Quadratic Surfaces

Ellipsoid

-all positive

-all variable² / coefficient²

=1

*looks like 3d ellipse

Hyperboloid of One Sheet

-one negative

-all variable²/ constant²

=1

axis is variable w negative coefficient

*looks like vase thats thin in middle

Hyperboloid of Two Sheets

-two negative

-all variable²/ constant²

=1

axis is variable w positive coefficient, no trace in perpendicular plane

*looks like two cones not touching in middle

Elliptic Cone

-one negative

-all variable²/ constant²

-equals 0

axis is negative coefficient

*looks like two cones that touch in the middle

Elliptic Paraboloid

-positive on both sides

-variable = variable²/ constant²

axis is variable raised to first power

*looks like 3d Parabola

Hyperbolic Paraboloid

-negative

-variable = variable²/ constant²

axis is variable raised to first power

*looks like saddle

Cylindrical to rectangular

x=rcostheta

y=rsintheta

z=z

spherical to rectangular

x= p sin(phi) cos(theta)

y = p sin(phi) sin (theta)

z= p cos (phi)

rectangular to cylindrical

r²=x²+y²

tan(theta) = y/x

z=z

spherical to cylindrical

r² = p² sin² (phi)

theta = theta

z = p cos (phi)

rectangular to spherical

p² = x² +y² + z²

tan (theta) = y/x

phi = arccos ( z/p)

cylindrical to spherical

p = sqrt (r²+z²)

theta = theta

phi = arccos (z / p)