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Home , Hyperboloid, Quadric

Section 12.6 A Survey of Surfaces

Conic Sections Quadric Surfaces Review: Conic Sections

In R2, conic sections are curves satisfying an equation of the form Ax 2 + Bx + Cy 2 + Dy + E = 0.

Ellipse

x − h 2 y − k 2 + = 1 a b

Hyperbola

x 2 − y 2 = a2 y 2 − x 2 = a2 Quadric Surfaces

In R3, a quadric is a surface satisfying an equation of the form Ax 2 + By 2 + Cz2 + Dxy + Exz + Fyz + ax + by + cz + d = 0.

A quadric surface is classified by the curves obtained by intersecting it with planes parallel to the xy, xz, and yz planes. (These intersection curves will always be conic sections.) Quadric Surfaces

There are many kinds of quadric surfaces (more than there are conics):

conic ; (including ); elliptic ; hyperbolic paraboloids; of one sheet; hyperboloids of two sheets;

What is important is knowing how to recognize a quadric surface from its equation or its cross-sectional curves, or vice versa. (The names aren’t so important.) The cross-sectional curves are all conic sections (defined by degree-2 ). To find a cross-sectional curve, set one of the variables to a constant. Conic

Surfaces with an equation containing only 2 of the three variables. Cylinders can “slide” along the axis of the missing variable.

z2 z2 + y 2 = 1 − + y 2 = 1 y = z2 9 9

All cross-sections are .

x 2 y 2 z 2 + + = 1 a b c

For example, intersecting the ellipsoid

x 2 z 2 + y 2 + = 1 2 3 with the z = 1 gives the

x 2 12 + y 2 + = 1 2 3 a, b, c = radii in x, y, z or directions x 2 8 + y 2 = 2 9 Elliptic

Cross-sections: two , one ellipse

x 2 y 2 z = + a b

Surface passes the Vertical Test (since z is a function of x and y).

When a = b, the horizontal intersections are circles. Hyperbolic Paraboloid (Saddle Surface) Cross-sections: two parabolas, one

x 2 y 2 z = − z a b

y x Plane Cross-sectional curve x = c (downward) y = c Parabola (upward) z = c z = 0 Two intersecting lines (passes Vert. Line Test)

z z z

−1 1 0

−1 0 1 x y y 1 x 0 x −1 y of One Sheet

Cross-sections: two hyperbolas, one ellipse

x 2 y 2 z 2 + = + 1 a b c

Cross-sections: two hyperbolas, one ellipse

x 2 y 2 z 2 + = a b c

Intersecting with z = 0 results in a single point.

Halves of the cone are called nappes. Nappes are the graphs of the functions r x 2 y 2 z = ±c + . a2 b2 Hyperboloid of Two Sheets

Cross-sections: two hyperbolas, one ellipse

x 2 y 2 z 2 + = − 1 a b c

If −c < z < c, there are no (x, y) solutions to the equation (because then the RHS is negative and the LHS has to be positive).

All solutions have z ≥ c (top sheet) or z ≤ −c (bottom sheet).

Each sheet is the graph of a function. Hyperboloids and Cones — Comparison

Hyperboloid Cone Hyperboloid of 1 sheet of 2 sheets

x2 y 2 z2 x2 y 2 z2 x2 y 2 z2 a2 + b2 = c2 + 1 a2 + b2 = c2 a2 + b2 = c2 − 1 Comparing Some Example 1: Sketch and identify the graphs of

(a) z = 3x 2 + 4y 2 (b) z = 2x 2 − 5y 2 Passes VLT? Yes Yes Intersect 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

4x 2 − y 2 + 9z2 + 4 = 0

Solution: Rewrite the equation as 4x 2 + 9z2 = y 2 − 4 or equivalently

3z 2 y 2 x 2 + = − 1. 2 2

Intersect with x = c Hyperbola Intersect with y = c (if |c| > 2) Ellipse Intersect with y = c (if |c| < 2) Nothing! Intersect with z = c Hyperbola

Hyperboloid of two sheets