7.2 Planes and Quadric Surfaces
Sketch planes in space ax + by + cz = d
Draw planes with different intercepts
Classify quadric surfaces in space (see pages 468,469)
y2
b2
z2
c2 = 0
2
2
Elliptic Paraboloid z = xa2 + yb2
2
2
Hyperbolic Paraboloid z = yb2 − xa2
2
2
2
Ellipsoid xa2 + yb2 + zc2 = 1
2
2
2
Hyperboloid of one sheet xa2 + yb2 − zc2 = 1
2
2
2
Hyperboloid of two sheets zc2 − xa2 − yb2 = 1
Elliptic Cone
x2
a2
+
−
General Equation for a plane ax + by + cz = d
To graph, it helps first to find the x, y, z intercepts. Example
3x + 2y − z = 4.
Find the x intercept (i.e. set y, z = 0) and so (4/3, 0, 0) is the
x intercept
Find the y intercept (i.e. set x, z = 0) and so (0, 2, 0) is the y
intercept
Find the z intercept (i.e. set x, y = 0 and so (0, 0, −4) is the
z intercept
Plot these 3 points in space and shade in the triangle between
them.
Caution!
Every plane does not have 3 intercepts! For example
2x + y = 3
Graph it! .
Quadric Surfaces: what is the difference here?
Elliptic cone
Elliptic paraboloid
Use traces to aid in graphing
Graph the following quadric surface
x2 +
.
y2
+ 9z 2 = 1
4
Standard form
To put the equation for one of the quadric surfaces into standard
form you will need to complete the square (in x, y or z.) Example:
Find the standard form of the equation
x2 + y 2 + 2y − z 2 + 4z = 4
What type of quadric surface is it? .