13.1 - Common Names for Equations and Graphs

Ellipsoid

x²/a² + y²/b² + z²/c² = 1

• if a=b=c, then the equation is x² + y² +z² = a², and the surface is a sphere

Elliptic cylinder

x²/a² + y²/b² = 1

• if a=b, the surface is a circular cylinder

Elliptic double cone

x²/a² + y²/b² = z²/c²

• if a=b, the surface is called a circular double cone

Elliptic paraboloid

x²/a² + y²/b² = z/c

• if a=b, the surface is called a circular paraboloid

Parabolic sheet (or parabolic cylinder)

z = ax²

• could replace z w/ x,y or x w/ z,y respectively (as long as the variables aren’t the same on both sides of the equation)

Hyperbolic paraboloid / Saddle function

y²/b² - x²/a² = z/c

Two hyperbolic sheets (or hyperbolic cylinder)

y²/b² - x²/a² = 1

Hyperboloid of one sheet

x²/a² + y²/b² - z²/c² = 1

• a circle if a=b

Hyperboloid of two sheets

z²/c² - x²/a² - y²/b² = 1

• a circle if a=b

Plane

a(x - x₀) + b(y - y₀) + c(z - z₀) = 0

• (a,b,c) is the normal vector N

• (x₀,y₀,z₀) is a point on the plane

• also has the form ax + by + cz = d

Line

r = r₀ + t(L)

• (x₀,y₀,z₀) is the initial point vector r₀

• (a,b,c) is L is the arbitrary direction vector of r scaled by t (some constant)

• P₀P = tL

Line (Parametric)

x = x₀ + at
y = y₀ + bt

z = z₀ + ct

• (x₀,y₀,z₀) is the initial point vector r₀

• (a,b,c) is L is the arbitrary direction vector of r scaled by t (some constant)

• P₀P = tL

Line (Symmetric)

(x - x₀)/a = (y - y₀)/b = (z - z₀)/c

• (x₀,y₀,z₀) is the initial point vector r₀

• (a,b,c) is L is the arbitrary direction vector of r scaled by t (some constant)

• symmetric equations eliminate t as a variable

• if a, b, or c = 0, write the equation separately

• P₀P = tL