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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. 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.
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
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
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
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
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
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
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 Two Sheets
The quadric surface with equation
− x^2 a^2
y^2 b^2
z^2 c^2
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
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
If |k| > c = 2, the horizontal plane z = k intersects the surface in the ellipse
x^2 + y^2 = k^2 − 1
