A Survey of Quadric Surfaces
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Section 12.6 A Survey of Quadric Surfaces Conic Sections Quadric Surfaces
Transcript of A Survey of Quadric Surfaces
Section 12.6A Survey of Quadric Surfaces
Conic SectionsQuadric Surfaces
Review: Conic Sections
In R2, conic sections are curves satisfying an equation of the form
Ax2 + Bx + Cy2 + Dy + E = 0.
Ellipse(x − h
a
)2
+
(y − k
b
)2
= 1
Hyperbola
x2 − y2 = a2
y2 − x2 = a2
Quadric Surfaces
In R3, a quadric surface is a surface satisfying an equation of the form
Ax2 + By2 + Cz2 + Dxy + Exz + Fyz + ax + by + cz + d = 0.
A quadric surface is classified by the curves obtained by intersecting itwith planes parallel to the xy , xz , and yz planes. (These intersectioncurves will always be conic sections.)
Quadric Surfaces
There are many kinds of quadric surfaces (more than there are conics):
conic cylinders; ellipsoids (including spheres); ellipticparaboloids; hyperbolic paraboloids; hyperboloids of one sheet;hyperboloids of two sheets; cones
What is important is knowing how to recognize a quadric surfacefrom its equation or its cross-sectional curves, or vice versa. (Thenames aren’t so important.)The cross-sectional curves are all conic sections (defined by degree-2polynomials).To find a cross-sectional curve, set one of the variables to a constant.
Conic Cylinder
Surfaces with an equation containing only 2 of the three variables.Cylinders can “slide” along the axis of the missing variable.
z2
9+ y2 = 1 − z2
9+ y2 = 1 y = z2
Ellipsoid
All cross-sections are ellipses.(xa
)2
+
(yb
)2
+
(zc
)2
= 1
For example, intersecting the ellipsoid(x2
)2+ y2 +
(z3
)2= 1
with the plane z = 1 gives the ellipse(x2
)2+ y2 +
(13
)2
= 1
or (x2
)2+ y2 =
89
a, b, c = radii in x , y , zdirections
Elliptic Paraboloid
Cross-sections: two parabolas, one ellipse
z =
(xa
)2
+
(yb
)2
Surface passes the Vertical Line Test(since z is a function of x and y).
When a = b, the horizontalintersections are circles.
Hyperbolic Paraboloid (Saddle Surface)Cross-sections: two parabolas, one hyperbola
z =
(xa
)2
−(
yb
)2
Plane Cross-sectional curvex = c Parabola (downward)y = c Parabola (upward)z = c Hyperbolasz = 0 Two intersecting lines
xy
z
(passes Vert. Line Test)
z
−101
yx
z
−10 1
y
x
z z
−101
yx
z
Hyperboloid of One Sheet
Cross-sections: two hyperbolas, one ellipse(xa
)2
+
(yb
)2
=
(zc
)2
+ 1
Cone
Cross-sections: two hyperbolas, one ellipse(xa
)2
+
(yb
)2
=
(zc
)2
Intersecting with z = 0 results in a single point.
Halves of the cone are called nappes.Nappes are the graphs of the functions
z = ±c√
x2
a2 +y2
b2 .
Hyperboloid of Two Sheets
Cross-sections: two hyperbolas, one ellipse
(xa
)2
+
(yb
)2
=
(zc
)2
− 1
If −c < z < c , there are no (x , y)solutions to the equation (because thenthe RHS is negative and the LHS has tobe positive).
All solutions have z ≥ c (top sheet) orz ≤ −c (bottom sheet).
Each sheet is the graph of a function.
Hyperboloids and Cones — Comparison
Hyperboloid Cone Hyperboloidof 1 sheet of 2 sheets
x2
a2 + y2
b2 = z2
c2 + 1 x2
a2 + y2
b2 = z2
c2x2
a2 + y2
b2 = z2
c2 − 1
Comparing Some Quadrics
Example 1: Sketch and identify the graphs of
(a) z = 3x2 + 4y2 (b) z = 2x2 − 5y2
Passes VLT? Yes YesIntersect with x = c Parabola (up) Parabola (down)Intersect with y = c Parabola (up) Parabola (up)Intersect with z = c Ellipse Hyperbola
(a) Elliptic paraboloid (b) Hyperbolic paraboloid
Example 2: Sketch and identify the graph of
4x2 − y2 + 9z2 + 4 = 0
Solution: Rewrite the equation as 4x2 + 9z2 = y2 − 4 or equivalently
x2 +
(3z2
)2
=(y2
)2− 1.
Intersect with x = c HyperbolaIntersect with y = c (if |c | > 2) EllipseIntersect with y = c (if |c | < 2) Nothing!Intersect with z = c Hyperbola
Hyperboloidof two sheets
