2 + J = 0 or Ax^2 + By^2 + Iz = 0

Ellipsoids

The quadric surface with equation

x^2 a^2

+

y^2 b^2

+

z^2 c^2

= 1

is called an ellipsoid because its traces are ellipses. For instance, the horizontal plane with z = k (−c < k < c) intersects the surface in the ellipse x 2 a^2 +^

y^2 b^2 = 1^ −^

k^2 c^2. Let’s graph^

x^2 4 +^

y^2 16 +^

z^2 9 = 1.

Set z = 0. Then x 2 4 +^

y^2 16 = 1.

Ellipsoids

The quadric surface with equation

x^2 a^2

+

y^2 b^2

+

z^2 c^2

= 1

is called an ellipsoid because its traces are ellipses. For instance, the horizontal plane with z = k (−c < k < c) intersects the surface in the ellipse x 2 a^2 +^

y^2 b^2 = 1^ −^

k^2 c^2. Let’s graph^

x^2 4 +^

y^2 16 +^

z^2 9 = 1. Set z = 0. Then x 2 4 +^

y^2 16 = 1. Set y = 0. Then x 2 4 +^

z^2 9 = 1.

Ellipsoids

The quadric surface with equation

x^2 a^2

+

y^2 b^2

+

z^2 c^2

= 1

is called an ellipsoid because its traces are ellipses. For instance, the horizontal plane with z = k (−c < k < c) intersects the surface in the ellipse x 2 a^2 +^

y^2 b^2 = 1^ −^

k^2 c^2. Let’s graph^

x^2 4 +^

y^2 16 +^

z^2 9 = 1.

Set z = 0. Then x 2 4 +^

y^2 16 = 1. Set y = 0. Then x

2 4 +^

z^2 9 = 1. Set z = 0. Then y

2 16 +^

z^2 9 = 1.

Ellipsoids

The quadric surface with equation

x^2 a^2

+

y^2 b^2

+

z^2 c^2

= 1

is called an ellipsoid because its traces are ellipses. For instance, the horizontal plane with z = k (−c < k < c) intersects the surface in the ellipse x 2 a^2 +^

y^2 b^2 = 1^ −^

k^2 c^2. Let’s graph^

x^2 4 +^

y^2 16 +^

z^2 9 = 1.

Set z = 0. Then x 2 4 +^

y^2 16 = 1. Set y = 0. Then x 2 4 +^

z^2 9 = 1. Set z = 0. Then y

2 16 +^

z^2 9 = 1. A couple more: Let’s do y = ± b 2 = ± 2. Then x 2 4 +^

z^2 9 =^

3

Ellipsoids

The quadric surface with equation

x^2 a^2

+

y^2 b^2

+

z^2 c^2

= 1

is called an ellipsoid because its traces are ellipses. For instance, the horizontal plane with z = k (−c < k < c) intersects the surface in the ellipse x 2 a^2 +^

y^2 b^2 = 1^ −^

k^2 c^2. Let’s graph^

x^2 4 +^

y^2 16 +^

z^2 9 = 1.

Set z = 0. Then x 2 4 +^

y^2 16 = 1. Set y = 0. Then x 2 4 +^

z^2 9 = 1. Set z = 0. Then y 2 16 +^

z^2 9 = 1. A couple more: Let’s do y = ± b 2 = ± 2. Then x

2 4 +^

z^2 9 =^

3

The six intercepts are (±a, 0 , 0), (0, ±b, 0), and (0, 0 , ±c).

Hyperboloids of One Sheet

The quadric surface with equation x^2 a^2

+

y^2 b^2

−

z^2 c^2

= 1

is called a hyperboloid of one sheet. The z-axis is called the axis of this hyperboloid. Let’s graph x^2 + y^2 − z 2 4 = 1.

Set z = 0. Then x^2 + y^2 = 1.

Hyperboloids of One Sheet

The quadric surface with equation x^2 a^2

+

y^2 b^2

−

z^2 c^2

= 1

is called a hyperboloid of one sheet. The z-axis is called the axis of this hyperboloid. Let’s graph x^2 + y^2 − z 2 4 = 1.

Set z = 0. Then x^2 + y^2 = 1. Set z = ±c = ± 2. Then x^2 + y^2 = 2.

Hyperboloids of One Sheet

The quadric surface with equation x^2 a^2

+

y^2 b^2

−

z^2 c^2

= 1

is called a hyperboloid of one sheet. The z-axis is called the axis of this hyperboloid. Let’s graph x^2 + y^2 − z 2 4 = 1.

Set z = 0. Then x^2 + y^2 = 1. Set z = ±c = ± 2. Then x^2 + y^2 = 2. Set y = 0. Then x^2 − z 2 4 = 1.

Hyperboloids of One Sheet

The quadric surface with equation x^2 a^2

+

y^2 b^2

−

z^2 c^2

= 1

is called a hyperboloid of one sheet. The z-axis is called the axis of this hyperboloid. Let’s graph x^2 + y^2 − z 2 4 = 1.

Set z = 0. Then x^2 + y^2 = 1. Set z = ±c = ± 2. Then x^2 + y^2 = 2. Set y = 0. Then x^2 − z

2 4 = 1. Set x = 0. Then y^2 − z 2 4 = 1.

Hyperboloids of One Sheet

The quadric surface with equation x^2 a^2

+

y^2 b^2

−

z^2 c^2

= 1

is called a hyperboloid of one sheet. The z-axis is called the axis of this hyperboloid. Let’s graph x^2 + y^2 − z 2 4 = 1.

Set z = 0. Then x^2 + y^2 = 1. Set z = ±c = ± 2. Then x^2 + y^2 = 2. Set y = 0. Then x^2 − z

2 4 = 1. Set x = 0. Then y^2 − z 2 4 = 1. So we have a decent idea of what a hyperboloid of one sheet looks like.

Hyperboloids of Two Sheets

The quadric surface with equation

− x^2 a^2

−

y^2 b^2

+

z^2 c^2

= 1

is called a hyperboloid of two sheets. The z-axis is called the axis of this hyperboloid. Let’s graph z 2 4 −^ x

(^2) − y (^2) = 1.

Hyperboloids of Two Sheets

The quadric surface with equation

− x^2 a^2

−

y^2 b^2

+

z^2 c^2

= 1

is called a hyperboloid of two sheets. The z-axis is called the axis of this hyperboloid. Let’s graph z 2 4 −^ x

(^2) − y (^2) = 1.

Traces in the xz- and yz-planes are the hyperbolas

−x^2 + z^2 4 = 1 and − y^2 + z^2 4

= 1

If |k| > c = 2, the horizontal plane z = k intersects the surface in the ellipse

x^2 + y^2 = k^2 − 1

Cones

The quadric surface with equation

z^2 = x^2 a^2

+

y^2 b^2 is called a cone. To graph the cone z^2 = x^2 + y

2 4 , find the traces in the planes z = ± 1 : the ellipses x^2 + y

2 4 = 1.

Elliptic Paraboloid

The quadric surface with equation z c

=

x^2 a^2

+

y^2 b^2 is called an elliptic paraboloid (with axis the z-axis) because its traces in horizontal planes z = k are ellipses, whereas its traces in vertical planes x = k or y = k are parabolas, e.g., the trace in the yz-plane is the parabola z = (^) bc 2 y^2. The case where c > 0 is illustrated (in fact z = x 2 4 +^

y^2 9 ).

Elliptic Paraboloid

The quadric surface with equation z c

=

x^2 a^2

+

y^2 b^2 is called an elliptic paraboloid (with axis the z-axis) because its traces in horizontal planes z = k are ellipses, whereas its traces in vertical planes x = k or y = k are parabolas, e.g., the trace in the yz-plane is the parabola z = (^) bc 2 y^2. The case where c > 0 is illustrated (in fact z = x 2 4 +^

y^2 9 ). The trace when z = 2 is x 2 4 +^

y^2 9 = 2.

Elliptic Paraboloid

The quadric surface with equation z c

=

x^2 a^2

+

y^2 b^2 is called an elliptic paraboloid (with axis the z-axis) because its traces in horizontal planes z = k are ellipses, whereas its traces in vertical planes x = k or y = k are parabolas, e.g., the trace in the yz-plane is the parabola z = (^) bc 2 y^2. The case where c > 0 is illustrated (in fact z = x 2 4 +^

y^2 9 ). The trace when z = 2 is x 2 4 +^

y^2 9 = 2. When x = 0, z = x 2 4 and when y = 0, z = y 2

Elliptic Paraboloid

The quadric surface with equation z c

=

x^2 a^2

+

y^2 b^2 is called an elliptic paraboloid (with axis the z-axis) because its traces in horizontal planes z = k are ellipses, whereas its traces in vertical planes x = k or y = k are parabolas, e.g., the trace in the yz-plane is the parabola z = (^) bc 2 y^2. The case where c > 0 is illustrated (in fact z = x 2 4 +^

y^2 9 ). The trace when z = 2 is x 2 4 +^

y^2 9 = 2. When x = 0, z = x 2 4 and when y = 0, z = y 2

When c < 0 , the paraboloid opens downwards.